Advanced Microeconomics
Final Examination
Sunday, January 8, 2023.
9:00 am-11:00 am
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1. (25’) The following figure shows the production set of a technology. Please answer
the following questions and explain your answer.
y2
Y
y1
(a) Does the production set satisfy possibility of inaction (i.e., 0 ∈ Y )?
(b) Does the production set satisfy no free lunch (i.e., Y ∩ RL+ ⊆ {0} )?
(c) Does the production set satisfy irreversibility (i.e., 0 ̸= y ∈ Y ⇒ −y ∈
/ Y )?
(d) Does the production set satisfy DRS (i.e., y ∈ Y ⇒ αy ∈ Y, ∀α ∈ [0, 1])?
(e) Does the input requirement set V (y2 ) = {y1 : (−y1 , y2 ) ∈ Y } satisfy convexity?
2. (25’) Consider a production economy with two consumers and one firm. Consumption sets are given by X1 = X2 = R2+ . Utility functions are
u1 (x11 , x21 ) = ln(x11 ) + ln(x21 ) + 8
u2 (x12 , x22 ) = ex2 ·x2 − 6
1
2
The firm uses good 1 to produce good 2, and the production set is
}
{
√
Y = (−y 1 , y 2 ) : y 2 ≦ y 1 , y 1 ≧ 0, y 2 ≧ 0 .
Suppose the firm is fully owned by consumer 1. Endowments w1 = (w11 , w12 ) > 0,
w2 = (w21 , w22 ) > 0. It is clear that zero prices cannot be a CE price. So, let
p = (p1 , p2 ) = (1, p) > 0.
√
(a) Determine the returns to scale of the production function y 2 = y 1 .
Hint: ∀t > 1, CRS: f (tx) = tf (x); DRS: f (tx) < tf (x); IRS: f (tx) > tf (x).
(b) Solve the firm’s profit maximization problem.
(c) Solve consumer 1’s utility maximization problem.
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(d) Solve consumer 2’s utility maximization problem.
(e) Write down the equations that determine the CE price. (You are not required
to solve for the CE price)
3. (20’) Consider a pure exchange economy with two consumers (A and B) and two
goods (good 1 and good 2). Initial endowments are given by
wA = (5, 5), wB = (5, 15).
Utility functions are
UA (x1A , x2A ) = ln (x1A ) + x2A , UB (x1B , x2B ) = (x1B )3 (x2B )2 .
Suppose consumers always consume strictly positive quantities of each good.
(a) Draw the Edgeworth box with initial endowment w and indifference curves
passing through w.
(b) Find the equation of the interior Pareto efficient allocations.
(c) At the prices given by p1 = p2 = 1, does market clearing condition hold?
(d) Solve for the competitive equilibrium prices (p∗1 , p∗2 ).
4. (30’) Consider a pure exchange economy with two consumers (A and B) and two
goods (good 1 and good 2). Initial endowments are given by
wA = wB = (1, 1).
Utility functions are
UA (x1A , x2A ) = x1A + x2A , UB (x1B , x2B ) = max{x1B , x2B }.
(a) Draw the Edgeworth box with initial endowment w and indifference curves
passing through w.
(b) Find the contract curve in the Edgeworth box.
(c) Find the set of core allocations in the Edgeworth box.
(d) At the prices given by p = (p1 , p2 ) > 0, find the demand functions.
(e) Solve for the competitive equilibrium allocation and equilibrium prices (p∗1 , p∗2 ).
(f) Is the equilibrium allocation Pareto efficient? Justify your answer.
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