Planteamiento del refrigerante

Michael Malcolm

June 18, 2011

There are two important classes of utility functions for which the calculus-based maximization techniques developed earlier in the unit cannot be applied. In this section, we will learn how to deal with both.

1

Perfect Complements

Left shoes and right shoes are 1-for-1 perfect complements. Right shoes and left shoes are useless unless you have them together. If left shoes areL and right shoes areR, then a utility function that represents these preferences is:

U(L, R) = min{L, R}

For example, ifL= 5 andR= 5, then the consumer’s utility level isU(L, R) = min{L, R}= min{5,5}= 5. If L = 7 but stillR = 5, then utility isU(L, R) = min{L, R} = min{7,5} = 5, which is the same as before. But this makes sense since the additional two left shoes give the consumer no extra utility if he has only 5 right shoes. Thus, the indifference curve forU = 5 contains the bundles (5,5), (5,6), (5,7), (6,5), (7,5), etc... Using the utility function above, all these bundles generate 5 units of utility. The indifference curve is represented graphically in figure 1.

Calculus-based techniques do not work for utility functions of this form because the utility function is not differentiable. Graphically, there is a ”sharp point” on the corner in figure 1, and so the derivative does not exist there.

Now, suppose that the price of a left shoe is PL = 4, the price of a right shoe isPR = 1 and that the consumer has an income ofY = 100. The obvious condition for the optimal bundle isL=R; otherwise the consumer is wasting money. Also, the budget constraint must hold:

4L+R= 100 But since L=Rat the optimal bundle, we can substitute.

4L+L= 100 5L= 100 L= 20 and sinceL=R, thenR= 20.

In general, when LandRare perfect, one-for-one complements, prices arePL andPR and income isY, then the budget constraint is:

Figure 1: Indifference curve for perfect complements

And since L=Rmust hold at the optimum, we substitute.

PLL+PRL=Y L(PL+PR) =Y L= Y PL+PR ⇒R= Y PL+PR

It is also possible to have perfect complements that are not in a 1-for-1 ratio. For example, a consumer needs 4 tiresT for every car C. Then a utility function that represents these preferences is

U(L, R) = min{T,4C}

Thinking about why this formulation makes sense, 1 car and 4 tires yields a certain level of utility (U = 4). 2 cars and 8 tires yields more utility (U = 8). You need 4 tires and 1 car to raise utility. An equivalent formulation would beU(L, R) = min{1

4T, C}.

We always find the optimality condition for utility functions of this variety by setting the two arguments equal. In this case, the optimality condition is:

T = 4C The budget constraint is:

PTT+PCC=Y Substituting in the optimality condition:

Figure 2: Indifference curve for perfect substitutes PT(4C) +PCC=Y C(4PT+PC) =Y C= Y 4PT+PC ⇒T = 4Y 4PT +PC

2

Perfect Substitutes

Consider PepsiP and CokeC, which are perfect 1-for-1 substitutes. Then a utility function that represents these preferences is:

U(P, C) =P+C

The additive formulation makes sense since one more unit of Pepsi always raises utility exactly as much as one more unit of Coke. The two are exactly substitutable. For example, any of the bundles (5,0), (4,1), (0,5), etc... will generateU = 5. The indifference curve is represented graphically in figure 2.

For a utility function like this, the solution will always be acorner solution, where the consumer spends all his money on one good or all his money on the other good. In general, our calculus-based techniques work only forinterior solutions where the consumer buys some combination of the two goods.

Now, suppose that the price of Pepsi is PP = 2, the price of Coke is PC= 2.5 and that the consumer’s income isY = 100. If the consumer spends all his income on Pepsi, then he will have 50 Pepsi and 0 Coke, so his utility would be:

U(P, C) =P+C= 50 + 0 = 50

If on the other hand he spends all his income on Coke, then he will have 0 pepsi and 40 Coke, so his utility would be:

U(P, C) =P+C= 0 + 40 = 40

Clearly, his utility is higher in the first case, so the optimal bundle is P = 50 and C = 0 – a corner solution where he consumes only Pepsi and no Coke.

In general consider a utility function over two goods x1 and x2 where the two commodities are perfect substitutes.

U(x1, x2) =x1+x2

The price of good x1 is P1 and the price of good x2 isP2. If the consumer buys only x1, then he can afford _PY

1 units ofx1, and so his utility would be:

U(x1, x2) =x1+x2= Y P1 + 0 =

Y P1

If the consumer buys only x2, then he can afford _PY₂ units of x2, and so his utility would be:

U(x1, x2) =x1+x2= 0 + Y P2

= Y P2 He will buy only x1 as long as his utility from this option is higher:

Y P1 > Y P2 P2Y > P1Y P2> P1

But this makes intuitive sense. If good x2 is more expensive, then he will spend all his income on x1, buying no x2. However, if P1 > P2, then he will spend all his income on x2, buying 0 units of x1. If the prices are the same, then any combination of the two that is on the budget line will yield the consumer the same utility. We can state the Marshallian demand correspondence formally as follows.

• IfP1< P2: x1= _PY

1 andx2= 0

• IfP1> P2: x1= 0 andx2=_PY

2

• IfP1=P2: anyx1 andx2such thatP1x1+P2x2=Y.

Suppose now that two goods are perfect substitutes, but not in a 1-for-1 ratio. For example, for a caffeine-addict, 2 cups of teaT is a perfect substitute for 1 cup of coffeeC. A utility function that represents these preferences is:

U(C, T) = 2C+T

This formulation makes sense because one cup of coffee raises utility by the same amount as two cups of tea. Again, the solution will be a corner solution where he spends all his income on either coffee or tea.

If the consumer purchases only coffee, he can afford Y

U(C, T) = 2C+T = 2 _Y PC + 0 = 2Y PC

If the consumer purchases only tea, he can afford _PTY cups of tea, and so his utility would be:

U(C, T) = 2C+T = 0 + _Y PT = Y PT

We can conclude that he will choose the option with only coffee when his utility is higher:

2Y PC > Y PT 2Y PT > Y PC 2PT > PC

In words, the consumer buys only coffee unless coffee is more than twice as expensive as tea, in which case he switches to buying tea only.

Another way to think about this and to get this same condition is to recognize that the consumer will buy coffee as long as the marginal utility per dollar spend on coffee exceeds the marginal utility per dollar spent on tea. In this case, sinceM UC= 2 andM UT = 1, we can conclude that the consumer will buy coffee if: M UC PC > M UT PT 2 PC > 1 PT 2PT > PC

which is the same as the condition derived above. Again, we can state the Marshallian demand corre- spondence formally as follows.

• IfPC<2PT: C= _PCY andT = 0 • IfPC>2PT: C= 0 andT = _PY

T

Unit 3.5: Revealed Preference