Macroeconomic Theory and Analysis V31.0013 Suggested Solutions for the First Midterm Question 1. Welfare Theorems (a) There are two households that maximize max i,g 1 + g 2 ) {c i,l i} (1) st : c i w(1 l i ) for each i =1, 2. Replacing the gardening technology we have The FOC s for household i implies max i,γl 1 + γl 2 ) {c i,l i } (2) st : c i w(1 l i ) MRS i = γu 2 (c i,γl 1 + γl 2 ) U 1 (c i,γl 1 + γl 2 ) = w = f 1 (2 l 1 l 2 ) (3) wherewehaveusedthefactthatfirms set w = f 1 (n) =f (2 l 1 l 2 ) to maximize profits. (b) The social planner maximizes the sum of utilities subject to the production possibility frontier, i.e. L = max {c 1,l 1,c 2,l 2 } U(c 1,γl 1 +γl 2 )+U(c 2,γl 1 +γl 2 )+λ [f(2 l 1 l 2 ) c 1 c 2 ] (4) Taking derivative with respect to {c 1,l 1,c 2,l 2 }, we obtain respectively U 1 λ = 0 γu 2 + γu 2 λf 1 = 0 (5) U 1 λ = 0 γu 2 + γu 2 λf 1 = 0 Combining the FOC s of the planner for household 1 (and similarly for household 2) 2γU 2 (c 1,γl 1 + γl 2 ) = f 1 (1 l 1 l 2 ) (6) U 1 (c 1,γl 1 + γl 2 ) which is different from the one obtained in the CE. Note that the social planner internalizes the effect of each household s choice of leisure on the other 1
household. This is a positive externality that the household ignores in the decentralized equilibrium. As a consequence, the First Welfare Theorem does not hold, that is the competitive equilibrium is not Pareto optimal. (c) To decentralize the social planner allocation notice that in the CE households have too little incentives to enjoy leisure, i.e. they overwork and underestimate the social value of their gardening activities. To achieve optimality, the government can impose a tax on labor income τ of exactly.5. Consider the problem of the household with this tax, the FOC would imply γu 2 (c i,γl 1 + γl 2 ) U 1 (c i,γl 1 + γl 2 ) γu 2 (c i,γl 1 + γl 2 ) U 1 (c i,γl 1 + γl 2 ) γ U 2 (c i,γl 1 + γl 2 ) (1 τ) U 1 (c i,γl 1 + γl 2 ) = (1 τ) w =(1 τ) f 1 (2 l 1 l 2 ) = (1 τ) f 1 (2 l 1 l 2 ) = f 1 (2 l 1 l 2 ) which equals the planner s condition (6) when τ =.5. The intuition is that for every household of type 1 who is gardening there is another household of type 2 who enjoys household 1 garden as much as type 1, so every household should be gardening twice as much. Question 2: CARA utility (a) The consumer s problem is ½ L = max 1 {c 1,,c 2 } γ e γc 1 1 1 1+ρ Taking FOC γ e γc 2 + λ y 1 + y 2 c 1 c ¾ 2 (7) e γc 1 = λ 1 1+ρ e γc2 = λ 1 Combining the FOC s, we obtain the Euler equation e γ(c2 c1) = 1+ρ eγ(c2 c1) = 1+ρ (b) Taking natural logs, we obtain γ(c 2 c 1 )=ln() ln (1 + ρ) Now use the approximation given in the statement of the problem, i.e., ln(1 + x) ' x to arrive at γ(c 2 c 1 )=r ρ that is c 1 = c 2 + ρ r γ 2 (8)
To solve for the optimal level of consumption we use the intertemporal budget constraint y 1 + y 2 = c 1 + c 2 {z } Y W Now use the Euler equation to replace c 1 in the above equation Y W = c 2 + ρ r γ + c 2 After some algebraic manipulation, we obtain the following expressions c 2 = Y w ρ r (9) 2+r γ c 1 = Y w ρ r + ρ r (10) 2+r γ γ (c) To determine under which conditions c 1 is bigger than c 2 we can take a look at the equation above (although it is enough to look at the Euler equation to answer this question). When ρ = r we have a constant consumption profile or perfect consumption smoothing. When ρ<r,then c 1 >c 2. Question 3: Borrowing constraint (a) Initially, AB in Figure depicts the consumer s budget constraint. The introduction of the tax results in a kink in the budget constraint, since the interestrateatwhichtheconsumercanlendisnowsmallerthantheinterest rate at which the consumer borrows. The kink occurs at the endowment, E. C 2 A D Y 2 E F B Y 1 C 1 3
C 2 A G J D H Y 2 Y 1 C 1 (b) The top panel of Figure 1 shows the case of a consumer who was a borrower before the imposition of the tax. This consumer is unaffected by the introduction of the tax. The bottom panel of Figure shows the case of a consumer who was a lender before the imposition of the tax. Initially the consumer chooses point G, and then chooses point H after the imposition of the tax. There is a substitution effect that results in an increase in first-period consumption and a reduction in second-period consumption, and moves the consumer from point G to point J. Savings also falls from point G to point J. Theincomeeffect is the movement from point D to point B, and the income effect reduces both first-period and second-period consumption, and increases savings. On net, consumption must fall in period 2, but in period 1, consumption may rise or fall. Figure 1 shows the case in which first-period consumption increases, which is a case where the substitution effect dominates. Question 4: Competitive Equilibrium The competitive equilibrium for this economy is a set of quantities {c 1,c 2,h 1,h 2,n 1,n 2,k 1,k 2 }, prices {w 1,w 2,r 1,r 2 } such that: 1. Given {s, w 1,r 1,w 2,r 2 } the household solves max 1, 1 h 1 )+βu(c 2, 1 h 2 ) {a,c 1,c 2,h 1,h 2 } s.t. c 1 + a = w 1 h 1 +ā (1 + r) c 2 = w 2 h 2 + a (1 + r) 4
2. Given {w 1,r 1,w 2,r 2 } the firm solves each period max {n 1,n 2,k 1,k 2 } {zf (k t,n t ) w t n t r t k t } 3. The labor market clears in each period and the equilibrium wage is given by w t n t = h t 4. The asset market clears and the equilibrium interest rate is given by r t ā = k 1 a = k 2 i.e. the supply of capital is fixedinthefirst period, but elastic in the second. 5. The goods market clears y 1 = c 1 + a ā = c 1 + s y 2 = c 2 a 5