University of Pittsburgh Department of Economics Econ 1720: Advanced Macroeconomics Handout 3

Transcription

1 University of Pittsburgh Department of Economics Econ 1720: Advanced Macroeconomics Handout 3 This handout presents how we can use all the results obtained in handouts 1 and 2 in order to characterize our competitive equilibrium. We ll do this in two different routes: following the definition of a competitive equilibrium, and then using the social planner s equivalence. 1 Competitive equilibrium In class we gave the following definition: Definition 1.1. A competitive equilibrium is a set of household quantities (C, l, N s ), firm quantities (N d, Y, π), a wage (w), and fiscal policies (G, T ) such that: 1. Given the wage and fiscal policies, the representative consumer chooses C, l to solve and N s = h l. C,l U(C, l) subject to C = w(h l) + π T C 0 0 l h, 2. Given the wage, the representative firm chooses N d to solve N d zf (K, N d ) wn d subject to N d 0, where Y = zf (K, N d ) and π = zf (K, N d ) wn d. 3. All markets clear: N d = N s C + G = zf (K, N d ) 4. The government s budget constraint is satisfied: G = T. 1

2 2 Characterizing the competitive equilibrium To characterize the competitive equilibrium, note that from the consumer s problem we get: U l (C, l) wu C (C, l) = 0 (2.1) wn s + π T C = 0. (2.2) From the firm s problem we get zf N (K, N d ) = w, (2.3) and finally, we need the labor market clearing condition and the government budget constraint: N d = N s (2.4) G = T. (2.5) Now let s try to simplify (2.1) (2.5). From (2.4), let N d = N s = N, and substitute in (2.2), (2.3). Then we get: Combining (2.1) and (2.7) gives wn + π T C = 0 (2.6) zf N (K, N) = w. (2.7) U l (C, l) zf N (K, N)U C (C, l). (2.8) Substituting the definition of π from the firm s problem into (2.6), along with (2.5): C = wn + Y wn G = zf (K, N) G. (2.9) Hence, our competitive equilibrium can be uniquely characterized by equations (2.10) and (2.11) below: U l (C, l) = zf N (K, N)U C (C, l) (2.10) C = zf (K, N) G. (2.11) Note that (2.10) and (2.11) form a system of two equations in two variables (C and l); hence, we can solve them together and obtain the equilibrium values (C, l ). In addition, the equilibrium wage rate can be obtained from w = zf N (K, N), (2.12) and aggregate output is given by Y = zf (K, N). (2.13) 2

3 Example 2.1. Let U(C, l) = C 1/2 l 1/2, h = 14, Y = zk 1/2 N 1/2, z = 1, K = 1, 225, G = 0. We want to find the competitive equilibrium. Note first the production function is given by Y = (1)(1, 225) 1/2 N 1/2 = 35N 1/2. With this, we can calculate equations (2.10) and (2.11) as: C = 17.5N 1/2 l C = 35N 1/2 Considering N = h l = 14 l, we can equate them: 17.5N 1/2 l = 35N 1/2 l = 2N l = 2(14 l) l + 2l = 28 3l = 28 so that l = 28/3 = It follows that N = 14 l = 14 28/3 = 14/3 = 4.67, C = 35N 1/2 = 35(14/3) 1/2 = and w = 17.5N 1/2 = 17.5(14/3) 1/2 = 8.1. Since G = 0, from the income-expenditure identity it follows that Y = C, and thus Y = Exercise 2.2. Repeat Example 2.1 but now assume G = 10. Go as far as you can into obtaining the competitive equilibrium allocation (C, l, N, w, Y ). Exercise 2.3. How are the results from Example 2.1 different from the results obtained under partial equilibrium in handouts 1 and 2? Are (C, l, N, w ) different? Discuss. 3

4 3 The Pareto optimal equivalence In class we introduced the concept of a social planner and showed the equivalence between a competitive equilibrium allocation and a Pareto optimal allocation. Here we present the social planner s problem more formally. Recall that the social planner s objective is to imize the representative consumer s utility subject to technology constraints. This is: C,l U(C, l) (3.1) subject to C = zf (K, h l) G C 0 0 l h. There are two ways to solve problem (3.1). The first one requires setting up a Lagrangian, while the other is simply to brute-force our way across. We discuss both approaches below. 3.1 Solution with the Lagrangian To solve (3.1), we will use the Lagrangian method. First, note that since we are assuming Inada conditions (which rule out corner solutions), we can disregard the non-negativity contraints. Instead of forcing the planner to respect the resource constraint, C = zf (K, h l) G, we let the planner choose consumption and leisure freely but penalize the planner with λ > 0 when violating the constraint. The planner s payoff, net of the penalty is given by the following Lagrangian: First-order conditions are given by: L(C, l, λ) = U(C, l) λ[c + G zf (K, h l)] (3.2) L(C, l, λ) C L(C, l, λ) l L(C, l, λ) λ Combining (3.3) and (3.4) gives = U C (C, l) λ = 0 (3.3) = U l (C, l) λzf N (K, h l) = 0 (3.4) = C G + zf (K, h l) = 0. (3.5) U l (C, l) zf N (K, h l)u C (C, l) = 0. (3.6) Hence, equations (3.5) and (3.6) give the solution to the planner s problem. Note that these equations are the same as (2.10) and (2.11)! This is just another way of seeing that both problems are equivalent. 4

