Handout: Competitive Equilibrium

1 Competitive equilibrium Handout: Competitive Equilibrium Definition 1. A competitive equilibrium is a set of endogenous variables (Ĉ, N s, N d, T, π, ŵ), such that given the exogenous variables (G, z, K, h): 1. Given {ŵ, π, T, h}, the representative consumer chooses (C, l ) to solve max C,l U (C, l) (1) subject to C = w(h l) + π T (2) C 0 (3) 0 l h, (4) The endogenous variables satisfy Ĉ = C (5) l = l (6) N s = h l (7) 2. Given {ŵ, z, K}, the representative firm chooses ( N d) to solve max N d zf (K, N d ) ŵn d (8) subject to N d 0, (9) The endogenous variables satisfy N d = π = zf (K, ) ŵ (10) (11) 3. The labor market clearing condition holds: N d = N s. (12) 4. The government s budget constraint is satisfied: T = G. (13) 1

First notice that we have two endogenous goods in this economy, the consumption good and labor. Remember that capital is exogenous. The market clearing condition stated above states only that the labor market clears. The reason for that is that Walras Law implies that if labor market clears, then the consumption good market must also clear. Definition 2. Walras Law asserts that in general equilibrium when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. To see that it is true, substituting the definition of profit into consumers budget constraint, we have Ĉ = ŵ N ( s + zf (K, N d ) ŵ N d) G. (14) Thus, if the labor market clears,i.e., N s = N d, and we obtain Ĉ + G = zf (K, N d ) (15) which is the consumption good market clearing condition. 2 Characterizing the competitive equilibrium Here our objective is to characterize the endogenous variables (Ĉ, N s, N d, T, π, ŵ) as functions of the exogenous variables (G, z, K, h). To characterize the competitive equilibrium, note that from the consumer s problem we get the following conditions U l (Ĉ, l) = wu C (Ĉ, l), (16) Ĉ = ŵ N s + π T, (17) which are the first order condition and the budget constraint respectively. From the firm s problem (assuming that the production function is concave) we get ŵ = zf N (K, N d ), (18) π = zf (K, N d ) ŵ N d (19) and finally, we need the market clearing condition, plus the government budget constraint: N d = N s (20) T = G. (21) 2

Now, substituting Equation (18) into Equation (16) and the fact that l = h N s we get U l (Ĉ, h N d ) = zf N (K, N d )U C (Ĉ, ĥ N d ) (22) Recall that the goods market clearing condition is Ĉ + G = zf (K, N d ) (23) Notice that Equation (22) and Equation (23) are two equations with only two endogenous variables, Ĉ and N d. Hence, solving this system of equations we get the equilibrium values for Ĉ, N d and N s. Then, the equilibrium level for ŵ and the profit π are given by Equation (18) and Equation (19). The lump-sum tax T is given by Equation (21). Here is the procedure to find the equilibrium allocation: 1. Write total output as a function of N d. Y = zf (K, N d ) (24) 2. Find the marginal product of labor. MP N = zf N (K, N d ) (25) 3. Write consumption C as a function of (N d, z, K, G). C = Y G = zf (K, N d ) G (26) 4. Write l as a function of N d. l = h N s = h N d (27) 5. Write the first order condition as Solve for N d. U l (zf (K, N d ) G, h N d ) U c (zf (K, N d ) G, h N d ) = zf N(K, N d ) (28) 3

6. Solve for other endogenous variables C = zf (K, N d ) G (29) N s = N d (30) w = zf N (K, N d ) (31) π = zf (K, N d ) zf N (K, N d )N d (32) T = G (33) 3 Example Let the utility function be U(C, l) = γ log C+(1 γ) log l, and the production function be zf (K, N) = zk 1 α N α. The exogenous variables are h, z, K, G. We want to find the competitive equilibrium (Ĉ, N s, N d, T, π, ŵ). Particularly, we assume that G = 0. Here is how to follow the procedure 1. Write total output as a function of N d. Y = zf ( K, N d) = zk 1 α (N d ) α (34) 2. Find the marginal product of labor. MP N = zf N (K, N d ) = αzk 1 α (N d ) α 1 (35) 3. Write consumption C as a function of (N d, z, K, G). C = Y G = zf (K, N d ) 0 = zk 1 α (N d ) α (36) 4. Write l as a function of N d. l = h N s = h N d (37) 5. The marginal rate of substitution is U l (C, l) (1 γ)c = U C (C, l) γl (38) Using step 3 and step 4, we have U l (C, l) (1 γ)c = U C (C, l) γl = (1 γ)zk1 α (N d ) α γ(h N d ) (39) 4

The marginal rate of substitution equals the marginal product of labor: Using Equation (39) and step 2, we have U l (C, l) U c (C, l) = zf N(K, N d ) (40) (1 γ)zk 1 α (N d ) α γ(h N d ) = αzk 1 α (N d ) α 1 (41) Solve for the labor demand N d : N d = αγ 1 γ + αγ h (42) 6. Solve for other endogenous variables ( ) C = zf (K, N d ) G = zk 1 α αγ α 1 γ + αγ h (43) N s = N d αγ = 1 γ + αγ h (44) ( ) w = zf N (K, N d ) = αzk 1 α αγ α 1 1 γ + αγ h (45) ( ) π = zf (K, N d ) zf N (K, N d )N d = (1 α)zk 1 α αγ α 1 γ + αγ h (46) T = G = 0 (47) 5

