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.
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..
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.
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).
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).
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.
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.
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.
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.
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.]
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.
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.
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
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.
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.
a & b
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.
a & b
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).
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).
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.
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
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
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.
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.
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.
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.
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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