Arithmetic Mean Essays and Research Papers

Instructions for Arithmetic Mean College Essay Examples

Title: Discussion Questions

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Essay Instructions: #1.
Explain how the concept of Diminishing Marginal Utility is the basis by which a good gets its value in the marketplace? As part of your answer explain the concepts of consumer?s preference and consumer?s surplus and how they contribute to the valuation process.


#2.
The shape of the short run variable cost curves is based upon the underlying notion of the Law of Diminishing Returns. Explain why for most industries, this law must eventually come into play.




Week 2 Lecture:

In the Week 1 lecture, we studied how a market system works. Our exploration of the concepts of reservation price, supply and demand, and elasticity, enable us to understand how the market system allocates resources to satisfy the most highly valued wants..

Consumer Preferences

From all the goods or services available to them, buyers choose a combination of items we call a market basket. Consumption of the bundle of goods in a market basket brings satisfaction to the buyer. Buyers choose between different bundles of goods, different market baskets, on the basis of the satisfaction they are expected to bring. Before we model how consumer's choose from among different market baskets, we need to first make explicit our assumptions about how consumer's behave. Our model of consumer's as buyers is based on four assumptions. Because these assumptions seem logically correct even without proof, they are often called axioms.

1. Axiom of Completeness
Given two market baskets, A and B, a consumer will know whether she prefers A to B (written as A ? B), does not prefers A to B (A ? B) or is indifferent between them (A I B).

When confronted with a choice between two market baskets, both of which contain desirable goods, a consumer will definitely know which is preferred, or will definitely know that s/he would be equally happy with either, i.e., that s/he would be indifferent between them. Note that such indifference does not imply that the consumer cannot choose between the two baskets. Nor does it imply that the consumer finds both baskets UNdesirable. Rather, indifference implies that both baskets are equally desirable. This state of indifference plays a crucial role in our model of consumer choice.

2. Axiom of Greediness
Given two market baskets, A and B, the consumer will always prefer the basket that has more of at least one item and no less of the other items.

For example, suppose basket A contains 2 loaves of bread and 3 bottles of wine, and basket B contains 4 loaves of bread and 3 bottles of wine. According to the axiom of greediness, consumers will always prefer basket B (i.e., B ? A) because it has more of one good (bread) and no less of any other good (wine).

3. Axiom of Transitivity
Given any three market baskets, A, B, and C, if a consumer indicates that A ? B, and B ? C, then, logically, the consumer will indicate that A ? C.

For example, if a consumer reveals that a basket of bread is preferred to a basket of cheese, and that a basket of cheese is preferred to a basket of wine, then, logically, s/he would choose a basket of bread over a basket of wine. This is an assumption about logical behavior. It states that consumers choose in a consistent, predictable way, and that their preferences do not change in "mid choice", as it were.




Indifference Curves



Armed with these three assumptions (the fourth assumption is listed below) we can begin to build a model of consumer choice. To keep the analysis as simple as possible we will place only two goods in each market basket, food (symbolized by F) and clothing (symbolized by C). Simplifying consumer choice in this way allows us to build and manipulate a two-dimensional graphical model. (Adding more goods will not change the results, but will make the analysis more complex and require the use of advanced mathemagics.)



We begin by constructing a two dimensional graph, with clothing (C units per period) on the y- (vertical) axis, and food (F units per period) on the x- (horizontal) axis. Any combination (F, C), makes up a single market basket. For example, the dot labeled A in Figure 1 below, indicates a market basket that contains 8 units of food and 25 units of clothing, i.e., A = (8, 25).



Figure 1





Consumption of this market basket gives the buyer a certain level of happiness. According to the Axiom of Greediness, any market basket that contains at least 8 units of food and more than 25 units of clothing would be preferred because it would lead to a higher level of satisfaction (see basket B, for example). Similarly, any market basket that contains at least 25 units of clothing and more than 8 units of food would also lead to a higher level of satisfaction (see basket C). Finally, any market basket that contains more of both goods, say, basket D, is clearly preferable to basket A. By the same Axiom, market baskets E, F and G would be not preferred to basket A because each contains less of one or both of the goods.



What about market baskets in the other two regions? Take, for example baskets x and y. We cannot say with certainty whether either of these baskets is preferred to or not preferred to basket A. It is possible that the consumer would be indifferent between basket A and either of these two baskets. In fact, the consumer may be indifferent between point A and any points in these two regions. Why indifferent? Because any market basket in those regions would contain more of one good but less of the other. Basket x, for example, contains more clothing but less food than basket A. Basket y contains more food but less clothing than A. Because more of a good increases satisfaction but less of a good decreases satisfaction (Axiom of Greediness), it may very well be that the consumer would be indifferent among these three baskets.

The thick, gray lines extending outward from basket A divide the chart into four quadrants. These lines help us clearly demarcate regions with market baskets that are preferred to A (see preferred region in Figure 2 below) or not preferred to A (see not preferred region). Until be have more information, the Axiom of Completeness allows that baskets in the other two regions may be indifferent to A (see regions of indifference).



Figure 2





Let's suppose that the consumer is indifferent among baskets A, x, and y. We can illustrate the set of market baskets that offer the same level of satisfaction as A by connecting them with a line going through basket A, as shown in Figure 3 below. This line shows all the market baskets offering the consumer the same level of satisfaction. Because a consumer is indifferent between any two of these baskets, this line is referred to as an indifference curve.



Figure 3





Because of the location of the regions of indifference, any line we draw will be negatively sloped. A negatively-sloped line tells us that if any units of C are removed from a basket, the consumer must be compensated with additional units of F in order to remain equally happy with the new basket.


Marginal Rate of Substitution

Now a negatively-sloped line can take on a number of smooth shapes. It can be linear, concave, or convex. The shape it takes tells us something about the rate at which one is willing to substitute clothing for one more unit of food.

A linear indifference curve has a constant slope. As a result, it tells us that the amount of clothing a consumer is willing to give up for one more unit of food is constant, regardless the amount of food s/he already has.
A concave (concave downward) indifference curve has a slope that increases as we raise the number of units of F. This tells us that the amount of clothing a consumer is willing to give up for one more unit of food increases the more food s/he already has.
A convex (concave upward) indifference curve has a slope that decreases as we raise the number of units of F. This tells us that the amount of clothing a consumer is willing to give up for one more unit of food decreases the more food s/he already has.
Which of these scenarios best describes the way a consumer would feel? Enter the fourth axiom:

4. Axiom of Substitution
The amount of good C one is willing to trade for one more unit of F diminishes as the number of units of food consumed increases.

Thus, the rate at which one is willing to trade C for one more F, diminishes. Why? Remember that along an indifference curve, one's level of satisfaction is unchanged because additional units of food are accompanied by fewer units of clothing. Moving downward along an indifference curve, one "pays" for an additional unit of food by giving up one's clothing. Because clothing is valued for the satisfaction it brings, the less clothing one has the greater the value of each remaining unit. Put another way, the opportunity cost, in terms of clothing given up, of getting more food, is increasing because the consumer is running out of clothing! How much one is willing to pay, i.e., how much clothing one is willing to trade for one more unit of food, varies according to one's relative preference for food over clothing. (We'll explore this point in the Excel workbook for this tutorial...)

For example, Figure 4 (below) shows a single indifference curve for a consumer. Six different market baskets are highlighted along the curve, each of which brings the consumer the same level of happiness, designated U1. The shape of the curve reveals that as the consumer moves from basket A (4, 45) to B (6, 30), s/he is willing to trade 7.5 C for 1 F. But as the composition of the basket changes from B (6, 30) to C (8, 22.75), s/he is now willing to trade only 3 1/8 C for 1 F. The reduction in willingness to trade continues in the same manner:

From C to D (10, 18.25) s/he is only willing to trade 2 1/4 C for 1 F;
From D to E (12, 15) s/he is only willing to trade 1 5/8 C for 1 F;
From E to F (14, 13) s/he is only willing to trade 1 C for 1 F.
Figure 4



Thus a consumer's willingness to substitute C for F diminishes with each additional unit of F received. This behavioral assumption, which we call the Axiom of Substitution, is usually referred to as the diminishing marginal rate of substitution.


Happiness is a Higher Indifference Curve

So far we have established a plausible way of modeling the satisfaction received by consuming a bundle of goods. The model proposes that a negatively sloped, convex line is all that is needed to show all the bundles, or market baskets, of two goods that leave a consumer equally happy.

But we also noted that baskets with more of at least one good, but no less of the other, will make the consumer better off. Similarly, baskets with less of at least one good, but no less of the other, will make the consumer worse off. For example, Figure 5, below, shows three indifference curves labeled U0, U1, and U2. It illustrates two additional features of the indifference curve model.

First, starting from basket A, we see that basket B has more of both food and clothing. As a result, consumption of the goods in that basket give the consumer more satisfaction than bundle A. Second, we have drawn an indifference curve through point B. This indicates that a number of other baskets could give the consumer the same level of satisfaction. This leads us to conclude that an indifference curve, such as U2, drawn above and to the right of another, indicates a higher level of satisfaction. Similarly, an indifference curve, such as U0, drawn below and to the left of another, indicates a lower level of satisfaction.



Figure 5





In fact, pick a point, any point in this (F, C) space, and one may draw an indifference curve through that point. As indifference curves move farther from the origin, the level of satisfaction from any bundle on that curve increases. Thus, happiness is a higher indifference curve!



The level of satisfaction, or happiness, is often referred to by economists as "utility". And the units we use to "measure" this happiness are called "utils", hence the symbol U on each indifference curve in Figures 4 and 5. Note that the term "measure" is used here to mean simply an ordinal ranking. Thus, higher indifference curves are given higher numbers to indicate that baskets on the higher indifference curve, e.g., U2, give a higher level of satisfaction than baskets on the lower one, e.g., U1.



