2. Basic Macroeconomic Model of General Equilibrium under Perfect Competition: Household faces decision about how much to consume and how much to work, it also has some capital that it rents out, max log(C) + xlog Lh) C,L,a, st. 0 = 10L}, + rKr, where C is consumption, L}, is labour supplied by household (so 1 Lb is leisure), Y is non-labour income, and m is the wage.1 There is a Firm with production function Y = L?K}_a, where Y is production, and L f is labour used in production (demanded), and Kf is captial used in production. Firms solve the prot maximization problem max Y wa rKf Y,Lf,Kf s.t. Y = L?K}'\" 1Since the Household does not like capital in an of itself it will always choose to rent out all the capital it possesses. note that prots equal revenues (Y, as price is normalized to one) minus costs of the inputs; labour de (wages paid to labour) and capital rKf (cost of renting the capital). General equilibrium will be given by market clearance, here that means the labour market clears L5 = Li. (a) Solve rms maximization problem to show that wage equals the marginal product of labour, ie. to = aL?_1K}_Q, and that the rental rate equals the marginal product of capital, ie. r = (l a)LjKf'a. (b) Solve households maximization problem to nd that people will choose to work Lg, = [1 X 3K al- (c) Show the following useful intermediate result: r / w = \"79%. (d) Imposing labour market clearance, L = L}, = L f, and capital market clearance, K = Kf = Kh, show that in general equilibrium L = aix' on r / w and L3,.) (Hint: combine previous results (e) Show that in general equilibrium the other economic outcomes in this economy are given by: K 2 K\3. (Optional, this question is slightly more advanced than. level of this course) Model of Equilibrium Effects Relating to Firm Size and Bribery Example From Lecture Notes: There are two rms, rm A and rm B. Firm A is more productive than Firm B. Both Firms produce a single good, namely y, which is sold for price p. Both Firms face the same Burocracy cost X , while Firm B is 'better connected' and so can avoid Burocracy costs by instead paying bribes which cost a fraction b of the Burocracy costs (Firm A is assumed to be unable to pay bribes due to being closer watched by NGOs). Both Firms aim to maximize prots. Total demand in the economy is given _ 0 if p > amm- D(p) _ { 3} otherwise (Normally in a general equilibrium model we would want demand to be endogenous, but for simplicity here we make it exogenous.) Firm A has cost function CA (3;), and Firm B has cost function of; (y).3 In a world in which no bribery was possible the maximization problem of Firm A is max\" PyA CAM) X 31A and the maximization problem of Firm B is minus P93 63(93) X '93 In a world in which bribery is possible the maximization problem of Firm A is max\" PyA CA (9A) XyA and the maximization problem of Firm B is "Miss Pys 63(9)?!) 5X93 We make the following assumptions about the cost functions: ' CHE?) 43(0) + bX o chm?) +25." 2 c'B((1 0037) | bX for some a E [0,1]. the existence of such an a: is ectuell im lied rst two \"sum tione 3" P P Market clearance requires that yA + 9'3 2 y. We assume that Social Welfare is the sum of the consumer surplus, the producer surplus, and the 'civil servant surplus'. The 'civil servant surplus' is the sum of Burocracy costs (their wages) and bribes. Let SW =Social Welfare, CS =Consumer Surplus, CSS =Civil Servant Surplus, and PS =Producer surplus; so SW 2 US + PS + 033. (a) For economy without bribery, show that 39,4 2 g and y3 = 0. (Hint: solve rms maximization problems assuming each produces more than zero to get a relationship between price and each of the rms marginal cost, compare results to rst of the assumptions about the cost functions.) sAssumed to be increasing and convex functions. Continuously differentiable. (b) For economy with bribery, show that yA = ay and yB = (1 - a)y. (Hint: as before, but now look at the first and second assumptions about the cost functions.) (c) Consumer Surplus: For economy without bribery, show that CS = (Pmar -(C'A(y) +X))y, while with bribery CS = (Pmar - (C'A(ay) + X))y. Which is larger? (Hint: from the demand function it follows that CS = (Pmar - P)y.) (d) Producer Surplus: For economy without bribery, show that PS = (CA(y) + X)y -CA(y) - Xy, while with bribery PS = [(C'A(ay) +X)ay-cA(ay) -Xay]+[(C's((1-Q)y) +bX)(1- a)y - CA((1 -a)y) - bX(1 -Q)y] (Hint: Producer Surplus is the sum of profits of Firm A and profits of Firm B) (e) Civil Servant Surplus: For economy without bribery, show that CSS = Xy, while with bribery CSS = Xay + bX(1 -a)y. (Hint: recall that Civil Servant Surplus is just sum of Burocracy costs and bribes paid) (f) Social Welfare: For economy without bribery, show that TS = Pmary -CA(y), while with bribery TS = Pmazy - CA(Qy) - CB((1 -Q)y) (g) Conclude that bribery reduces Social Welfare. Discuss who is better off and who is worse off due to existence of bribery (ie. What happens to surpluses of Consumers? Firm A? Firm B? Civil Servants?) (h) Discuss in a few sentences what the model suggests about general equilibrium effects of the kind of micro-level empirical results on bribery that we mentioned in the lecture slides, namely "The [micro-level/ evidence about the effect of bribery on economic growth is mixed. Some find it harmful while others believe it helps via a grease the wheels effect. This column argues that the ambiguity can be explained by divergent effects of the mean and dispersion of corruption. A high bribery-mean retards productivity growth of firms, but a high bribery-dispersion facilitates performance of weak firms." (Source)