4 The Pareto optimal equivalence Definition 3. An allocation is Pareto optimal if there is no way to rearrange production or to reallocate goods so that someone is made better off without making someone else worse off. Since in our economy there is only the representative consumer, Pareto optimality requires simply that given the technology of production the utility of the representative agent be maximized. In class we introduced the concept of a social planner. The social planner s problem is to maximize consumer welfare given the technology and the resource constraints. Thus, the Pareto optimum is the allocation that a social planner would choose. Definition 4. A social planner s problem is to solve max C,l U (C, l) (48) subject to C = zf (K, h l) G C 0 0 l h. Note that in the social planner s problem, the social planner chooses consumption, leisure and labor supply (there is no labor demand since there is no firm) directly. There is no market, no tax system, no transfer. Also note that this is not consumers problem. The Lagrangian function for the social planner s problem is First-order conditions are L (C, l) = U(C, l) + λ[zf (K, h l) (C + G)] (49) L C = U C(C, l) λ = 0 (50) L l = U l(c, l) λzf N (K, h l) = 0 (51) L = zf (K, h l) (C + G) = 0 λ (52) These lead to U l (C, l) = zf N (K, h l)u C (C, l) (53) Hence, once again using the identity N = h l we arrive at U l (C, h N) = zf N (K, N)U C (C, h N) (54) 6

5 Comparative statics Let us now study how the model react to a change of the exogenous variables. We assume the utility function is U(C, l) = γ log C + (1 γ) log l, and the production function is zf (K, N) = zk 1 α N α. We have shown that the allocation, i.e., the consumption C, the total labor supplied l, are the same as in the competitive equilibrium and the social planner s problem. Therefore, we can either solve the competitive equilibrium or solve the social planner s problem. Supposing we decide to solve the social planer s problem, the solution needs to satisfy the following conditions: U l (C, h N) U c (C, h N) = zf N(K, N) (55) C + G = zf (K, N) (56) If we use the particular functional form we have assumed, we will obtain (1 γ)c γ(h N) = αzk1 α N α 1 (57) C + G = zk 1 α N α (58) We can eliminate C using the resource constraint to get (1 γ)(zk 1 α N α G) γ(h N) = αzk 1 α N α 1 (59) 5.1 Government spending We first consider the effects of a change of the government spending G. Intuitively, when government spending increases, if consumers provide the same labor as before, they have to cut their consumption spending. Naturally, consumers would like to provide more labor so that they do not have to cut their consumption too much. Therefore, we should expect an increase of labor supply and a decrease of consumption. To show the intuition more formally, we can rewrite the Equation (59) as (1 γ)(zk 1 α N α G) = γαhzk 1 α N α 1 γαk 1 α N α, (60) which is ((1 γ) + γα)zk 1 α N α γαhzk 1 α N α 1 = (1 γ)g (61) 7

Note that ((1 γ) + γα)zk 1 α N α is increasing in labor N, and γαhzk 1 α N α 1 is decreasing in N. Therefore, the left hand side is increasing in N. When there is an increase in the government spending G, to main the equality of Equation (61), there has to be an increase of N. Recall that C = zk 1 α N α G Equation (60) can be rewritten as (1 γ)c = γαhzk 1 α N α 1 γαk 1 α N α (62) Note that the right-hand side γαhzk 1 α N α 1 γαk 1 α N α is decreasing in N. We have just shown that after an increase of government spending, N has to increase. As a result, the consumption has to decrease. Now we turn to the competitive equilibrium. Since the lump-sum tax T = G, after an increase of the government spending, T increases. The marginal product of labor is decreasing in N and the wage rate w equals to the marginal product of labor. We know that the wage rate will decrease after G increases. However, the profit π will increases. 5.2 Technology Now imagine the government spending remains the same, but there is an increase of the technology level, z. Equation (61) can be rewritten as zk 1 α [ ((1 γ) + γα)n α γαhn α 1] = (1 γ)g (63) Note that ((1 γ) + γα)n α γαhn α 1 is increasing in N. As a result, when z increases, N has to decrease. Write Equation (61) differently, zk 1 α N α [ ((1 γ) + γα) γαhn 1] = (1 γ)g (64) When N decreases, ((1 γ) + γα) γαhn 1 decreases. Since the government spending does not change, we must have the total output zk 1 α N α increases, with it, the consumption increases. 8

If we think of the competitive equilibrium, it is easy to conclude that the lump-sum tax does not change. The wage rate w = αzk 1 α N α 1 increases for two reasons: one is that z increases, and the other is N decreases. For the profit π, we have π = zk 1 α N α wn (65) = zk 1 α N α αzk 1 α N α (66) = (1 α)zk 1 α N α (67) Since total output increases, the profit also increases. Here, there are two competing forces at work. When the technology is better, the marginal product of labor is higher, and the wage rate is higher. Consumers want to provide more labor. On the other hand, the profit is larger, wealthier consumers want to provide less labor. In our example, the second force dominates. 9

6 Distortionary income tax Definition 5. A competitive equilibrium is a set of endogenous variables (Ĉ, N s, N d, τ, π, ŵ), such that given the exogenous variables (G, z, K, h): 1. Given {ŵ, π, T, h}, the representative consumer chooses (C, l ) to solve max C,l U (C, l) (68) subject to C = (1 τ)w(h l) + π T (69) C 0 (70) 0 l h, (71) The endogenous variables satisfy Ĉ = C (72) l = l (73) N s = h l (74) 2. Given {ŵ, z, K}, the representative firm chooses ( N d) to solve max N d zf (K, N d ) ŵn d (75) subject to N d 0, (76) The endogenous variables satisfy N d = π = zf (K, ) ŵ (77) (78) 3. The labor market clearing condition holds: N d = N s. (79) 4. The government s budget constraint is satisfied: τŵ N s = G. (80) The first order conditions are 10

U l (C, h N) U c (C, h N) = (1 τ)zf N(K, N) (81) C + G = zf (K, N) (82) 11