The Budget Constraint

Earlier in this lecture, we studied that a consumer seeks a bundle of goods, a market basket, the consumption of which generates the maximum happiness (utility). Graphically, such a basket is found on the indifference curve as far from the origin as possible. Of course these goods are not free, and one's money income is not limitless. So one's goal of maximizing utility is subject to the constraint imposed by one's budget. In this tutorial we will develop a model of the consumer's budget constraint.

Suppose a consumer has a fixed periodic money income, represented by I, and must pay a price, Pf, for each unit of food, and a price, Pc, for each unit of clothing. The consumer's budget can be represented by the identity shown in equation (1),

,

where Pf*F indicates the amount of money income I spent on food, and Pc*C is the amount of money income spent on clothing. Equation (1) simply indicates that the sum of spending on food and clothing must equal the amount of money income available.

To add this constraint to our model, we need to rewrite it in a form that can be plotted in (F, C) space. We can do that by simply rewriting equation (1) so that C is the dependent variable.



...subtracting Pc*C from both sides.

...multiplying both sides by -1.


...dividing both sides by Pc.



...rearranging terms.


Now we have an equation that can be plotted in (F, C) space, where C is the dependent variable, F is the independent variable, and I, Pf, and Pc are parameters with known values:



Let's read equation (2). First of all, C(F) tells us that the amount of clothing one can buy depends on the amount of food one has purchased. The y-intercept term, I/Pc, indicates the amount of clothing that may be purchased, given income and the price of clothing, if one buys no food (i.e., if F = 0). The slope term, Pf/Pc, indicates the amount of clothing that must be given up in order to free up enough money to buy one more unit of food.

Let I = $80 per period, Pc = $2 per unit, and Pf = $5 per unit. Given these parameter values, equation (2) would be written

,

which is an equation of a straight line (e.g., y = a +b*x), where a = I/Pc, and b = -Pf/Pc. A graph of this equation is shown as B1 in Figure 6 below. The vertical intercept is 40 units/t. It is the amount of clothing that may be purchased given an income of $80 and a price of clothing equal to $2 per unit (e.g., 40 = $80/$2). How is the horizontal intercept, 16, calculated? [Hint: The horizontal intercept, 16, is the amount of food that may be purchased with an income of $80 and a price of food of $5 per unit.]



Figure 6





The slope is equal to 5/2 = 2.5 units. It tells us that one must give up 2.5 units of clothing in order to free up enough cash to buy one unit of food. How's that? Well, a unit of clothing is priced at $2. So, "selling off" 2.5 units of clothing would bring you $5:

2.5 * $2 = $5,

which is the price of a unit of food. Thus, selling 2.5 units of clothing gives you enough money to buy 1 unit of food.



Changes in Parameter Values

Now let's explore what happens to the budget line when income or the prices of the goods changes.

Increases (decreases) in income result in an outward (inward) shift in the budget line parallel to the original. This is because more (less) income expands (contracts) the set of affordable baskets of food and clothing. On the graph, as income rises (falls), both the vertical intercept, I/Pc, and horizontal intercept, I/Pf, rise (fall). Why? Because each intercept term shows the maximum amount of a good one can by given one's income and the price of the good. Thus as income rises (falls), the maximum amount of a good one can afford to buy rises (falls), assuming the prices of the goods remain constant.

A decrease (increase) in the price of food lowers (raises) the slope of the budget line (Pf/Pc) by raising (lowering) the horizontal intercept (I/Pf), leaving the vertical intercept (I/Pc) unaltered. Similarly, decreases (increases) in the price of clothing raise (lower) the slope of the budget line by raising (lowering) the vertical intercept, leaving the horizontal intercept unaltered. In both cases, a decrease (increase) in price expands (contracts) the set of affordable baskets of food and clothing.



Consumer Choice

Thus far, we have learned that a consumer seeks a market basket of goods which generates the maximum level of utility. Then, we learned that one's goal of maximizing utility is subject to the constraint imposed by one's money income and the prices of the goods.

Stated formally, the consumer's objective is to maximize utility subject to a budget constraint. That is,



The function u(F, C) describes the level of satisfaction ("utility") received when the bundle of goods (F*, C*) is consumed. This form of the utility function allows us to manipulate the relative strength of the consumer's preference for F or C by changing the parameters a and b.

Stating the consumer choice problem this way makes explicit the fact that, whereas one desires to maximize the satisfaction from a bundle of goods (Max u(F, C)), one is constrained in by one's income and the prices of the goods (i.e., by one's budget).

The complete model of consumer choice is shown in Figure 7 below. Graphically, the consumer's utility-maximizing choice must meet two conditions:

The market basket chosen must place the consumer on the highest indifference curve attainable;
The market basket must be located on the budget line (i.e., no "saving").
By the Axiom of Greediness, any basket on U1 is preferred to any basket on U or U0. But attainment of U1 is blocked by the budget constraint.


At points C or D, for example, the budget constraint is satisfied, but U0 is not the highest level of utility attainable. There are a number of indifference curves higher than U0. In fact any point on B1 that lies between bundles C and D would place the consumer on a higher indifference curve.


Market basket A exhausts the consumer's income ($5*8 + 20*$2 = $80) and results in a higher level of utility than bundles C, D, or any other.

Figure 7





Therefore, this graphical model shows that the maximum attainable level of satisfaction takes place when the budget line and indifference curve are tangent. At the point of tangency, A, income is exhausted, and:







The slope of the budget line
=
The slope of the highest IC attainable

|Pf/Pc|
=
|MRSfc|

The rate at which one can trade C for F
=
The rate at which one is willing to trade C for F

MC (of one more unit of F)
=
MB (of one more unit of F)




In contrast, at point C, MRSfc > Pf/Pc. So the rate at which the consumer is willing to trade C for F, exceeds the amount s/he is required (by the market) to trade. So if 15 units of C is traded for 6 units of F -- that's 2.5C for 1F, what the market requires -- the consumer will be better off as they move to point A on indifference curve U1.

Similarly, at point D, MRS < Pf/Pc. The rate at which the consumer is willing to trade F for C, exceeds the amount s/he is required (by the market) to trade. So if 6 units of F is traded for 15 units of C -- that's 1C for 2/5F, what the market requires -- the consumer will be better off as they move to point A on indifference curve U1.



Some Mathematics (This section is optional but just understand the results)

Remember that the consumer's objective is to maximize the satisfaction received from the consumption of a market basket of goods, subject to the constraint that spending on the goods does not exceed income, i.e.,



Graphically, we were able to determine that this constrained maximum occurs at the point where the slope of the budget line, Pf/Pc, equals the slope of the indifference curve, referred to as the marginal rate of substitution (MRS). In this section we will learn how to calculate the MRS.

Once again, the function u(F, C) describes the level of satisfaction ("utility") received when the bundle of goods (F*, C*) is consumed. An indifference curve simply describes all the combinations of F and C that generate a constant level of utility, U1, to the consumer. So, the change in utility along an indifference curve is equal to zero. But as you give up some clothing, for example, your utility level changes (drops). The change in utility from a one unit change in clothing is called marginal utility (MUc). So the size of the change in total utility from a reduction in C depends on the size of the change in C and on the consumer's MUc, e.g., DC*MUc.

Similarly, when you get more food, your utility level changes again (rises). This time the change in total utility from an increase in F is equal to DF*MUf.

Mathematically, the two changes cancel each other out:

.

Subtracting DC*MUc from both sides,
.

Divding both sides by DF and by MUc, and cancelling similar terms,
.




So the slope of an indifference curve is measured by
.




Now all we have to do is figure out how to calculate marginal utilities. As mentioned above, marginal utility measures the additional satisfaction one receives from consumption of another unit of a good. Mathematically, marginal utility is the partial derivative of a total utility function. Starting with the utility function noted above:

,

differentiating with respect to F, gives us MUf =


differentiating with respect to C, gives us MUc =



To calculate MRS = MUf/MUc





One of the principle reasons for studying the consumer choice model is to discover what it tells us about the shape of an individual consumer's demand curve. After all, we can use the consumer choice model to vary the price of a good and watch what happens to the quantity of that good consumed. Well, that's all the information we need to construct a demand schedule -- two prices and the associated quantities demanded.

Individual Demand

Let's manipulate the consumer choice model and see how we can generate an individual consumer's demand curve. Figure 1 below shows the consumer choice model, with food and clothing as the two goods, where Pf = $2 per unit, Pc = $2 per unit, and income = $80 per period. The consumer has attained the highest satisfaction possible given her preferences, the prices of the two goods, and her income, with F* = 20 units, C* = 20 units, and u(F*, C*) = 400 "utils".



From its initial rate of $2 per unit, we'll raise the price of food to $8 per unit. Because the law of demand states that quantity demanded varies inversely with price, c?teris paribus, we must hold the other variables constant. So we'll freeze preferences (i.e., no change in a or b in the utility function), keep PC at $2 per unit, and keep income at $80 per period. Figure 2 below shows the new budget constraint this consumer faces.



Now let the consumer adjust to this new budget reality. We already learned that he consumer's objective is to maximize utility subject to a budget constraint. Graphically, this means that the consumer's utility-maximizing choice must meet two conditions:

The market basket chosen must place the consumer on the highest indifference curve attainable;
That same market basket must be located on the budget line (i.e., no "saving").
To reach this objective, the consumer chooses a combination of the two goods such that the budget line and indifference curve are tangent.

Figure 3 shows the result. Drawing a new indifference curve (U1) just tangent to the new budget constraint (B1) shows the consumer's new optimum choice of food (F* = 5 units) and clothing (C* = 20 units), with U1(F*, C*) = 200 "utils".



We now have just enough information to derive the consumer's individual demand curve. Table 1 summarizes this information.


Table 1

Pf
QfD
u(F*, C*)
PC
Income
Pref's
a & b

$2/unit
20 units/t
400 "utils"
$2/unit
$80/pd
0.5
0.5

$8/unit
5 units/t
200 "utils"
$2/unit
$80/pd
0.5
0.5


Draw a graph with Price on the vertical axis (in $/unit) and quantity on the horizontal axis (units of food/t).

