# Homework 2

```National University of Singapore
Department of Economics
EC2101 Microeconomic Analysis I
Semester 1 AY 2014/2015
Homework 2
Cost Minimization and Perfect Competition
Due date: 30 Oct 5 pm
Question 1 A firm’s production function is Q = min(2L,5K ) . The price of labor is
w > 0 and the price of capital is r > 0 .
a) Derive the input demand functions for labor and capital respectively.
b) Derive the long-run total cost function.
c) Redo part a) and b) for the production function Q = 2L + K . For simplicity, assume
w ≠ 2r .
Question 2 A firm’s production function is given by Q = (L +1)K . The price of
labor is 9 and the price of capital is 1.
a) Suppose in the short run, capital is fixed at K = 4 . What is the equation of the
firm’s short-run total cost curve?
b) At what output level is L = 1 a cost-minimizing choice of labor in the long run? To
produce this level of output, how much capital does the firm use in the long run?
c) In the long run, what is the cost-minimizing input choice when the firm produces 1
unit of output?
Question 3 In a perfectly competitive industry, every firm faces the short-run total
cost curve characterized by STC(Q) = Q 2 + 20Q +10 . Assume all fixed costs are sunk
in the short run. Currently there are 60 firms in the market. The market demand curve
is D(P) = 900 − 20P .
a) What is the short-run supply curve for each firm?
b) What is the short-run equilibrium price in this market?
c) At the equilibrium, what is the producer surplus for each firm? What is the profit of
each firm?
Question 4 Suppose the market for cotton is perfectly competitive. All firms are
identical and the long-run total cost of production for each firm is given by
! 1
# Q 2 + w, Q > 0
LTC(Q,w) = " 2
#
0,
Q=0
\$
where w is the wage paid to the general manager. Each firm requires one and only one
5000
general manager. The market demand curve for cotton is D(P) =
.
P
1
National University of Singapore
Department of Economics
EC2101 Microeconomic Analysis I
Semester 1 AY 2014/2015
a) What is the long-run equilibrium price of cotton? At the long-run equilibrium, how
much cotton does each individual firm produce? How many firms will be in the
market in the long-run equilibrium? (All of your answers will be in terms of w.)
w
. Since each
4
firm only hires one general manager, the market demand for general managers is
equal to the number of firms in the market. Assuming the market for cotton reaches
the long-run equilibrium, what is the equilibrium wage for general managers? (Hint:
at what wage is the supply of general managers equal to the demand of general
managers?) Using the equilibrium wage, determine the long-run equilibrium price of
cotton and the number of firms in the long-run equilibrium.
b) Suppose the market supply curve for general managers is S(w) =
c) Suppose the market demand for cotton increases to D(P) =
20000
. The market
P
w
. Assuming the market for cotton
4
reaches the long-run equilibrium, what is the new equilibrium wage for general
managers? Using the new equilibrium wage, determine the long-run equilibrium price
of cotton and the number of firms in the long-run equilibrium.
supply curve for general managers is still S(w) =
2
```