Dirk Bergemann Department of Economics Yale University Economics 121b: Intermediate Microeconomics Midterm Suggested Solutions 2/8/12 1. (a) The equation of the indifference curve is given by, (x 1 + 2) (x 2 + 3) = U x 2 = U x 1 + 2 3 Therefore the slope of the indifference curve is given by, dx 2 U = dx 1 (x 1 + 2) 2 Hence the MRS between good 1 and 2 is, (b) See Figure 1. MRS 1,2 = dx 2 U = dx 1 (x 1 + 2) 2 (c) The utility function is, u (x 1, x 2 ) = (x 1 + 2) (x 2 + 3) Therefore, the marginal utilities of good 1 and 2 are, MU 1 = U x 1 = x 2 + 3 > 0 MU 2 = U x 2 = x 1 + 2 > 0 Hence the utility function is strictly increasing in x 1 and x 2. (d) The utility maximization problem is given by, max x 1,x 2 (x 1 + 2) (x 2 + 3) subject to p 1 x 1 + p 2 x 2 I In this optimization exercise the endogenous variables are x 1 and x 2 and the exogenous variables are prices p 1, p 2 and income I. 1
Figure 1: indifference curve (e) We have noted in part (c) of this question that the utility function is strictly increasing. Hence increasing the consumption of any of the goods raises the utility of the consumer. Hence given any amount of income the utility maximizing consumer will exhaust all his income purchasing the 2 goods. If he is left with positive amount of money after maximizing his utility then he can always purchase any one of the goods with the money left and consuming that additional amount of that good will raise his utility as it is always increasing. Hence we can restrict attention to the equality constraint: p 1 x 1 + p 2 x 2 = I. 2. (a) First, we shall calculate the first-order condition through the method of substitution. We solve the budget constraint in terms of x 2 and substitute it into the utility function. We have now gone from a constrained twovariable optimization problem to an unconstrained one-variable optimization problem. The consumer now seeks to maximize the following: (( ) ) I p1 x 1 u(x 1 ) = (x 1 + 2) + 3. To find the maximum, we take the first derivative and set it to zero. p 2 2
(b) This is the first-order condition: ( du = (x 1 + 2) p ) 1 + dx 1 p 2 (( I p1 x 1 p 2 ) ) + 3 = 0. Second, we shall calculate the first-order condition through the Lagrangean method. This time, we change the problem from a constrained two-variable optimization to a unconstrained three-variable optimization problem, where the utility function is augmented with a penalty for violating the budget constraint: L(x 1, x 2, λ) = (x 1 + 2)(x 2 + 3) + λ(i p 1 x 1 p 2 x 2 ). We seek to maximize the augmented utility function L(x 1, x 2, λ) with respect to x 1 and x 2 while minimizing it with respect to λ. To do so, we take the partial first derivatives with respect to each variable and set them to zero. These are the first-order conditions: (c) Using the method of substitution: L x 1 = (x 2 + 3) λp 1 = 0, L x 2 = (x 1 + 2) λp 2 = 0, L λ = I p 1 x 1 p 2 x 2 = 0. 2p 1 x 1 + I + 3p 2 2p 1 = 0 p 2 p 2 2p 1 x 1 = I + 3p 2 2p 1 x 1 = I 2p 1 + 3p 2 2p ( 1 ) I p I 2p1+3p 2 1 2p 1 x 2 = p 2 = I + 2p 1 3p 2 2p 2 3
Using the Lagrangean method: x 1 = λp 2 2 x 2 = λp 1 3 I λp 1 p 2 + 2p 1 λp 1 p 2 + 3p 2 = 0 λ = I + 2p 1 + 3p 2 p 1 p 2 x 1 = I 2p 1 + 3p 2 2p 1 x 2 = I + 2p 1 3p 2 2p 2 (d) The own-price elasticity of a good x is the percentage change in the Marshallian demand of x over the percentage change in its price. In other words, own-price elasticity shows the percentage change in x given a one-percent change in the price of x. Mathematically, For the given utility function, ɛ i,i = x i. x i p j ɛ 1,1 = = (2p 1)( 2) (I 2p 1+3p 2)(2) 4p 2 1 I 2p 1+3p 2 2p 2 1 I 3p 2 I 2p 1 + 3p 2 (e) The indirect utility function is found by plugging Marshallian demand into the utility function: ( ) ( ) I 2p1 + 3p 2 I + 2p1 3p 2 v(p 1, p 2, I) = + 2 + 3 2p 1 2p 2 = (I + 2p 1 + 3p 2 ) 2 4p 1 p 2 By the envelope theorem, v I evaluated at (x 1, x 2). Remember that the Marshallian demand is treated as fixed in this case. Therefore, I = L v I = λ = I + 2p 1 + 3p 2 p 1 p 2 4
Figure 2: Slutsky equation 3. (a) Differentiate x i (p, E (p, U)) = h i (p, U) on both sides with respect to p j using the chain rule: Using the fact that E x i + x i I E = h i. = x j and rearranging we get x i = h i }{{} SE x i I x j }{{} IE where the total (uncompensated) price effect appears on the LHS and the RHS consists of the compensated price effect (substitution effect, SE) and the income effect (IE). (b) See Figure 2. (c) (1) We can analyze whether a good is a Giffen good, i.e., whether its own-price effect is positive. As the Slutsky equation shows, this can only be the case if this good is also an inferior good (has a negative, 5
income effect) since the compensated own-price effect is always negative. (2) The Slutsky equation also allows us to analyze whether two goods are net subsitutes or complements. For example, two goods are gross substitutes if the uncompensated price effect is positive and they are net substitutes if the compensated price effect is positive. Since we cannot observe the compensated price effect in consumption data, we can plug the observed uncompensated price effect and the income effect into the Slutsky equation to infer whether two goods are net substitutes or complements. 6