Plot the first price-quantity combination (Pf = $2/unit and Qfd = 20 units/t).
Plot the second price-quantity combination (Pf = $8/unit and Qfd = 5 units/t).
Connect the two points and you have a line representing the consumer's demand for food!
Your graph should look like the one in Figure 4.





Let's examine the properties of this demand line.

An increase in price, P, from $2 to $8 per unit, causes a decrease in the quantity demanded, Q, from 20 to 5 units per time period. Similarly, a decrease in price would cause an increase in the quantity demanded.
The level of utility that can be attained changes as we move along the demand curve -- an increase in P (from $2 to $8) causes a decrease in the maximum indifference curve attainable, which implies a decrease in satisfaction (from 400 "utils" to 200 "utils"). Similarly, a decrease in P causes an increase in the maximum indifference curve attainable, which implies an increase in satisfaction.
At every point on the consumer's demand curve, the consumer is maximizing U by satisfying the condition that MRS - Px/Py = 0, subject to the constraint that Income = Px*X + Py*Y.
There are two implications that may be derived from this last point.

The MRS varies along D (the relative value of Food falls as one buys more).
How much one is willing to pay for a good varies along an individual's demand curve(the more Food one has the less one is willing to pay for another unit of it), i.e., the consumer's reservation price falls as the quantity already consumed per period increases.
Shifting the Demand Curve

So far we've been able to establish that the consumer choice model generates an individual consumer's demand curve that shows an inverse relationship between price and the quantity demanded, c?teris paribus, as predicted by the law of demand.

But what if c?teris is not paribus? In Tutorial 3, we saw that if one of the non-price determinants of demand (i.e., P-PINE) is changed while price, and everything else, is kept constant, the demand curve will shift. Question 2 in the Consumer Demand Workbook is devoted to this issue, so I'll just outline the steps here.

Figure 4 showed the demand curve generated by manipulating the price of food in the consumer choice model, holding income and all other parameters in the model constant. So now we'll change income and then measure the quantity of food demanded at each of the two prices used in the previous example.

Suppose income rises to $100 per period, and Pf = $2 per unit and Pc = $2 per unit. Draw a new graph of the consumer choice model showing the consumer at the optimum bundle of F and C. Because the budget line has shifted outward, I am confident that the new consumption of food will be greater than 20 units per period -- let's say it's 25 units per period.

Now, once again, raise the price of food to $8 per unit, holding the other variables constant. Modify your graph of the consumer choice model to show the consumer at the new optimum bundle of F and C. Because the budget line has shifted outward, I am confident that the new consumption of food will be greater than 5 units per period -- let's say it's 6 units per period. Table 2 shows the results of this little experiment.

Table 2

Pf
QfD
u(F*, C*)
PC
Income
Pref's
a & b

$2/unit
20 units/t
400 "utils"
$2/unit
$80/pd
0.5
0.5

$8/unit
5 units/t
200 "utils"
$2/unit
$80/pd
0.5
0.5

$2/unit
25 units/t
800 "utils"
$2/unit
$100/pd
0.5
0.5

$8/unit
6 units/t
300 "utils"
$2/unit
$100/pd
0.5
0.5


Add this new data on price and quantity demanded to your previous graph of the consumer's demand line.

Plot the first price-quantity combination (Pf = $2/unit and Qfd = 25 units/t).
Plot the second price-quantity combination (Pf = $8/unit and Qfd = 6 units/t).
Connect the two points and you have a line representing the consumer's new demand for food when income increased!
Your graph should look like the one in Figure 5.



The consumer choice model thus predicts that an increase in income, c?teris paribus, will shift the demand curve to the right

Substitution and Income Effects

As I write this, gasoline prices are rising. This makes many Americans angry and they demand that their government do something to this. Let's look at a short run response. (We'll examine long run responses later.)

Many people act as if they mistakenly believe that when the price of a good rises, consumers of that good are forced to either pay up or go without. In fact, when the price of a good increases, even gasoline, which has a price inelastic demand, one buys a little less of it -- people don't just pay up or do without it all together. Higher prices force one to look for substitutes. What's a substitute for gasoline? Gasoline at other filling stations whose prices are lower, of course. Also, shorter trips, less time spent "flooring it"!, sharing a ride, even trading in your gas-guzzling Ford Explorer for a more fuel efficient Honda CRX... And this is an important economics lesson -- there are substitutes for everything!

Well then, when the price of something rises we look for substitutes. This is one reason why demand curves slope downward. The substitution effect of an increase in the price of a good leads to less of that good being purchased as one pursues more of a relatively cheaper substitute. The other reason quantity demanded varies inversely with price is called the income effect. As the price of a good rises, one's real income (income divided by the price level) falls. So one cannot afford to buy as many goods in total as before. Using the same example, one may buy a little less gasoline and eat out less often.



Substitution and Income Effects in the Consumer Choice Model

When price rises, quantity demanded falls because of the substitution effect and the income effect. Let's use the consumer choice model to separate the decrease in quantity demanded into two parts, one for each reason.

Sketch out the consumer choice model showing the consumer at an optimum bundle of two goods, Food and Clothing. Here are the initial parameter values:

I = $80/period; Pc = $2 each; Pf = $2 each; C* = 20 units/period; and F* = 20 units/period. Assume Food is a normal good.

If we raise the price of Food from $2 to $8 per unit, c?teris paribus, the budget line becomes steeper, rotating leftward along the x-axis. Now we know the consumer is made worse off when a price increases. What if we were able to offset that higher price with higher income (just as an experiment)? If we raise income, we can eliminate the income effect of the price increase. Our goal would be to raise income enough that the consumer would be just as happy as before the price increase. I wonder if there would be any change in the consumption bundle if we eliminate the income effect?

As we raise income, the budget line shifts outward, parallel to the original. When it is tangent to the initial indifference curve, we know the consumer will be just as happy as before. Does this increase in income leave the consumer with the same consumption bundle as before? No! See Figure 6 below. After increasing income we see that at the new point of tangency (of U1 and B1) the consumption bundle has less Food (10 units/period) and more Clothing (40 units/period).

Figure 6


By increasing the consumer's income we were able to eliminate the income effect of the price increase. We can now see that when the price of a good increases, AND real income held constant, the consumer substitutes more clothing for less food. (If that seems odd remember that there are only two goods in the entire consumption set of this consumer...) So the drop in Food consumption from 20 to 10 units per period is due entirely to the substitution effect of the price increase.

[Note: The previous three paragraphs deserve to be read carefully!]

Now let's take that income away (it was only an experiment after all!). Figure 7 shows the result. The budget line drops parallel to its new position and becomes tangent to a new, lower indifference curve at F* = 5 units/period and C* = 20 units/period. So now the drop in Food consumption from 10 to 5 units is due entirely to the income effect (real income is the only thing that changed).

Figure 7


As price increases from $2 to $8 per unit, the total effect on the consumption of Food is a drop from 20 to 5 units per period. This is what we saw in the first part of Tutorial 6. Now we know that this drop takes two steps. The drop in Food consumption from 20 to 10 units per period is due to the substitution effect of the price increase and the drop from 10 to 5 units per period is due to the income effect of a price increase.

The market demand for goods is determined by the sum of all the individual consumers' demand curves. That is, for any given price, we add up the quantity each individual consumer is willing and able to buy. The sum of those quantities tells us how much all demanders in a market are willing and able to buy at that price. Technically, this summation is referred to as a horizontal summation because we are keeping price constant (on the vertical axis) and summing along the horizontal axis the quantities for all buyers. This is illustrated in Figure 8 below.

Figure 8


The first three demand curves belong to individual buyers in this market. Buyer 1 is willing and able to purchase 0 units at a price of $225 each, Buyer 2 is willing and able to purchase 25 units at that price, and Buyer 3 is willing and able to purchase 50 units. Because all three buyers are in the market for this good, the quantity demanded in this market at a price of $225 each is (0 + 25 + 50) = 75 units per period.

If we continue this process for all the prices, we can derive the market demand curve for the good.

To this point in the course we have focused most of our attention on the demand side of the competitive market system. Now we turn our analytic eye on its complement, the supply side.

The Technology of Production

Nicholas Georgescu-Roegen once said that the study of economics is about how we transform resources into garbage -- noting, of course, that we get consumption benefits in between. Well, in order to have goods and services to consume we must send various "inputs" into some sort of production process and out of that process comes the goods and services we demand (along with production byproducts, e.g., pollution, that we don't demand).

The wide variety of inputs used in the production process can be grouped into three categories:

Natural Resources (R) -- nature made inputs.
Labor (L) -- physical and mental work.
Capital (K) -- human made inputs.
Note that technology is not an input per se. Technology is "how we get something done", and should not be confused with capital, as most people do. At times technology is embodied in new capital -- as information technology is embodied in computers. But more often new technology reveals itself simply as a reorganization of productive resources -- as when Henry Ford championed the assembly line method of producing automobiles. Regardless of its form, new technology almost always makes inputs more productive.



Our model of the production process starts with a production function:

Q = ?(R, L, K),



where Q symbolizes the output, the goods and services, of the production process, R, L, & K are the inputs defined above, and technology is determined outside the model.

If we hold the amount of natural resources (R) fixed, then the production function reduces to a form we can model with a three-dimensional graph:

Q = ?(L, K).



This equation is a special type, known as a Cobb-Douglas production function after its two creators.



Production with One Variable Input (Labor)

Although viewing the 3-D form of the production function may give us a bit more complete view, it is not a model that is easily reconstructed or manipulated graphically without the aid of a computer. What would happen if we simplified the model further? For purposes of simplicity and to aid in graphical modeling, we will look at a firm's production process in the short run. By definition, the short run is an amount of time such that at least one of the inputs used in the production process is fixed. [Note: We will use the term short run several more times in this course, and each time, in true Humpty Dumpty fashion, we will make it mean what we want it to mean, neither more nor less. It will be up to you to keep track of the term's current meaning...]

