1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 13.3
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 13.3(1): Question
purpose: a simple model of choice in the presence of non-convexity
method: carefully describe attainable set; find possible equilibria for consumers; compare them using Pareto criterion.

3. Ex 13.3(1): Production possibilities
x1 units of good 1 must cost F + mx1 of good 2
F is fixed cost
m is marginal cost
There are R2 units of good 2 available in the economy
So the maximum possible amount of good 1 is:

4. Ex 13.3(2): Question
method:
Simple sketches in (x1, x2)-space

5. Gas: production possibilities
Commodity space x2
Endowment of “other goods”
Fixed cost of gas production
Attainable set
Max possible amount of gas R2 l F
Constant MC of gas production
Isoquant1-4 m l x1
[R2F]/m

6. Ex 13.3(2): Preferences From the utility function We can check the MRS
goes to 0 as x1 goes to 
for a given x1 MRS is the same for all x2
MRS is high for high a and vice versa

7. Ex 13.3(2): Indifference curves
Low value of a
High value of a
U = x2 when x1 = 0
Isoquant1-4 x1

8. Ex 13.3(3): Question method:
Check points on each of the two families of indifference curves

9. Ex 13.3(3): max utility, high a
Attainable set
A typical IC
Reservation IC
IC where MRS = MRT = m R2 l
U(x*) > U(0, R2)
Isoquant1-4 x* l x1

10. Ex 13.3(3): max utility, low a U(0, R2) > U(x*)
Attainable set
A typical IC
Reservation IC
IC where MRS = MRT = m R2 l
U(0, R2) > U(x*)
Isoquant1-4 x* l
For formal comparisons of two utility levels see next part x1 x1

11. Ex 13.3(4): Question method:
Compute consumer’s utility-maximising equilibrium

12. Ex 13.3(4): interior solution
Marginal rate of substitution is:
Interior solution where MRS = m
So at interior solution we have:
This implies:

13. Ex 13.3(4): utility Maximised utility at interior solution:
a[1 - exp (x1*)]+x2* :
Substituting in the value of (x1*, x2*) utility is:
Utility at corner (0, R2) is just R2
So (x1*, x2*) represents a global maximum if
U(x1*, x2*) >U (0, R2)
which implies

14. Ex 13.3(4): implement interior solution
Attainable set
Indifference Curves
Interior optimum
A fee schedule R2 l F
Two part tariff:
Fixed charge F
price per unit m
Isoquant1-4 x* l x1

15. Ex 13.3(4): points to remember
Fixed cost implies nonconvexity of attainable set
Nonconvexity implies two possible solutions
which is relevant?
depends on preference parameter
Need to check utility levels to find PE
Two-part tariff can be used to implement
induces consumer to choose optimum
covers production costs