Solution:
(a) A consumption bundle that consists of 4 units of garlic and 7 units of oil.
1
(b) AE. Oil is relatively more expensive.
(c) Point C
(d)
max
x,y
U(x, y) = 2 lnx+ 3 ln y s.t. 2x+ 3y = 100
∂U(x, y)/∂x
∂U(x, y)/∂y
=
2y
3x
=
2
3
2x+ 3y = 100
(e) Now garlic becomes relatively cheaper since the slope of the budget line is flatter.
Question 2. The following diagram shows the cost minimization problem of a producer.
L
K
Y = 300
Y = 200
Y = 100
(a) A point in this space has the coordinates (4, 7), what does it mean?
(b) Suppose the producer wants to produce 200 units of the good. At which point in this
space should he choose? (You need to label it on the diagram).
(c) Suppose the production function is Y (L,K) = 2 lnL + 3 lnK and the cost function is
c(L,K) = 2L + 3K. Formulate the problem that the producer is facing in part (b) and
write down the equations to solve for the optimal. (You don’t need to solve for it).
(d) Does the production function Y (L,K) = 2 lnL+ 3 lnK exhibit constant return to scale?
Solution:
(a) An input bundle that consists of 4 units of labour and 7 units of capital
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(b)
L
K
Y = 300
Y = 200
Y = 100
(c)
min
K,L
2L+ 3K s.t. 2 lnL+ 3 lnK = 200
∂Y/∂L
∂Y ∂K
=
2
3
2 lnL+ 3 lnK = 200
(d) No, it exhibits increasing return to scale instead.
Y (aL, aK) = 2 ln (aL) + 3 ln (aK)
= 2 ln a+ 2 lnL+ 3 ln a+ 3 lnK
= 5 ln a+ 2 lnL+ 3 lnK
= ln a5 + lnL2 + lnK3
= ln (a5L2K3)
aY (L,K) = a(2 lnL+ 3 lnK)
= a ln (L2K3)
= ln (L2K3)a
Y (aL, aK)− aY (L,K) = ln (a5L2K3)− ln (L2K3)a
= ln
[
a5L2K3
(L2K3)a
]
> 0
Question 3. Consider the following market demand and supply curve:
Supply: P = 50 + 2Q
Demand: P = 1400−Q
(a) Find the market equilibrium.
(b) Find the sizes of consumer surplus and producer surplus at the equilibrium.
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Solution:
(a) Set Qs(P
∗) = QD(P ∗). Solve for 12P
∗ − 25 = 1400 − P ∗. The market equilibrium is
P ∗ = 950, Q∗ = 450.
(b) CS = 0.5× (1400− 950)× 450 = 101250, PS = 0.5× (950− 50)× 450 = 202500.
Question 4. Consider the following Edgeworth box diagram.
(a) What are the total endowments of good x and good y respectively?
(b) What is the initial endowment of good x that agent 1 has? What is the initial endowment
of good y that agent 2 has?
(c) Suppose both agents 1 and 2 want to consume 8 units of good x. Is it feasible?
Solution:
(a) 14; 8.
(b) 10; 4
(c) No. The total supply of good x is only 14 units, which is less than the total demand
of 16 units.
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