If we hold the amount of natural resources (R) and the amount of capital (K) fixed, then the production function reduces to a form we can model with a two-dimensional graph:

Q = ?(L | K),

where the vertical bar | indicates "given", so the statement reads, "Output is a function of Labor, given a fixed amount of Capital" --



The Marginal Product of Labor

When there is only one input that is allowed to vary, we can measure that inputs contribution to output, i.e., its productivity. If we differentiate the production function we will have an equation describing the marginal productivity of labor, MP(L). MP(L) measures the additional output generated by one additional unit of Labor, every other input held constant. Mathematically,



Q?(L | K) = MP(L) = 2 * L-0.5



The Law of Diminishing Returns?

What do you see in the plot of the Marginal Product of Labor? As more labor is hired MP(L) falls. What's going on here? Well, first you need to be clear on the difference between total and marginal. Marginal Product refers to the addition to output from hiring one more unit of labor. So a falling, but positive, MP(L) merely indicates that total output is rising at a decreasing rate.



Ok, but why does MP(L) decline as more labor is employed? Because capital is fixed, workers must share it as they attempt to produce the good. The more workers there are sharing this fixed amount of capital, it is inevitable that, at some point, each new workers' addition to output per hour (i.e., MP(L)) will fall because they will waste time waiting for capital to be free for their use.



This diminishing MP(L) is referred to as the law of diminishing returns. It refers to the fact that, in the short run, as the amount of one variable input increases, all other inputs held constant, the addition to output from each new input diminishes. This production law cannot be avoided -- if it could, we could feed the world from a flower pot! But remember that all other inputs and technology are frozen -- only one input is allowed to vary.



The Average Product of Labor

There is another way to measure the productivity of our one variable input. We can divide the total amount of goods produced by the amount of input used to produce it, to find the equation describing the average productivity of labor AP(L). AP(L) measures the arithmetic mean of the output generated by all units of Labor used to this point. Mathemagically,

AP(L) = Q(L | K)/L = 4 * L-0.5



What do you see in the plot of the Average Product of Labor? As more labor is hired AP(L) falls. What's going on here? This is due more to arithmetic than economics. As long as marginal product lies below average product, it will pull the latter down. Why? Well, look at your average course grade. If you do better than your average on a quiz (a marginal assignment) your class average will rise. If you do worse than your average on a quiz, your class average will fall. Similarly, if MP(L) > AP(L), AP(L) will rise; if MP(L) < AP(L), AP(L) will fall.

Production with Two Variable Inputs: Isoquants


Various combinations of L and K have been highlighted that would generate an output level Q = 10 units/t (see the Table below).

Output Q = 10 units/t

Labor (L/t)
Capital (K/t)

4
25

5
20

10
10

20
5

25
4


This reveals a fact of fundamental importance in the theory of production. It is possible, using the production method modeled by the Cobb-Douglas production function, to substitute less of one input for more of another and still keep output constant. In the case illustrated by the Data, we can increase the amount of labor used in the production process from 10 to 20 units/t, and simultaneously drop the amount of capital used from 10 to 5 units/t with no change in output. There is more than one way to produce a given level of output!

Diminishing Returns Again?

So what have we got? The Data help us see that, for a given level of output, when L is increased, K can be decreased. These lines of constant output are referred to as isoquants.

The slope of the isoquant reveals how much K may be given up when one more L is used to produce a given level of Q. The absolute value of the rate at which L may be substituted for K is referred to as the Marginal Rate of Technical Substitution between L and K (MRTSLK). Let's work with this mathematically and see what we find...

MRTS is measured as the negative of the slope of an isoquant (which, of course, is negative). So

MRTS = ?DK/DL > 0

for a given level of Q, where DK is the "rise" and DL is the "run". Looking at the table, you will see that as the amount of labor used in production increases from 4 to 5 units/t, the amount of capital that may be "retired", keeping output constant at 10 units/t, is 5 (from 25 to 20 units/t). So MRTS is 5 here. But when the amount of labor used in production increases from 10 to 20 units/t, the amount of capital that may be "retired", keeping output constant at 10 units/t, is also 5 (from 10 to 5 units/t). So MRTS is 0.5 here. So MRTS diminishes as the quantity of labor hired increases. But why?

Let's see. Normally, when a firm increases the amount of one input used, keeping the other constant, output increases. In this example, an increase in L will cause an increase in Q = MPL*DL. That is, the additional amount of labor hired, DL, will raise output by an amount equal to the additional output that extra labor makes (i.e., its marginal product, MPL).

Similarly, when the firm decreases the amount of one input used, keeping the other constant, output decreases by an amount equal to the additional output that input would have produced. In this example, a decrease in K will cause a decrease in Q = MPK*DK. That is, the amount of K laid off, DK, will decrease output by an amount equal to the marginal product of that capital, MPK.

If we substitute more labor for less capital along an isoquant, then the change in output would be zero -- the additional output created by the new labor (MPL*DL) would be exactly offset by the loss of output resulting from the decrease in capital (MPK*DK). Thus:

DQ = 0; MPL*DL + MPK*DK = 0

Rearranging terms:

MPL*DL = ?MPK*DK

?DK/DL = MPL/MPK

MRTS = MPL/MPK

So we have found that the rate at which L may be substituted for K is equal to the ratio of their respective marginal products. Earlier we found that the marginal rate of technical substitution declines as we move along the isoquant. How does knowing how to calculate MRTS answer the question of why MRTS declines as more L is substituted for K?

Well, in the beginning of this lecture we learned that the marginal product of a single variable input eventually declines as more of that input is used with all other inputs held constant. This "law of diminishing returns" was due to the fact that that variable input, say labor, had to share the fixed amount of capital that was available. It is that sharing that promotes diminishing returns. Turning this around, as less labor is used marginal product rises because less sharing of capital is required.

So as you move left to right along our isoquant, L increases, causing MPL to decline, and K increases, causing MPK to increase. If MRTS = MPL/MPK, then as you move left to right along an isoquant the numerator decreases (e.g., 5, 4, 3, 2, 1) while the denominator increases (e.g., 1, 2, 3, 4, 5), resulting in a smaller value for the ratio (e.g., 5, 2, 1, 0.5, 0.2).

Returns to Scale

To keep our model of the production process as simple as possible (but not simpler!) we will assume there are only two inputs available to produce goods. Varying both inputs varies the scale of operations, and, by definition, the scale of operations can change only in the long run. That is, the short run is characterized as that time period during which at least one input is fixed. The long run is characterized as that time period during which all inputs can vary. For a small, T-shirt shop, in might take half a year to vary all of its inputs, so the long run might be 6 months. For a large, automobile producing plant, the long run might be five years.

What might happen to output if both inputs were to double? There are three possible outcomes:



Double inputs and output increases by double ("constant returns to scale").

Double inputs and output increases by more than double ("increasing returns to scale").

Double inputs and output increases by less than double ("constant returns to scale").



What you should find is that when:

a + b = 1, doubling both inputs causes output to double, so the firm is experiencing "constant returns to scale".
a + b > 1, doubling both inputs causes output to more than double, so the firm is experiencing "increasing returns to scale".
a + b < 1, doubling both inputs causes output to less than double, so the firm is experiencing "decreasing returns to scale".
Measuring Cost: Which Costs Matter

Economic Costs versus Accounting Costs

Both economists and accountants look at costs, but each takes a slightly different perspective. Accountants are retrospective. They are trained to look at explicit costs, costs involving direct, out-of-pocket payments for wages, salaries, materials, property rentals, etc. Economists are forward looking.



They are trained to look at opportunity costs -- the value of something in its next best use. To economists, all costs are opportunity costs -- wages, while an explicit cost, must be at least as high as the opportunity cost of purchasing labor in a competitive market. But in addition to explicit costs, economists consider it just as important for a firm to look at the opportunity cost of the owners implicit salary (what the owner could make using her skills in their next best use), and the opportunity cost of using up capital equipment (actual depreciation -- not as an accounting entry like "straight line" or "sum of the year's digits").

The importance of this difference in perspective can be illustrated by looking at sunk costs, one category of explicit costs. A sunk cost is an expenditure that has already been made and cannot be recovered. From a retrospective point of view, the cost of all the inputs purchased must be taken into consideration when making production decisions. But from a forward looking point of view, if the purchased input has no alternative uses, then it has no opportunity cost -- the expenditure is "water under the bridge" and so should not be considered in making production decisions.

Cost in the Short Run

From Production Function to Total Cost Function

The production function Q(L | K) shows the level of output from a given amount of labor, holding capital fixed. Figure 1 shows a production function based on the Cobb-Douglas form, where capital, K, has been fixed at 40 units per period. Notice that as more variable inputs are applied to the production process, output increases at a decreasing rate. Recall that this is due to the law of diminishing returns. The production function in Figure 1 shows us that if the firm employs about 200 units of labor they will produce a total of 500 units of output.

Figure 1


Invert the production function, i.e., Q-1(L | K), and Figure 2 now shows the amount of labor required for a given level of output. I call this a Variable Input Requirement (VIR) curve. In this case it reveals that, to produce an output of 500 units/t, 156.25 units of labor are required. Thus, the shape of the VIR curve mirrors the shape of the production function shown in Figure 1. Thus, the shape of the VIR function too is the result of the law of diminishing returns.

Figure 2



If we know the amount of labor required to produce various levels of output, then multiplying that amount by the wage rate would tell us the variable cost of production. If the wage rate, w, is $10 per period, then w*VIR = VC(Q), as shown in Figure 3. For an output of 500 units/t, the variable cost is the amount of labor required, 156.25 units, times the $10 wage rate, for a variable cost of $1562.50. Notice that the shape of VC(Q) is the same as the shape of the VIR function, only steeper. That implies that the shape of the variable cost curve too results from the law of diminishing returns.

