Problems for the …rst seminar: Theory of the
…rm
ECON4230 Microeconomic Theory –Fall semester 2010.
Solutions to the problems will be presented at the seminars in week 36.
Please direct any question to Kjell Arne Brekke (Room ES1032, Tel 228 41169,
E-mail: k.a.brekke@econ.uio.no)
Problem 1 Varian 1.8
Problem 2 Varian 2.2
Problem 3
Consider the produktion function
f (x1 ; x2 ) = (min(x1 ; x2 ))a where 0 < a
1. For which values of a will this exhibit constant, increasing and
decreasing returns to scale?
Let (p; w1 ; w2 ) be the pro…t function
2. Is the pro…t function well de…ned in the cases a < 1, a = 1 and a > 1?
In the following we assume that a < 1.
3. Find pro…t maximizing demand and supply functions
Problem 4
Consider a production function of the form
f (x1 ; x2 ) = xa1 + xa2
with 0 < a.
1. For which values of a will this exhibit constant, increasing and
decreasing returns to scale?
In the following assume that a
1:
2. What is the TRS
1
3. What is the elasticity of substitution?
Now consider the CES production function
g(x1 ; x2 ) = (xa1 + xa2 )1=a
4. What is the return to scale for the production function g?
5. Show that for any isoquant of f is also an isoquant of g and vice versa
6. Without any further calculation, can you –based on the calculations
above –give TRS and the elasticity of substitution for the CES
production?
Problem 5
Consider a production function of the form
f (x1 ; x2 ) = x21 + x22
and suppose that p = 1 and factor prices are w = (1; 1). To maximize pro…t,
we may look at the …rst order condtions. The …rst order conditions are
@f
@x1
@f
p
@x2
p
= w1
= w2
Show that this yields the solution candiate:
x1 = x2 = 1=2
Is this the pro…t maximizing choice? (Check the second order condition.)
2