1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 7.10
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. 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.

3. 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

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

5. 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:

6. Ex 7.10(2): Question
Method
Use solution of part 1 to derive excess-demand functions
Find equilibrium by setting ED to zero

7. 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

8. Ex 7.10(2): Equilibrium price ratio
For equilibrium, set excess demands to 0:
From this we get:
So equilibrium price ratio is

9. 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

10. 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:

11. Ex 7.10(3): Question Method Use equilibrium price to find profits…
…and wage income in equilibrium

12. Ex 7.10(3): Equilibrium incomes
Profits in equilibrium:
Wages in equilibrium:
Profit/Wage ratio is 1
independent of A

13. 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