Figure 3


There are two categories of inputs, variable (as labor is in this model) and fixed (as capital is in this model). Suppose the firm's capital stock costs $500. Because that fixed cost does not change with output, we can add it to variable costs to get the total cost of various levels of output. This is shown in Figure 4. For 500 units of output, variable costs are shown to be $1562.50, we said fixed costs are $500, so total cost is $2062.50.

Figure 4




Now we have explored the connection between the firm's short run production function discussed earlier and it's short run total cost function. The linchpin is the variable input requirement function -- the inverse of the production function. We found that the shape of the total and variable cost curves followed directly from the shape of the total product curve. The cause of that shape is the law of diminishing returns.



Marginal and Average Costs

The graphical models in Figures 1 - 4 are useful for seeing the connection between production and costs. But the economic approach to decision making is based not on an analysis of totals (e.g., total cost) but on an analysis of marginals (e.g., marginal cost). Remember, economists are "forward looking". Every problem is looked at by comparing the cost of an additional step with the benefits of that additional step. So the cost information of most importance to an economist is the marginal cost of production.

Marginal cost shows the additional cost of producing one more unit of output. [Later we'll see how to compare it to the additional benefit of one more unit of output...]

As was the case with marginal utility and marginal product, marginal cost is equal to the change in cost (total or variable) divided by the change in output. Mathematically, marginal cost, MC(Q), is the derivative of the total cost (or variable cost) function.

MC(Q) = dVC(Q)/dQ
(i.e., marginal cost is found as the derivative of VC(Q) with respect to Q).

Two other important measures of the cost of production are the average total and average variable costs. Like any average, average total cost, ATC(Q), is simply the sum of total costs divided by the number of units of output produced:

ATC(Q) = TC(Q)/Q.

Similarly, average variable cost, AVC(Q), is simply the sum of variable costs divided by the number of units of output produced:

AVC(Q) = VC(Q)/Q.

The Determinants of Short-Run Costs -- Details

Several times we have seen that the reason for the shape of the cost curves is the law of diminishing returns to a variable input. In this section we will explore that relationship mathematically.

According to the law of diminishing returns, because capital is fixed in the short run, workers must share it as they attempt to produce the good. The more workers there are sharing this fixed amount of capital, it is inevitable that, at some point, each new workers' addition to output per hour (i.e., MP(L)) will fall because they will waste time waiting for capital to be free for their use. This has important implications for the firm's marginal cost.

If MPL is diminishing, then the firm must hire increasing amounts of labor to keep output going up at the same rate. This implies that marginal cost will (eventually) be positive and increasing.

To see this, let's work through the algebra. Suppose the firm hires L at a fixed wage rate, w, then VC(Q) = w ? L(Q), where L(Q) is the variable input requirement function. Marginal cost, we said earlier, is the derivative (i.e., slope) of the variable cost function. Thus:






Because MC is inversely related to MPL, a decreasing MP (which is guaranteed by the law of diminishing returns) causes an increasing MC.

The law of diminishing returns also affects the shape of the average variable cost, AVC(Q), and average total cost, ATC(Q), curves in the short run. Let's see how:







Ditto for average total cost... Once again this tells us that if AP(L) is increasing, then AVC(Q) must be falling. If AP(L) is decreasing, then AVC(Q) must be rising.

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Essay Instructions: DQ #1
Explain why an economy's output, in essence, is also its income?


DQ#2
Why do national income accountants compare the market value of the total outputs in various years rather than the physical volumes of production? What problem is posed by any comparison over time of the market values of various total output? How is the problem resolved?

************************************************************************CLASS LECTURE PASTED BELOW
*******************************************************

Without measures of economic aggregates like GNP, macroeconomics would be in a sea of unorganized data.

from Paul A. Samuelson and William Nordhaus,

Economics, 13th edition (New York: McGraw Hill Book Company, 1989)



______________________________________________________________________________

In the last class, we looked at market behavior for individual products or microeconomics However, if we wanted to look at an economy as a whole we need to aggregate all of the prices and quantities to determine the macoeconomic level of the economy.



The focus for this chapter is to present the concept of Gross Domestic Product (a.k.a GDP) which plays a pivotal role in determining the current state of the United States economy. But we need to understand the concepts of stocks and flows when talking about national income measures such as GDP:



Stocks is a variable measure at a point in time while flow is a variable measured per unit of time. Suppose you have a credit card and each month you run up a bill that exceeds repayments to the credit card company by $100. Thus, your deficit each month is $100. This is a flow. Stock is the amount of debt during a particular month.



In terms of economic variables, flow variables would be GDP and income while stock would be capital and wealth.



Gross Domestic Product (GDP) is a measure of the goods and services produced by the labor and property located in the United States.



Prior to 1992 the United States used GNP or the Gross National Product: (The United States converted to GDP because most of the international community used GDP)



Gross National Product (GNP) is a measure of the goods and services produced by the labor and property supplied by US residents.



What is the difference between GNP and GDP? GDP measures the production of goods and services on US soil while GNP measures the production of goods and services produced anywhere by the US residents. For example, Ford build a plant in Belgium to produce cars. This would be included in the calculation of GNP. However, this would not be included in GDP since it was not manufactured on US soil.



GNP was devised by Simon Kuznets who is also known as the "father of national income accounting."



Kuznets developed the first set of national accounts at the start of the Great Depression because policy-makers wanted more estimates concerning the originating from the various sectors of the economy and to develop future trends in the economy. These accounts are called the National Income and Product Accounts (NIPA). These NIPA aggregates are analogous to the income state of the firm because the NIPA?s reveal economic activity for a period of time, i.e., a year, a quarter etc.



Both GNP and GDP are structured by having 4 sectors. These sectors are groups of decision-making units that are engaged in the same type of transactions and are affected by and respond to economic developments in a similar manner. For example, consumers buy all types of goods and services (Lender?s bagels, gasoline, dental care, memberships to health clubs, ovens, VCR?s cellular phones etc). and if a recession should occur, the consumption of these goods would decline as many consumers may be laid off and do not have much disposable income to spend on goods. Thus, the level of GDP would decline.



GDP is an aggregate measure of the economy. The components of GDP are as follows:



1. Personal Consumption Expenditures (PCE)?this component represents the consumption of goods and services by consumers. PCE can be broken down into

a. Nondurable goods?goods that last a short period such as food and clothing

b. Durable Goods?goods that last a long time such as cars, tv?s, VCR?s etc.

c. Services? work done for consumers by individuals and firms such as haircuts, rug-cleaning, landscaping services, psychological counseling etc.

2. Gross Private Domestic Investment (GPDI)-- part of current output used to increase the capital stock and produce more output in the future



Note: This does not include a firm?s acquisition of stocks, bonds, and other financial instruments since the focus in on economic investments, not financial investments. when a company you purchased stock from uses the money you gave them to buy machines and other items, this is called investment.



This sector can be divided as follows:

a. Business fixed investment-purchase of new plants and equipment by firms

b. Residential fixed investment? the purchases of new housing by households

c. Inventory investment?when a firm produces but is unable to sell these goods



3. Government (G)?goods and services purchased by the public sector. By government, this includes federal, state, and local governments. For example, the Department of Defense buys tanks for its arsenal; government builds highways; they purchase supplies etc.

4. The international sector (Xn) this sector is the difference between exports and imports. Exports are added to GDP. Suppose your German cousin bought a Ford Contour. This is counted in GDP and not as PCE or the other sectors.



Imports are subtracted out. Suppose your cousin in Texas bought a Toyota Camry made in Japan. This Toyota will appear in US consumption but it is not part of our accounts so it needs to be taken out.



Remark: If we assume a closed economy then we have three sectors: PCE, GPDI, and G. If we have an open economy, we have the four sectors discussed above.



Remark 2: The largest component of GDP is PCE.



Suppose I bought a Ford Excursion (the 19 footer). When I purchased this, I am paying for the costs of the final good. What about the paint, metal, glass, tires etc. that were used to make the product? How are these items treated in the national accounts? Since Ford bought these items to make the Ford Excursion, these items would not be counted in the calculation of the GDP. That is, intermediate goods (paint, metal, glass, tires etc) and they are not part of GDP, just the final product is calculated into GDP. If we included the intermediate goods in the calculation of GDP, we would have a problem of "double-counting" and GDP would be overstated.



They are based in the idea that the amount of economic activity that occurs during a

period of time can be measured in terms of:



THE PRODUCT APPROACH

The amount of output produced, excluding output used up in intermediate stages

of production.



THE INCOME APPROACH

The incomes received by the producers of output, i.e., interest, profits, etc.



THE EXPENDITURE APPROACH

The amount of spending by the ultimate purchasers of output. This is called



All three approaches give identical measurements of the amount of current economic activity.



Example: Suppose we have the following hypothetical data from Microstats and Sourceone:



Microstats:

wages paid to employees 70,000

taxes paid to government 20,000

equipment purchased from Sourceone 40,000

Revenue from sale of software 370, 000





Sourceone:

wages paid 10,000

taxes paid 5,000

revenue from sale of equipment 60,000

to public 20,000

to Microstats 40,000



Solution: We shall illustrate the concept of the three approaches:



1. The Product Approach:

Microstats has a total revenue of $370,000



the product approach makes use of the idea of value-added which is the value of the output minus the value of the inputs it purchases from other producers (in this case Sourceone).



Now the value-added int his case would be 370,000-40,000=330,000. This is the estimate for value-added.

The value-added for Sourceone is equal to the total revenue since they do not purchase inputs from any other producer. Thus, value added is 60,000.



The total value-added is 390,000 in the economy (330,000+60,000)

Thus, the product approach gives a GDP of $390,000.



