Macroeconomic Theory and Analysis Suggested Solution for Midterm 1

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1 Macroeconomic Theory and Analysis Suggested Solution for Midterm February 25, 2007 Problem : Pareto Optimality The planner solves the following problem: u(c ) + u(c 2 ) + v(l ) + v(l 2 ) () {c,c 2,l,l 2 } s.t. Setup the Lagrangian as c + c 2 π ( l ) + π 2 ( l 2 ) = Y L = {c,c 2,l,l 2,λ} {u(c ) + u(c 2 ) + v(l ) + v(l 2 ) (2) λ(c + c 2 π ( l ) π 2 ( l 2 ))} with FOCs u c λ = 0 u c2 λ = 0 v l π λ = 0 v l2 π 2 λ = 0

2 and the slackness condition: λ(c + c 2 π ( l ) π 2 ( l 2 )) = 0 By combining the first order conditions we can easily show that: the two agents consume the same quantity of consumption good: if we combine the FOCs w.r.t. consumption we get u c2 = u c. Given the specification of the utility function, the above equality implies c 2 = c at the optimum agent consumes less leisure than agent 2 this results is also obtained by combining the FOCs w.r.t. leisure. At the optimum it must be the case that v l v l2 = π π 2 > since we have assumed π > π 2. Given the functional form of v(.), the above inequality implies l < l 2. The solution s properties depend on the fact that the utility function is separable in consumption and leisure and that the planner values agent utility and agent 2 utility in the same way (i.e. same Pareto weights on agent and agent 2). As a consequence, it is optimal for the planner to assign the same quantity of consumption good to the agents in the economy. Separability in consumption and leisure is also crucial for the decision of the planner to make the more productive agent (the agent with π ) work more than the less productive one (the agent with π 2 < π ). 2 Question 2: Competitive Equilibrium 2. State the problem of the household and the problem of the firm Let s start from the household. The choice variables are c, c 2, l, l 2 and a, while ā is given.the problem that the individual household solves is: subject to: c,c 2,l,l 2 U(c, c 2, l, l 2 ) = u(c, l ) + βu(c 2, l 2 ) (3) c + a w ( l ) + ( + r )ā + φ + d (4) 2

3 c 2 w 2 ( l 2 ) + ( + r 2 )a + φ 2 + d 2 (5) In the household s budget constraints we have to include dividends because we assumed that the firm s has a decreasing return to scale production function, as a consequence the Euler theorem will not apply and there will be profits in this economy The firm chooses (k t, n t ) to imize profits net of the profits tax: π t = ( τ)[f(k t, n t ) w t n t (r t + δ)k t ] for t =, Define a competitive equilibrium A competitive equilibrium for this economy is an allocation (c, c 2, l, l 2, a, k, k 2, n, n 2 ), a price system (w, w 2, r, r 2 ) and a government policy (τ, φ, φ 2 ) such that, given ā: Given the government policy, the price system and the initial endowment, the allocation (c, c 2, l, l 2, a) solves the household s problem ; Given the government policy and the price system, the allocation (k, k 2, n, n 2 ) solves the firm s problem capital markets clear: ā = k and a = k 2 and r t = f kt δ for t =, 2; labor markets clear: n t = ( l t ) and w t = f nt for t =, 2; goods markets clear: c + (k 2 ( δ)k ) = y and c 2 = ( δ)k 2 + y 2 ; government budget constraint is balanced: τπ t = φ t for t =, 2 d t = π t for t =, 2 3 Question 3: Two periods economy with borrowing constraint 3. The unconstrained problem The household solves the following problem: 3

4 c,c 2 log c + β log c 2 (6) s.t. c + c 2 + r y + y 2 + r L = {c,c 2,λ} {log c + β log c 2 λ(c + c 2 + r y y 2 )} (7) + r Since the functional form for the agent s utility is nice, we just need to consider the FOCs in order to solve for the optimal quantities: c β c 2 = = λ λ + r and the slackness condition: λ(c + c 2 + r y y 2 + r ) = 0 By looking at the first order conditions w.r.t. consumption today and consumption tomorrow we can recover that 2 = βc +r. Since the budget con- c straint holds with equality, we can substitute c 2 with βc +r and derive that the optimal consumption in period is: c = + β (y + y 2 + r ) Since in equilibrium c 2 +r = βc, the optimal consumption in period is: c + r = β + β (y + y 2 + r ) 4

5 3.2 The constrained problem If the household cannot borrow more than a fraction α of the discounted endowment next period, then it will consume less than the optimal quantity in period whenever c > y +. +r If we want to characterize the parameters region where the borrowing constraint binds we need to solve the above inequality in terms of α. c > y + + r < c + r y + r + r < < α < + β (y + y 2 + r ) y y 2 + β + r β + β y + β β + β y + r y 2 So, if α satisfies the above inequality than the borrowing constraint will bind, otherwise the constrained solution will coincide with the unconstrained solution. When the borrowing constraint binds, the agent will consume all his period endowment, y, plus the imum amount of consumption good that he can borrow, namely. In period 2 the agent will consume the leftover +r amount of period 2 endowment given by ( α)y 2 5