ECON 4010: Advanced Microeconomic Theory
Final Exam answers.
Fall 2009
SECTION A
1. Not necessarily. Exception: corner solutions.
2. False. Indirect utility function is homogenous of degree zero. If prices and income
double, the budget constraint and utility remains the same.
3. False. For inferior goods it is the other way round.
4. False. As wealth increases, marginal utility of potential loss from the gamble falls
but so does the marginal utility of winning the gamble. So the net result is
indeterminate.
5. True if non-credible threat or sub-game-imperfect equilibrium is ruled out.
Example battle of sexes. If the wife moves first, boxing/boxing is not a sub-game
perfect equilibrium.
6. False. Marginal products for linearly homogenous functions are inversely related
to the relative amount of the factor used.
7. True. The Shepard’s lemma is used to derive the conditional factor demand
functions, which allows us to find the relationship between inputs and output.
Example: C = 2(wv)1/2q. Taking the partial derivatives wrt w & v gives the
conditional factor demands: L = (v/w)1/2q, K = (v/w)-1/2q. Thus, L/q = q/K or q =
(KL)1/2.
8. True. Follows from the envelope theorem (Hotelling lemma): the partial
derivative of the profit function with respect to product price gives the product
supply function.
9. True only for constant cost industry (firm’s cost curves are independent of the
number of firms).
10. True. Follows from the fact that MC = MR = P( 1 + 1/e ).
11. Not necessarily. Exception: first degree price discrimination.
12. True. P = MC under Bertrand duopoly when firm’s product and cost are identical
but P > MC under Cournot duopoly.
SECTION B:
1. a) 2,2 and 1,1
b) Let q (1-q) be the probability column plays left (right).
EU (row) for top = q (2) + (1-q) (-1) = 3q – 1
EU (row) for bottom = q (-1) + (1-q) (1) = 1 – 2q
Equating (i) and (ii): q* = 2/5
(i)
(ii)
Similraly, let p (1-p) be the probability row plays top (bottom)
EU (column) for left = p (2) + (1-p) (-1) = 3p – 1
(iii)
EU (column) for right = p (-1) + (1-p) (1) = 1- 2p
(iv)
Equating (iii) and (iv), p* = 2/5
c) Top, left
2. a) Profit Π = P ( L1/2 + K1/2 ) – wL – vK
FOC for maximizing Π wrt L and K:
∂ Π/∂ L = P ½L-1/2 – w = 0 or L = (p/(2w))2
∂ Π/∂ K = P ½K-1/2 – v = 0 or K = (p/(2v) 2
Substituting in the profit function and simplifying: Π = (P2/4) (1/w + 1/v)
The function is homogenous of degree 1. Multiplying p,w,v by t raises Π by t.
It is convex in p because ∂ 2Π/∂ P2 = 1/2 (1/w + 1/v) > 0.
b) The supply function by Hotelling lemma (envelop theorem):
∂ Π/∂ P = (1/w + 1/v) (P/2)
3. a) Elasticity of substitution: σ = ∂ ln ( K/L)/∂ ln (RTS)
ln RTS = ln {(∂ q/∂ L)/ (∂ q/∂ L)} = ln (K/L)
Thus: σ = 1
Elasticity of output wrt l: Є = ∂ ln q /∂ ln L
ln q = ¼ lnL + ¼ ln K or ∂ ln q = ¼ ∂ ln L
Thus: Є = ¼
b) LRTC = C = wL + vK
FOC for cost minimization gives: w/v = K/L or K = L(w/v)
Substituting in the cost equation C = 2wL
Substituting in the production fn: q = L1/4 (L(w/v))1/4 = L1/2(w/v)1/4
or L = (v/w)1/2 q2. Substituting back in the cost equation:
C = 2 w1/2 v1/2 q2
SRTC = Cs = vK + wL = v + wL
Short run production function: q = L1/4 or L = q4.
Substituting: Cs = v + w q4
c) if v = w = 1, C = 2q2 and Cs = 1 + q4
Clearly, C = Cs when q = 1. This is the output level for which K = 1 is the cost
minimizing quantity of capital.
4. a) Firms supply curve: P = MC = 2q or q = ½ P
Market supply curve: 100q = Q = 50P
b) Market equilibrium: equating quantity demanded and supplied, 200 – 50P =
50P or P* = 2 and Q* = 100
c) Supply curve with tax: Q = 50 (P – 1), where P is tax inclusive price.
Equilibrium with tax: 200 – 50P = 50 (P – 1) or P* = 2.5 and Q* = 75 ie
consumers pay 2.5 and producers receive 1.5. Deadweight loss = ½ (25 x .5) + ½
(25 x .5) = 12.5
5. a) In equilibrium MC = 4 = MR = 15 – q. So, q* = 11 and p* = 9.5
b) Third-degree price discriminating equilibrium requires: MR1 = MR2 = MC.
MR1 = 15 – q1 = 4. So, as before, q1* = 11 and P1* = 9.5
MR2 = 20 – 4q2 = 4. So, q2* = 4 and P2* = 12
c) Aggregating market 1 and 2: Q = q1 + q2 = 40 – (5/2) P or P = 16 – (2/5)Q.
Setting MR = 16 – (4/5)Q = MC = 4, P* = 10, Q = 15, q1* = 10 and q2* = 5.
6. a) P = MC = 0
b) Firm 1’s profit function: Π = ( 100 - q1 - q2 ) q1
FOC for profit maximization: ∂ Π/∂q1 = 100 - 2q1 - q2 = 0 or q1 = 50 – ½ q2
c) Following the procedure in b) Firm 2’s reaction/best response function is:
q2 = 50 – ½ q1. Solving the reaction functions, q1 = q2 = 33.33.
d) P = 0. Each firm’s dominant strategy is to cut price until profits reach zero.