2. The Income Approach? measures economic activity by adding all income received including wages, taxes, and after-tax profits:



Wage Income (70,000+10,000)= 80,000

Taxes (20,000+5,000) = 25,000

Profits (240,000+45,000) = 285,000





390,000



3. The Expenditure Approach: measures economic activity by adding the amount spent by the final users of the output.



Microstats sales to the public 370,000

Sourceone sales to the public 20,000



390,000



The 3 approaches are equal. Why? By definition the amount sold to the public must equal the amount paid by the public so the product approach and the expenditure approaches are the same.



The money received by the producers must go somewhere: either to the people, the owners, or the government, so the income expenditure equals the expenditure.



With these three approaches being equal, this is called the Fundamental Identity of National Income Accounting.



Why use the market value of goods?

It helps with aggregation. It takes into account differences in the relative economic

importance of different goods and services.

--market value does not help with goods and services which are not sold in formal

markets (eg. Services provided by a homemaker. Or babysitting services provided

by an uncle do not count in GDP although payments to a day-care center would).



Capital goods: is a good that is itself produced and then is used to produce other goods, unlike an intermediate good it doesn?t get all used up in the same period that it itself was produced. Capital goods are counted as final goods and are thus included in GDP.



Inventories: Inventories are stocks of unsold finished goods, goods in process, and raw materials held by firms. Inventory investment is the amount by which inventories increase during the year (it can be negative). Inventory investment is treated as a final good and as such is included in GDP.



Expenditure Approach to Measuring GDP.

GDP contains four components and by summing these four sectors is the expenditure approach.



Now let?s look at the Income Approach to measuring GDP.

National Income = the sum of:

compensation of employees: wages, salaries, employer benefits, employer

contributions to social security

proprietor?s income: income of the self-employed

rental income

corporate profits

net interest: interest earned by individuals from businesses and foreign

sources minus interest paid by individuals

Indirect business taxes: sales and excise taxes

depreciation: the value of capital that wears out during the period over

which economic activity is being measured

net factor payments



Here is a numerical example which illustrates the calculations involved. From these data calculate the GDP, NDP, and NI.

Net rental income of persons - 24.1

Depreciation 669.1

Compensation of employees 3780.4

Personal Consumption expenditures 4378.2

Sales and excise taxes 525.3

Business transfer payments 28.7

Statistical discrepancy 2.3

Gross Private domestic investment 882.0

Exports of goods and services 659.1

Net subsidies of government business 9.0

Government purchases of goods and services 1148.4

Imports of goods and services 724.3

Net interest 399.5

Proprietor's income 441.6

Corporate profits 485.8

Net factor income from rest of world 5.7



a. GDP = C+I+G+EX-IM = 4378.2 + 882.0 + 1148.4 + 659.1 - 724.3 = 6343.4

b. NDP = GDP - depreciation = 6343.4 - 669.1 = 5674.3 [NDP is the net domestic product which is the GDP-capital consumption allowance or depreciation]

c NI via expenditure method:

NI = NDP - indirect business tax and nontax liability

- business transfer payments

- statistical discrepancy

+ government subsidy

+ net factor income from rest of world

= 5674.3 ? 525.3 - 28.7 ? 2.3 + 9.0 + 5.7 = 5132.7

NI via adding the incomes of different people

NI = compensation of employees + proprietors' income

+ net rental income

+ corporate profits

+ net interest

= 3780.4 + 441.6 + 24.1 + 485.8 + 399.5 = 5131.4



Index Numbers



Index numbers are tools used to measure the relative movement in a price, quantity, or expenditures or some aggregation of prices, quantities, or expenditures.



Nominal GDP (or current dollars) represents the total money value of goods and services produced in the current year, i.e., the year 1990 only has items produced in 1990.



Nominal GDP is simply an aggregation of all goods and services produced in a given year which is described mathematically as





where pi is the price of the ith good/service and qi is the amount of the quantity of the ith good/service.



Real GDP ( or constant dollars) these values are in terms of market prices of a previous period dollars. In other words, taking 1999 prices and converting them to 1982 prices. To find real GDP, we use the following formulation:



GDP Deflator is a weighted average of the price indexes used to deflate GDP.



Why do we want to deflate the nominal GDP? The growth rate of nominal GDP captures the increases in the goods/services and the prices for a given year. Thus, it becomes difficult to compare the nominal GDP between 1999 and 1949 because prices have risen quite a bit. Real GDP tries to correct these changes in prices by holding prices constant across time.



When an index number is based on the price and quantity of a single commodity, this is a simple index number.



Definition: A simple index number is based on the relative changes (over time) in the price and quantity of a single commodity.



A simple price index can be constructed as





(1)





That is, (1) tells us the averaging of the price ratios for two time periods over the collect of N items purchased by the population. (1) is called the Carli index as created by Carli in 1764. If we assigned weights to the price ratios in A, this is called the Sauerback index which can be given as





(2)

To determine the changes in the prices and quantities of a single commodity, we first need to define the base period. This is important because we can measure changes in other periods with the price and quantities during the base period, However, these two indexes are deficient to use because these indexes are really simply arithmetic mean indexes. Thus, this deficiency may be seen to stem from the actual probability distributions of the prices and the empirical evidence suggest that the data for prices may be right skewed which would make the arithmetic indexes upwardly biased. A remedy for this would be to take logarithms of the prices which would be close to a normal distribution. This index would look like





(3)



(3) basically looks like a geometric mean index and this index is known as the Jevons index. This index preserves the distribution of the price ratios or this index is log-normally distributed when the price ratios are mutually independent, log-normal random variables. Also note that the price ratios are in the same time period



Let?s illustrate the idea of a simple index. The calculation of a simple index number is given as follows:







where It is the index number at time t

Yt is the time series value at time t

Yo is the time series value at the base period



Example: Given the following silver prices calculate the index for 1975 and 1985 using 1972 as the base year.



Year
Price of Silver (per ounce)

1970
1.771

1971
1.546

1972
1.684

1973
2.558

1974
4.078

1975
4.419

1976
4.353

1977
4.620

1978
5.440

1979
11.090

1980
20.633

1981
10.481

1982
7.950

1983
11.439

1984
8.141

1985
6.192










The silver price has risen by 162.4% between 1972 and 1975 and by 267.70% between 1972 and 1985.



Note: The index number for the base period is always 100. In out preceding example, the 1972 index number is given as follows:







Note: The value of the index numbers is always measured as a percentage and is based out of 100.



Composite Index Numbers



Definition: A composite index number represents combinations of the prices or quantities of several commodities.



Definition: A simple composite index is a simple index for a time series consisting of total price or total quantity of 2 or more commodities.



The drawback of the simple composite index is that the quantity of the commodity that is purchased during each period is not taken into account when only price totals are used to calculate the index. To remedy this, the weighted composite price index is used:



Definition: A weighted composite price index weights the prices by quantities purchased prior to calculating totals for each time period. The weighted totals are then used to compute the index as the unweighted totals are used for simple composite indexes.



There are three types of weighted composite price index. Note that b is the base year and c is the current year, and N is the total items in the index



1. The Laspeyres Quantity Index or Fixed Weight:









2. The Paasche Quantity Index or the Variable Weight Index:









3. The Fisher Quantity Index?this is simply the geometric mean between the Laspeyres and the Paasche indexes. The Fisher?s Ideal Index incorporates quantity information from both time periods







Now let?s proceed to explain each of these indices.

1. The Laspeyres Index uses the base period quantities as the weights because the prices in each time period should be compared as if the same quantities were purchases each period were purchased during the base period.



2. The Paasche Index uses the current year quantities to base period prices at current purchase levels. This index uses weights of the quantities for the period in which the index value represents.



3. The Fisher Quantity Index (also called the chain-weighted index) is the geometric mean of the Laspeyres and the Paasche indexes. It is called a chained weighted index because we calculate it not by taking an average of the current year and the base year, but rather of the current year and the year before.



It becomes best to illustrate with an example of the above concepts:



Example: (Seattle Example) Suppose we have the economy of Seattle where three items are produced: airplanes, computers, and Starbucks coffee. Below is the table with the quantities produced and their market prices for 1996 and 1997.






1996

1997



Quantity
Price
Quantity
Price

Airplanes
10
1000
20
1500

Coffee
100
100
100
200

Computers
10
500
15
500




Calculate the growth in Real GDP using the fixed weight index, the variable weight index, and the chain-weighted index setting 1996=100.



Solution:



1. The fixed weight index is given as follows:









so the growth rate of GDP under the fixed weight index is 50%.



2. The variable weight index is given as follows:











so the growth rate of GDP under the variable fixed weight index is 43.75%



3. The Fisher Index is as follows:







so the growth rate is 1.468*100=146.80 is 46.80%



Remark: A chained weighted index can be given as

C_1 * F_{1,2} * F_{2,3} *......*F_{t-1, t} ~=~C_t

where C1 is the nominal value or the quantity of the base year and Ct is the chained quantity in real terms in year t.



For example, we wanted to know what the chain weighted real GDP is for 1997 with 1996 as the base year, first find the nominal value for 1996 and multiply by the Fisher index between the two years . Nominal GDP is calculated as (10*1000+100*100+10*500=25000) then multiplying 25,000 by the Fisher Index we have 25000*1.468=36700.



How do we measure prices?



1. The Consumer Price Index (CPI) attempts to measure the change in prices of the typical basket of goods that a household would buy in an urban area and consumers living in rural areas are not included in the CPI. The CPI comprises of two components: (1) housing and (2) commodities/services. For the housing component, the target population is the collection of all residential units (occupied by owners and renters) in urban areas of the United States. For the commodities/services, the target population is the collection of all retail and service outlets at which the consumers shop (including catalogs, telemarketing operations, and internet shopping sites). The underlying assumption of the CPI is that consumers substitute freely in response to changes in relative prices; however, across strata, the assumption is that there is no substitution occurs when relative prices change.





The government uses the CPI to measure inflation (the Fed responds to the changes in the CPI but conducting the appropriate monetary policy, i.e., increase or decrease the money supply.) and benefits from transfer programs (Social Security) are indexed to the CPI so that the benefits do not decline in real value.

