ECON 311
Prof. Hamilton
Fall Quarter, 2010
NAME:__________________________
FINAL EXAM
200 points
1. (30 points). A consumer derives utility u ( X , Y ) from consuming two goods (X and Y) whose prices are PX
and PY. She has a fixed income of $I. Draw a 2-panel diagram that shows demands for each good, X* and
Y*, at an initial level of prices and income in the upper panel and a point on the demand function for X* at
the initial price level in the lower panel. Suppose the price of good X increases from PX0 to PX1 > PX0.
Carefully label the income effect and substitution effect of the price change for good X in the upper panel of
your diagram. In the lower panel of your diagram, draw the ordinary demand curve and the compensated
demand curve associated with each indifference curve. Carefully label the regions in the lower panel of your
diagram that represent compensating variation, equivalent variation, and the change in consumer surplus.
Y
U0
I = Px1X + PyY
U1
IE
X1*
I = Px0X + PyY
SE
X1c X0*= X0c
X
$/X
Px1
C
A
Px
B
0
X*(Px, Py*, I*)
Xc(Px, Py*, U0*)
X (Px, Py*, U1*)
c
X1*
X1c
X0*= X0c
Compensating variation (CV) is area A + B + C. Equivalent variation is area A. The change in consumer
surplus is area A + B
2. (30 points) The production function for a firm that produces 17" flat panel displays in the home computer
market is given by q = f ( L, K ) = L0.5 K 0.5 , where L is labor and K is capital. The firm sells flat panel
displays for $p per unit in a competitive market, and pays a wage rate of w for labor and a rental rate of r =
$20 for capital. The firm has K = 4 units of capital and cannot increase capital in the short run.
a. Calculate the firm's supply function for flat panel displays. Graph. Show the effect of an increase in the
wage rate (w) on supply.
With K = 4, q = 2L0.5 => q2 = 4L
=>
L(q) = q2/4
cost function: c(q) = wL(q) + r(4)
=>
c(q) = wq2/4 + 80
π = pq – c(q) = pq –wq2/4 – 80
qs = 2p/w
FOC: p - wq/2 = 0 =>
An increase in the wage rate shifts supply leftward; less quantity is supplied at any price (p)
$/q
qs(w1) = 2p/w1
qs(w0) = 2p/w0
q
b. Calculate the firm's demand for labor. Graph. Show the effect of an increase in the price of flat panel
displays (p) on labor demand.
From part (a): Ld = (qs)2/4 = (2p/w)2/4 = 4p2/(4w2) =>
Ld = p2/w2
An increase in the output price shifts labor demand rightward; more labor is demanded at any wage (w
$/L
w*
Ld = p12/w2
Ld = p02/w2
L
3. (30 points) The production function for a firm that produces 17" flat panel displays in the home computer
market is given by f ( L, K ) = L0.5 K 0.5 , where L is labor and K is capital. The firm sells flat panel displays for
$p per unit in a competitive market, and pays a wage rate of w = $20 for labor and a rental rate of r = $20 for
capital. Both L and K can be varied by the firm in making flat panel displays.
a. Draw a diagram that shows the cost-minimizing input mix for the firm (L*, K*). Describe the condition that
must be met at a cost-minimizing input mix and provide intuition on why this condition must hold.
At a cost-minimizing input mix, the marginal product per dollar must be equated across all factors of
production: MPK / r = MPL / w
K
C/r
K*
q = q0
L*
C/w
b. Does this production function exhibit increasing, decreasing, or constant returns to scale?
Suppose we started with K=1 and L=1 and then doubled the use of each factor. Would output double?
With K=1, L=1: Q = (1)(1) = 1.
With K = L = 2: Q = (21/2)(21/2) = 41/2 = 2
Output exactly doubles in response to a doubling of both factors, so the production function is characterized by
constant returns to scale
c. Derive the firm's total cost function.
Firm’s Problem (FP):
FOC: (1)
(2)
(3)
min. C = wL + rK
s.t.
0.5 0.5
= 20L + 20K + λ[Q - L K ]
0
d /dL = 20 - λ0.5L-0.5K0.5 =
-0.5 0.5
d /dK = 20 - λ0.5K L
=
0
0.5 0.5
d /dλ =
Q-L K
=
0
Q = L0.5 K 0.5
⇒
L=K
From (1), (2)
K/L = 1
which means the firm should employ equal amounts of labor and capital at the cost-minimizing input mix.
Substituting L =K into the production function, we find
Q = (K)0.5K0.5 = K ⇒ Q = K => K* = Q
L* = K*
⇒
L* = Q
The cost function is: c(Q) = 20L* + 20K* = 20Q + 20Q = 40Q => c(Q) = 40Q
4. (40 points) Suppose short-run total cost for a firm that produces textbooks is c(q) = 100 + 5q + q2.
a. What is the firm’s profit maximizing output level, q*, when p* = $45? Calculate the firm’s average cost and
profit level at q*.