5 3.2 General solution We can still solve problem (3.1) without a Lagrangian, but it becomes a bit more complicated, notation-wise. If we substitute the technology constraint into the utility function, we get a one-variable optimization problem: l U[zF (K, h l) G, l]. We can obtain the first-order condition with the chain rule (note that we need to be careful in keeping track of the partial derivatives): U C [zf (K, h l) G, l][zf N (K, h l)]( 1) + U l [zf (K, h l) G, l](1) = 0 which simplifies to zf (K, h l)u C (C, l) = U l (C, l). (3.7) Hence the solution to (3.1) is given by equation (3.7) plus the technology constraint C = zf (K, h l) G. But these are the same as equations (3.5) and (3.6), which are in turn equal to (2.10) and (2.11), so we arrive at the same result! Example 3.1. Assume the same conditions as in Example 2.1 hold. We want to obtain the solution using the social planner s problem. Hence, we set up the Lagrangian: First-order conditions are given by: From the first two equations we get: Substituting in the third equation: L = C 1/2 l 1/2 λ[c 35(14 l) 1/2 ]. (1/2)C 1/2 l 1/2 λ = 0 (1/2)C 1/2 l 1/2 λ[35(1/2)(14 l) 1/2 ] = 0 C + 35(14 l) 1/2 = 0. C = 17.5l(14 l) 1/2 35(14 l) 1/2 = 17.5l(14 l) 1/2 2(14 l) 1/2 = l(14 l) 1/2 2(14 l) = l 28 = 3l and thus l = 28/3 = 9.33, the same as in Example 2.1. It is a routine exercise to check that C = Exercise 3.2. Repeat Example 3.1 using the methodology presented in Section 3.2. Show that the same results are obtained. 5

6 4 Tax distorted competitive equilibrium (TDCE) Let s extend the concept of competitive equilibrium with distortionary taxes. consider the example of labor income taxes. Here, we Definition 4.1. A tax distorted competitive equilibrium is household quantities (C, l, N s ), firm quantities (N d, Y, π), a wage (w), and fiscal policies (G, T, τ) such that: 1. Given the wage and fiscal policies, the representative consumer chooses C, l to solve here N s = h l. C,l U(C, l) subject to C = w(1 τ)(h l) + π T C 0 0 l h, 2. Given the wage, the representative firm chooses N d to solve N d zf (K, N d ) wn d subject to N d 0, where Y = zf (K, N d ) and π = zf (K, N d ) wn d. 3. All markets clear: N d = N s C + G = zf (K, N d ) 4. The government s budget constraint is satisfied: 4.1 Characterization G = T + τwn s. To characterize the TDCE, note that from the consumer s problem we get: From the firm s problem we get U l (C, l) w(1 τ)u C (C, l) = 0 (4.1) w(1 τ)n s + π T C = 0. (4.2) zf N (K, N d ) = w, (4.3) and finally, we need the labor market clearing condition and the government budget constraint: N d = N s (4.4) G = T + τwn s. (4.5) 6

7 As before, we can simplify to get: U l (C, l) = zf N (K, N)(1 τ)u C (C, l) (4.6) C = zf (K, N) G (4.7) Note that (4.6) and (4.7) form a system of two equations in two variables (C and l); hence, we can solve them together and obtain the equilibrium values (C, l ). In addition, the equilibrium wage rate can be obtained from w = zf N (K, N), (4.8) lumpsum taxes are given by and aggregate output is given by T = G τwn, (4.9) Y = zf (K, N). (4.10) Example 4.2. Let U(C, l) = C 1/2 l 1/2, h = 14, Y = zk 1/2 N 1/2, z = 1, K = 1, 225, G = 0, and τ = 1/5. We want to find the tax-distorted competitive equilibrium. Note first the production function is given by Y = (1)(1, 225) 1/2 N 1/2 = 35N 1/2. With this, we can calculate equations (4.6) and (4.7) as: C = 35 2 N 1/2 4 5 l = 14N 1/2 l C = 35N 1/2 we can equate them (and substitute N = 14 l): 14N 1/2 l = 35N 1/2 2l = 5N 2l = 5(14 l) 2l + 5l = 70 7l = 70 so that l = 10. It follows that N = 14 l = = 4, C = 35N 1/2 = 35(4) 1/2 = 70 and w = 17.5N 1/2 = 17.5(4) 1/2 = Since G = 0, from the income-expenditure identity it follows that Y = C, and thus Y = 70. From the government s budget constraint, we obtain the lumpsum tax, T = τwn = 7. Note that compared to the competitive equilibrium in Example 2.1, leisure (10 > 9.33) is higher, consumption and output is lower (70 < 75.61), the wage (8.75 > 8.1) is higher. Furthermore, the utility of the household in the TDCE is U(C, l) = C 1/2 l 1/2 = 70 1/2 10 1/2 = 26.46, which is lower than the utility of the household in the CE, U(C, l) = C 1/2 l 1/2 = [ 35(14 28/3) 1/2 ] 1/2 (28/3) 1/2 =