The CPI measures price changes by buying a market basket of goods at different times. Three known

problems are:





1. Tastes may change over time but same basket is bought.

2. Relative prices may change but the same basket is bought

3. Price changes involve subtle income changes that give rise to income effects that will alter the mix of goods bought.





2. The GDP Deflator is used to convert nominal GDP to real GDP.



For the CPI, it is a fixed weight index which is also known as the Laspeyres Index while the GDP Deflator is a variable weight index or the Paasche Index. However, unlike the earlier versions of the fixed and variable indexes, we want to keep the quantities constant at either a current or base year and let prices change. The variations of the fixed and variable indexes are given as follows:





GDP does not include nonproductive non-market activities which are not included in GDP, i.e., fixing things around the home, housework etc. The data from these activities do not exists; therefore cannot be included in GDP.



Anything that shifts from sector to another sector in the national accounts such as transfer payments (Social Security benefits, welfare benefits etc) is not included into GDP. Also suppose you buy a 1966 Ford Mustang in 1999. Would this sale be included in GDP? No because when the Ford Mustang was purchased in 1966 it was included into GNP and this transaction is simply a transfer of ownership from one person to another.



GDP is not the perfect measure of a society?s well-being because GDP does not account for the major developments of this century, i.e., increase in leisure time, improved quality of goods etc.



More importantly, GDP measures economic activity that involves an exchange of money. Thus, it leaves out much that people value and that serves basic needs such as unpaid work in households caring for children and elderly, the free time devoted to family and community activities; and it also leaves out the contributions of the natural habitat such as fresh air, fertile soil, clean water etc.



GDP fails to incorporate the distinction between transactions that contribute to well-being and those that diminish well-being. That is, GDP operates like an income statement that adds expenses to income instead of subtracting them. That is, GDP does not distinguish between progress and regression. GPI is a measure of the well-being of the nation expressed in economic terms which includes the value of market and nonmarket activity within one set of national accounts.



For example, suppose you buy a care in 1999. GDP in 1999 increases. then you need to buy gasoline and pollutes the air. GDP increases. Additional cars increase traffic and accidents so more money is spent on maintenance, insurance, and police. GDP increases. If you are involved in an accident, this results in repair bills and medical expenses so GDP increases. From this example, GDP keeps increases without subtracting the ?bad" items." Thus, a true accounting measure would offer the net benefits which subtract out the bads. The Genuine Progress Indicator (GPI) is such a measure. GPI includes more than 20 aspects of our economic lives which the GDP ignores. GPI is calculated by the organization of Redefining Progress located in California. Redefining Progress used data available from various sources to use in the GPI.



Here are some of the factors that are included in GPI which are not included in GDP:



1. Household and volunteer work ?GDP does not include caring for the children, the elderly, volunteering out time around the neighborhood etc.

2. Crime. GDP counts the money spend on deterring crime and the damages associated with it. Realistically, paying for deterring crime and the damages associated with them should be treated as costs which should be subtracted.

3. The distribution of income. Few people every experience increase in wealth so the GPI would show the extent in which the whole population actually shared in the increase.

4. Resource Depletion and degradation. GDP treats environmental degradation and natural resource depletion as a gain while GPI subtracts the costs of environmental degradation and mineral depletion.

5. Loss of leisure. If one has to work two or more jobs to stay ahead, they are not really better off since they have less time with their families, pursuing other interests etc. GDP basically assumes that time is worth nothing.





For a detailed discussion about the GPI, please consult the following:

"If the GDP is Up, Why is America Dowm?" The Atlantic Monthly, October 1995 which is available at .com/atlantic/election/connection/ecbig/gdp.htm)

Cobb, Clifford, Ted Halstead and Jonathan Rowe. (1995). The Genuine Progress Indicator: Summary of Data and Methodology. Redefining Progress.

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Title: Financial Analysis of McDonalds

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Essay Instructions: There are Three Parts to this Assignment that has to do completely with the McDonald's Fast Food Corp. Every time I refer to something as company I mean McDonald's Fast Food Corp.

First Part
I need you to answer all questions in a Question and answer format not as a term paper. It has to do with some math work so please show calculations.

Example :
1) answer to question 1
2) answer to question 2
3) answer to question 3
and so on for the following Questions:

1)Using monthly returns for the last five years, compute your company’s “beta.” Use the S&P 500 Index as a proxy for the market to compute monthly market returns for the last 60 months.

2)What was your company’s holding period return over the last year? Explain your computation. (Hint: yahoo.com has historical prices.)

3)What was the market’s (S&P 500) holding period return over the last year? Explain your computation. (Hint: yahoo.com has historical prices.)

4)What was your company’s most recent annual ROE? What was your company’s average annual ROE over the most recent 5 years?


5)Provide an estimate of your company’s “required rate of return.”

6)What were annual sales for the most recent year available? What was your company’s average annual sales revenue growth rate for the most recent five year period? Find both the arithmetic mean and geometric mean sales growth rates.

7)What was your company’s most recent annual EPS? Were annual EPS trending up, down or holding steady over the last 5 years? Answer the last question by referring to actual EPS numbers.

8)What was your company’s most recent annual net profit margin (i.e. net income/sales)? Has the net profit margin been improving, deteriorating or holding steady over the last five years?

9)Does your company pay a dividend? If yes, what was the annual dividend per share paid to common shareholders in 2009? What were annual dividends five years ago?

10)In your opinion, who is your company’s main competitor? Provide your response in a few sentences (i.e. not more than one paragraph).
11)Can you identify the major risk to your industry?

12)Can you identify the major opportunity for your industry?

13)If yes, can you cite support for your views in 1 and 2?

14)Can you summarize the outlook for your industry?

15)How does your company’s most recent ROE and use of debt compare with its main competitor?

(Hint: The Standard & Poors NetAdvantage Industry Reports are useful. Also, Mergent Online is good.)


Part 2

Answer these questions in a term paper format 1-2 pages

1. Identify and explain your company’s main strength.
2. Identify and explain your company’s biggest weakness.
3. Identify and explain the biggest opportunity enjoyed by your company.
4. Identify and explain the chief threat faced by your company.


Part 3

Answer this as a separate term paper

Submit a final 2-3 page recommendation as to the investment action that should be taken in relation to your company’s stock. This final assignment should include a recommendation to buy, sell or hold the company’s stock and a rationalization of this recommendation.

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Title: Statistics

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Essay Instructions: Specifically write the term paper based ONLY on the following article:

What this handout is about
The purpose of this handout is to help you use statistics to make your argument as effectively as possible.
Introduction
Numbers are power. Apparently freed of all the squishiness and ambiguity of words, numbers and statistics are powerful pieces of evidence that can effectively strengthen any argument. But statistics are not a panacea. As simple and straightforward as these little numbers promise to be, statistics, if not used carefully, can create more problems than they solve.

Many writers lack a firm grasp of the statistics they are using. The average reader does not know how to properly evaluate and interpret the statistics he or she reads. The main reason behind the poor use of statistics is a lack of understanding about what statistics can and cannot do. Many people think that statistics can speak for themselves. But numbers are as ambiguous as words and need just as much explanation.

In many ways, this problem is quite similar to that experienced with direct quotes. Too often, quotes are expected to do all the work; are treated as part of the argument, rather than a piece of evidence requiring interpretation (see our handout on how to quote.) But if you leave the interpretation up to the reader, who knows what sort of off-the-wall interpretations may result? The only way to avoid this danger is to supply the interpretation yourself.


But before we start writing statistics, let's actually read a few.

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Reading statistics
As stated before, numbers are powerful. This is one of the reasons why statistics can be such persuasive pieces of evidence. However, this same power can also make numbers and statistics intimidating. That is, we too often accept them as gospel, without ever questioning their veracity or appropriateness. While this may seem like a positive trait when you plug them into your paper and pray for your reader to submit to their power, remember that before we are writers of statistics, we are readers. And to be effective readers means asking the hard questions. Below you will find a useful set of hard questions to ask of the numbers you find.

1. Does your evidence come from reliable sources?
This is an important question not only with statistics, but with any evidence you use in your papers. As we will see in this handout, there are many ways statistics can be played with and misrepresented in order to produce a desired outcome. Therefore, you want to take your statistics from reliable sources (for more information on finding reliable sources, please see our handout on evaluating print sources). This is not to say that reliable sources are infallible, but only that they are probably less likely to use deceptive practices. With a credible source, you may not need to worry as much about the questions that follow. Still, remember that reading statistics is a bit like being in the middle of a war: trust no one; suspect everyone.

2. What is the data's background?
Data and statistics do not just fall from heaven fully formed. They are always the product of research. Therefore, to understand the statistics, you should also know where they come from. For example, if the statistics come from a survey or poll, some questions to ask include:

Who asked the questions in the survey/poll?
What, exactly, were the questions?
Who interpreted the data?
What issue prompted the survey/poll?
What (policy/procedure) potentially hinges on the results of the poll?
Who stands to gain from particular interpretations of the data?

All these questions are a way of orienting yourself toward possible biases or weaknesses in the data you are reading. The goal of this exercise is not to find "pure, objective" data but to make any biases explicit, in order to more accurately interpret the evidence.

3. Are all data reported?
In most cases, the answer to this question is easy: no, they aren't. Therefore, a better way to think about this issue is to ask whether all data have been presented in context. But it is much more complicated when you consider the bigger issue, which is whether the text or source presents enough evidence for you to draw your own conclusion. A reliable source should not exclude data that contradicts or weakens the information presented.

An example can be found on the evening news. If you think about ice storms, which make life so difficult in the winter, you will certainly remember the newscasters warning people to stay off the roads because they are so treacherous. To verify this point, they tell you that the Highway Patrol has already reported 25 accidents during the day. Their intention is to scare you into staying home with this number. While this number sounds high, some studies have found that the number of accidents actually goes down on days with severe weather. Why is that? One possible explanation is that with fewer people on the road, even with the dangerous conditions, the number of accidents will be less than on an "average" day. The critical lesson here is that even when the general interpretation is "accurate," the data may not actually be evidence for the particular interpretation. This means you have no way to verify if the interpretation is in fact correct.