Profit: π = pq – c(q) = 45q – 5q –q2 – 100
(or 45 = 5 + 2q) => 2q = 40 => q* = 20
=> FOC: p – c′ = 0
The firm’s average cost at q* is: AC = 100/q + 5 + q = 100/20 + 5 + 20 = $30
The firm’s equilibrium profit level at q* is: π* = 45(20) – 5(20) – (20)2 – 100 = $300
b. Draw a diagram that shows marginal cost, average cost, and profit when the price of textbooks is p* = $45.
$/q
MC
AC
45
25
profit
qmes = 10
q* = 20
Quantity
c. Calculate the minimum efficient scale of the firm and the long-run competitive equilibrium price. Show
these values on your diagram.
In the long run competitive equilibrium, P = MC and P = AC (zero profit). The point where MC = AC is the
minimum efficient scale of the firm, qMES. Since AC = c(q)/q = 5 + q + 100/q
qMES solves: MC = 5 + 2q = 5 + q + 100/q
The market price solves P=MC=AC, or
=> q = 100/q
PLR = 5 + 2(10) = $25
P
=> q2 = 100
=> qMES = 10
5. (40 points) Consumers in the cell phone market derive benefits from cell phone consumption (q) according to
the utility function u(q) = 100q - 0.5q2. The price of cell phones is p. The cost function for cell phones is given
by c(q) = 10q + 2q2.
a. Calculate the socially optimal quantity of cell phones, q*.
Social Planner’s Problem (SP):
max. W = u(q) - c(q) = 100q - 0.5q2 - 10q - q2
FOC:
dW /dq = 100 - q - 10 - 2q
=
=> 90 = 3q
=> q* = 30
0
b. Calculate the equilibrium price, pc, and quantity, qc, of cell phones in a perfectly competitive market.
Provide a graph that shows pc, qc, and the social optimum, q*.
Consumer’s Problem (CP): max. CS = 100q - 0.5q2 - pq
FOC: dCS / dq = 100 - q - p = 0 ⇒
Firm’s Problem (FP):
pd = 100 - q (inverse demand)
max. π(q) = pq - c(q) = pq - 10q - q2
FOC: dπ /dq = p - 10 - 2q = 0
=> ps = 10 + 2q (inverse supply)
Market equilibrium: p = pd = ps: 100 - q = 10 + 2q => qc =q* = 30
pc= 100 - qc => pc= $70
$/q
MC + t
p^=75
MC
pc=70
DWL
MB
q^ =25
qc = q* = 30
quantity (q)
c. Suppose the government imposes a unit tax of $15 per cell phone. Derive the after-tax price and quantity of
cell phones p^, q^. Show the after-tax competitive equilibrium on your graph.
Consumer’s Problem (CP): As above,
pd = 100 - q (inverse demand)
Firm’s Problem (FP): max. π = pq - c(q) - tq = pq - 10q - q2- 15q
FOC: dπ /dq = p - 10 - 2q - 15 = 0 => ps = 25 + 2q (inverse supply)
Market equilibrium: p = pd = ps: 100 - q = 25 + 2q => q^ = 25
p^ = 100 - q^ => pc= $75
5d. Derive the deadweight loss (or excess burden) of the tax in the cell phone market. Identity the region that
represents deadweight loss on your graph.
DWL = 1/2(30 - 25)(15) = 37.5
6. (30 points). Suppose Nokia and Samsung compete in a Cournot duopoly market for cell phones with the cost
functions c(qn) = 50qn for Nokia and c(qs) = 50qs for Samsung. Player’s view Nokia and Samsung cell
phones to be perfect substitutes and inverse demand for cell phones is P = 650 − Q, where Q = qn + qs is the
total quantity of cell phones produced by both firms.
a. Calculate the reaction functions, qnR and qsR. Show the equilibrium quantity of cell phones produced by each
firm as the intersection of two reaction functions on a graph.
Nokia’s Profit:
πn = [650– qn – qs]qn – 50qn
FOC: dπn /dqn = 650 – 2qn – qs – 50 = 0 =>
Reaction function:
R
qn = 300 – 1/2qs
2qn = 600 - qs
Similarly:
qsR = 300 – 1/2qn
qn
600
qsR
300
Cournot equilibrium
200
qn R
200
300
600
qs
b. Find the equilibrium quantity of cell phones produced in a Cournot duopoly. What is the market price of a
cell phone in a Cournot duopoly?
Cournot equilibrium is the intersection of the reaction functions.
qn = 300 – 1/2[300 - 1/2qn] = 150 + 1/4qn => 3/4qn = 150
Market price: P* = 650 – Q* = 650 – 400
=> qn* = 200 = qs* => Q* = 400
=> P* = $250
THE END. Have a great Winter Break!