MICROECONOMICS Principles and Analysis Frank Cowell Exercise 7.10 MICROECONOMICS Principles and Analysis Frank Cowell November 2006
Ex 7.10(1): Question purpose: construct and solve a GE model with production method: use solution of Ex 6.4 optimal net-outputs and profits of the firm(s). Then work out optimal consumption of capitalists and workers.
Ex 7.10(1): Background from Ex 6.4 Production function is [ q1]2 + [ q2]2 + Aq3 ≤ 0 where qi is net output of good i good 3 is labour A is a positive constant From solution to Ex 6.4 we have net output of good 1: q1 = ½Ap1 net output of good 2: q2 = ½Ap2 where prices are denoted in terms of good 3 Also from solution to Ex 6.4 maximised profits are [ p1]2 + [ p2]2 P = A 4
Ex 7.10(1): Capitalists’ consumption Utility of capitalists is given as For Cobb-Douglas preferences there are constant expenditure shares. Here expenditure shares are same for the two goods they spend equal amounts on the two goods Capitalists’ income consists just of profits So capitalists’ demand for good i is
Ex 7.10(1): Workers’ consumption Utility of workers is given as They maximise this given the budget constraint: Substituting for x1w, this is equivalent to maximising The FOC for a maximum is: This yields optimal labour supply: So, from the budget constraint:
Ex 7.10(2): Question Method Use solution of part 1 to derive excess-demand functions Find equilibrium by setting ED to zero
Ex 7.10(2): Excess demand Good 1 Good 2 demand by capitalists: x1c = ½P[p1]1 demand by workers: x1w = ½[p1]2 net output: q1 = ½Ap1 Good 2 demand by capitalists: x2c = ½P[p2]1 demand by workers: 0 net output: q2 = ½Ap2 There are no stock of goods 1 or 2 So excess demands are: good 1: x1c + x1w q1 = ½P[p1]1 +½[p1]2 ½Ap1 good 2: x2c q2 = ½P[p2]1 ½Ap2
Ex 7.10(2): Equilibrium price ratio For equilibrium, set excess demands to 0: From this we get: So equilibrium price ratio is
Ex 7.10(2): Profits and prices This only gives us the price ratio p1 / p2 To solve completely we need to express profits in terms of prices From Ex 6.4 profits are [ p1]2 + [ p2]2 P = A 4 We already know that p2 = p1/3 Substituting in, profits in terms of p1 are [ p1]2 + ⅓[ p1]2
Ex 7.10(2): Equilibrium price We know that the equilibrium condition for good 1 is Pp1 + 1 = A[ p1]3 But profits P are [ p1]2 + ⅓[ p1]2 P = A 4 So equilibrium condition for good 1 is: p1 A + 1 = A[ p1]3 Solving this we get:
Ex 7.10(3): Question Method Use equilibrium price to find profits… …and wage income in equilibrium
Ex 7.10(3): Equilibrium incomes Profits in equilibrium: Wages in equilibrium: Profit/Wage ratio is 1 independent of A
Ex 7.10: Points to remember In part 1 clearly identify what determines each agent’s income Use the answer to each part to build the solution for the next