There is generally a comparison implied in the use of statistics. How can you make a valid comparison without having all the facts? Good question. You may have to look to another source or sources to find all the data you need.

4. Have the data been interpreted correctly?
If the author gives you her statistics, it is always wise to interpret them yourself. That is, while it is useful to read and understand the author's interpretation, it is merely that??"an interpretation. It is not the final word on the matter. Furthermore, sometimes authors (including you, so be careful) can use perfectly good statistics and come up with perfectly bad interpretations. Here are two common mistakes to watch out for:


Confusing correlation with causation. Just because two things vary together does not mean that one of them is causing the other. It could be nothing more than a coincidence, or both could be caused by a third factor. Such a relationship is called spurious.

The classic example is a study that found that the more firefighters sent to put out a fire, the more damage the fire did. Yikes! I thought firefighters were supposed to make things better, not worse! But before we start shutting down fire stations, it might be useful to entertain alternative explanations. This seemingly contradictory finding can be easily explained by pointing to a third factor that causes both: the size of the fire. The lesson here? Correlation does not equal causation. So it is important not only to think about showing that two variables co-vary, but also about the causal mechanism.


Ignoring the margin of error. When survey results are reported, they frequently include a margin of error. You might see this written as "a margin of error of plus or minus 5 percentage points." What does this mean? The simple story is that surveys are normally generated from samples of a larger population, and thus they are never exact. There is always a confidence interval within which the general population is expected to fall. Thus, if I say that the number of UNC students who find it difficult to use statistics in their writing is 60%, plus or minus 4%, that means, assuming the normal confidence interval of 95%, that with 95% certainty we can say that the actual number is between 56% and 64%.

Why does this matter? Because if after introducing this handout to the students of UNC, a new poll finds that only 56%, plus or minus 3%, are having difficulty with statistics, I could go to the Writing Center director and ask for a raise, since I have made a significant contribution to the writing skills of the students on campus. However, she would no doubt point out that a) this may be a spurious relationship (see above) and b) the actual change is not significant because it falls within the margin of error for the original results. The lesson here? Margins of error matter, so you cannot just compare simple percentages.


Finally, you should keep in mind that the source you are actually looking at may not be the original source of your data. That is, if you find an essay that quotes a number of statistics in support of its argument, often the author of the essay is using someone else's data. Thus, you need to consider not only your source, but the author's sources as well.

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Writing statistics
As you write with statistics, remember your own experience as a reader of statistics. Don't forget how frustrated you were when you came across unclear statistics and how thankful you were to read well-presented ones. It is a sign of respect to your reader to be as clear and straightforward as you can be with your numbers. Nobody likes to be played for a fool. Thus, even if you think that changing the numbers just a little bit will help your argument, do not give in to the temptation.

As you begin writing, keep the following in mind. First, your reader will want to know the answers to the same questions that we discussed above. Second, you want to present your statistics in a clear, unambiguous manner. Below you will find a list of some common pitfalls in the world of statistics, along with suggestions for avoiding them.

1. The mistake of the "average" writer
Nobody wants to be average. Moreover, nobody wants to just see the word "average" in a piece of writing. Why? Because nobody knows exactly what it means. There are not one, not two, but three different definitions of "average" in statistics, and when you use the word, your reader has only a 33.3% chance of guessing correctly which one you mean.

For the following definitions, please refer to this set of numbers:
5, 5, 5, 8, 12, 14, 21, 33, 38


Mean (arithmetic mean)

This may be the most average definition of average (whatever that means). This is the weighted average??"a total of all numbers included divided by the quantity of numbers represented. Thus the mean of the above set of numbers is 5+5+5+8+12+14+21+33+38, all divided by 9, which equals 15.444 (Wow! That is a lot of numbers after the decimal??"what do we do about that? Precision is a good thing, but too much of it is over the top; it does not necessarily make your argument any stronger. Consider the reasonable amount of precision based on your input and round accordingly. In this case, 15.6 should do the trick.)
Median

Depending on whether you have an odd or even set of numbers, the median is either a) the number midway through an odd set of numbers or b) a value halfway between the two middle numbers in an even set. For the above set (an odd set of 9 numbers), the median is 12. (5, 5, 5, 8 < 12 < 14, 21, 33, 38)
Mode

The mode is the number or value that occurs most frequently in a series. If, by some cruel twist of fate, two or more values occur with the same frequency, then you take the mean of the values. For our set, the mode would be 5, since it occurs 3 times, whereas all other numbers occur only once.

As you can see, the numbers can vary considerably, as can their significance. Therefore, the writer should always inform the reader which average he or she is using. Otherwise, confusion will inevitably ensue.


2. Match your facts with your questions
Be sure that your statistics actually apply to the point/argument you are making. If we return to our discussion of averages, depending on the question you are interesting in answering, you should use the proper statistics.

Perhaps an example would help illustrate this point. Your professor hands back the midterm. The grades are distributed as follows:


Grade # Received
100 4
98 5
95 2
63 4
58 6




The professor felt that the test must have been too easy, because the average (MEDIAN) grade was a 95.

When a colleague asked her about how the midterm grades came out, she answered, knowing that her classes were gaining a reputation for being "too easy," that the average (MEAN) grade was an 80.

When your parents ask you how you can justify doing so poorly on the midterm, you answer, "Don't worry about my 63. It is not as bad as it sounds. The average (MODE) grade was a 58."

I will leave it up to you to decide whether these choices are appropriate. Selecting the appropriate facts or statistics will help your argument immensely. Not only will they actually support your point, but they will not undermine the legitimacy of your position. (Think about how your parents will react when they learn from the professor that the average (MEDIAN) grade was 95.) The best way to maintain precision is to specify which of the three forms of "average" you are using.

3. Show the entire picture
Sometimes, you may misrepresent your evidence by accident and misunderstanding. Other times, however, misrepresentation may be slightly less innocent. This can be seen most readily in visual aids. Do not shape and "massage" the representation so that it "best supports" your argument. This can be achieved by presenting charts/graphs in numerous different ways. Either the range can be shortened (to cut out data points which do not fit, e.g., starting a time series too late or ending it too soon), or the scale can be manipulated so that small changes look big and vice versa. Furthermore, do not fiddle with the proportions, either vertically or horizontally. The fact that USA Today seems to get away with these techniques does make them OK for an academic argument.



Charts A, B, and C all use the same data points, but the stories they seem to be telling are quite different. Chart A shows a mild increase, followed by a slow decline. Chart B, on the other hand, reveals a steep jump, with a sharp drop-off immediately following. Conversely, Chart C seems to demonstrate that there was virtually no change over time. These variations are a product of changing the scale of the chart. One way to alleviate this problem is to supplement the chart by using the actual numbers in your text, in the spirit of full disclosure.



Another point of concern can be seen in Charts D and E. Both use the same data as charts A, B, and C for the years 1985-2000, but additional time points, using two hypothetical sets of data, have been added back to 1965. Given the different trends leading up to 1985, consider how the significance of recent events can change. In Chart D, the downward trend from 1990 to 2000 is going against a long-term upward trend, whereas in Chart E, it is merely the continuation of a larger downward trend after a brief upward turn.



One of the difficulties with visual aids is that there is no hard and fast rule about how much to include and what to exclude. Judgment is always involved. In general, be sure to present your visual aids so that your readers can draw their own conclusions from the facts and verify your assertions. If what you have cut out could affect the reader's interpretation of your data, then you might consider keeping it.

4. Give bases of all percentages
Because percentages are always derived from a specific base, they are meaningless until associated with a base. So even if I tell you that after this reading this handout, you will be 23% more persuasive as a writer, that is not a very meaningful assertion because you have no idea what it is based on??"23% more persuasive than what?

Let's look at crime rates to see how this works. Suppose we have two cities, Springfield and Shelbyville. In Springfield, the murder rate has gone up 75%, while in Shelbyville, the rate has only increased by 10%. Which city is having a bigger murder problem? Well, that's obvious, right? It has to be Springfield. After all, 75% is bigger than 10%.

Hold on a second, because this is actually much less clear than it looks. In order to really know which city has a worse problem, we have to look at the actual numbers. If I told you that Springfield had 4 murders last year and 7 this year, and Shelbyville had 30 murders last year and 33 murders this year, would you change your answer? Maybe, since 33 murders are significantly more than 7. One would certainly feel safer in Springfield, right?

Not so fast, because we still do not have all the facts. We have to make the comparison between the two based on equivalent standards. To do that, we have to look at the per capita rate (often given in rates per 100,000 people per year). If Springfield has 700 residents while Shelbyville has 3.3 million, then Springfield has a murder rate of 1,000 per 100,000 people, and Shelbyville's rate is merely 1 per 100,000. Gadzooks! The residents of Springfield are dropping like flies. I think I'll stick with nice, safe Shelbyville, thank you very much.

Percentages are really no different from any other form of statistics: they gain their meaning only through their context. Consequently, percentages should be presented in context so that readers can draw their own conclusions as you emphasize facts important to your argument. Remember, if your statistics really do support your point, then you should have no fear of revealing the larger context that frames them.

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Important questions to ask (and answer) about statistics

Is the question being asked relevant?
Do the data come from reliable sources?
Margin of error/confidence interval??"when is a change really a change?
Are all data reported, or just the best/worst?
Are the data presented in context?
Have the data been interpreted correctly?
Does the author confuse correlation with causation?

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Conclusion
Now that you have learned the lessons of statistics, you have two options. Use this knowledge to manipulate your numbers to your advantage, or use this knowledge to better understand and use statistics to make accurate and fair arguments. The choice is yours. Nine out of ten writers, however, prefer the latter, and the other one later regrets his or her decision.


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