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Assignment 11 ECON 211.3 (03): Intermediate Microeconomics TI: 2020/21 part 2 of the final exam; due on December 17 th; submit on Blackboard Question 1: General Equilibrium So far, we have mostly been looking at one agent (a household or a firm) or one market at a time. We have taken the rest of the environment as given and haven't really asked where the behaviour of other agents or the prices in other markets come from. Now, we will see what happens if we model a whole (simple) economy at once and solve for everything simultaneously - everyone's optimal choices and all market prices. This means we are solving a general equilibrium model rather than a partial equilibrium one. You can imagine that this will be a little more complicated. Here's the setting: The economy consists of one (representative) household and two firms. The household owns the firms (and is entitled to the capital income and profits they generate), the household works (supplies labour) and earns labour income, and it enjoys consuming two types of goods, A and B. Firm A produces good A and firm B produces good B. Both firms as well as the household are price takers. The firms use different Cobb-Douglas production technologies with production factors capital k and labour n. For firm A, this is 2A = SA(KA,nA) = }king, and firm B's technology is TB = fB(KB, nB) = king. The firms sell their output in the goods market at prices pA and PB and hire as much labour as they need in the labour market at a factor price (wage) w. The amounts of capital the firms use have already been determined by prior investment. Firms don't directly pay a factor income to the owners of the capital they use; instead, they pay dividends to the owner of the firm (the household), who's also the owner of all capital operated by the firm. For firm i = A, B, these dividends d; are whatever is left of the firm's revenue after paying labour costs, di = P.I; - un; The household has preferences over the two goods A and B that are described by the utility function u(TA, TB) = CAT. The household supplies n = 24 units of labour in total. It can spend all its labour and capital income on the two goods, PATA + PBTB = m, where m = wn + dA + dB. The predetermined capital levels of the firms are KA = 27 and kg = 256. The labour market clears if nA + ng = n. We want to solve for a number of endogenous variables, including the prices of the two goods (PA. PB), the wage rate w, the quantities of the goods supplied (IA, TB), the labour input of each firm ( nA, ng) and the dividends paid to households ( dA, da). a) For each firm i = A, B, find the optimal amount of labour input n; as a function of output I;, the output price p; and the wage. b) Solve the household's optimization problem and find the demand a, of each good i = A, B as a function of the household's available resources m and the price of that good, pi. c) Eliminate w from the two equations you got in part a). Solve the resulting equation for PAZA d) Eliminate m from the two equations you got in part b). Solve the resulting equation for PALA