# Discussions on Intermediate Microeconomics

Intermediate Microeconomics
Midterm 2: Review problem
Exercise 1
Consider a two-agent economy with two goods, x and y. Agents’ utility functions are
identical: uA(x, y) = uB(x, y) = min{x, 2y}. Initial endowments are eA = (1, 1), eB =
(3, 1).
1. Normalizing price of good x to 1, find the set of all competitive equilibria in this
economy.
Solution:
We want a set of all allocations (xA, yA), (xB, yB) and price vectors (1, p) such
that both agents maximize their utility subject to a budget constraint and markets
clear.
Agents’ optimization
Given the preferences, xi = 2yi must always hold for agent i’s optimal consumption bundle, i = A, B.
Budget constraints for the two agents are:
1 · 1 + p · 1 = 1xA + pyA
1 · 3 + p · 1 = 1xB + pyB
Combine this with optimality conditions to get
1 + p = (2 + p)yA yA = 1 + p
2 + p
, xA = 2yA = 2(1 + p)
2 + p
3 + p = (2 + p)yB yB = 3 + p
2 + p
, xB = 2yB = 2(3 + p)
2 + p
Market clearing
Total endowment in the economy e = eA + eB = (4, 2).
Check that market clearing holds for allocations we found above:
xA + xB = 2(1 + p)
2 + p
+
2(3 + p)
2 + p
=
8 + 4p
2 + p
= 4
yA + yB = 1 + p
2 + p
+
3 + p
2 + p
=
4 + 2p
2 + p
= 2
Hence, set of competitive equilibria is
(xA, yA) = (2(1 + p)
2 + p
,
1 + p
2 + p
), (xB, yB) = (2(3 + p)
2 + p
,
3 + p
2 + p
), (1, p), where p is any
non-negative number.
1
2. What is the set of Pareto optimal allocations?
Solution:
Set of Pareto optimal allocations will coincide with location of kinks for agent A:
xA = 2yA. In this case, as long as no endowment is wasted, agent B’s bundles
are also located on the kinks of her indifference curves: xB = 4 xA = 4 2yA =
2(2yA) = 2yB. For bundles outiside of the kink line, we can always increase one
agent’s utility without decreasing the other’s by selecting the appropriate bundle
on the kink line.
3. Is the bundle (xA, yA) = (1/2, 1/4) Pareto optimal? Can it constitute a competitive equilibrium for some level of prices p given initial endowment?
Solution:
Yes, it is Pareto optimal (1/2 = 2 · 1/4). It cannon be a competitive equilibrium
for any positive price p, since it would require xA = 2(1 + p)
2 + p
=
1 2
, but it only
holds for p = 2/3.
4. Suppose we can force agent B to give agent A a units of good x in exchange for
a units of good y. Can the the bundle from the previous question constitute a
competitive equilibrium after such exchange happens for some a and some price
p?
Solution: Essentially we’re changing agents’ endowments to eA = (1 + a, 1
a), eB = (3 a, 1 + a) before bilateral trade happens.
Rewriting agents’ budget constraints and keeping the optimality condition xi =
2yi we can obtain:
(xA, yA) = (2(1 + a + p(1 a))
2 + p
,
1 + a + p(1 a)
2 + p
)
(xB, yB) = (2(3 a + p(1 + a))
2 + p
,
3 a + p(1 + a)
2 + p
)
Take a = 1 (we could have chosen different a, but we just want one that works)
xA = 4
2 + p
, yA = 2
2 + p
xA = 1/2 = 4
2 + p
p = 6
at this price level yA = 1/4, xB = 7/2, yB = 7/4, so market clearing holds,
and (xA, yA) = (1/2, 1/4) constitutes competitive equilibrium for a = 1, p = 6
(equivalently, for endowment eA = (2, 0), eB = (2, 2) and price p = 6).
2
Exercise 1
Consider a consumer who lives for two periods. he consumes a single consumption
good in each period. The price of the good is p for t = 1,2; his consumption is
denoted as ct and income It. Lastly, we know that the consumer can borrow or save
at an interest rate of δ and his utility function is U(c1,c2,c3) = u(c1) + δu(c2),
where u is a continuous, weakly increasing, and strictly concave function where
u(0) = 0.
(1) What is the consumer’s budget constraint?
(2) What is the optimal consumption stream for the consumer?
Solution
(1) The discounted value of the consumption stream (c1,c2,c3) must be less than
or equal to the discounted value of his income stream. In other words,
pc1 + δ pc2 I1 + δI2
(2) Set up the Lagrangian:
L = u(c1)+ δu(c2)+ λ(I1 + δI2 pc1 δ pc2)
Then the first order condition is:
L
c1 = u0(c1)λ p = 0 (1)
L
c2 = δu0(c2)λδ p = 0 (2)
L
∂ λ = I1 + δI2 pc1 δ pc2 = 0 (3)
By equations (1) and (2), we get u0(c1) = u0(c2) = λ p. Therefore, c1 = c2. Plug
this into equation (3), we get c1 = c2 = I1+δI2
(1+δ)p.
Exercise 2
Consider an exchange economy where consumers A and B have endowments eA =
(4,0) and eB = (0,4), where x > 0 is a constant. The two consumers have CobbDouglas utility functions, uA(x1 A,x2 A) = (x1 A)α(x2 A)1α and uB(x1 B,x2 B) = (x1 B)β(xB 2)1β
(1) Draw the Edgeworth box of this economy
(2) What is the equation for the contract curve?
(3) What is the competitive equilibrium?
Solution
1
(2) At Pareto efficient allocations, we have MRSA = MRSB. So we have 1αα x xA 2 A
1
=
β
1β
xB
2
xB
1
=
β
1β
4xA
2
4xA
1
. Rearrange the equation and we get the equation of the contract
curve, which is
xA
2 =
4(1α)βx1 A
4α(1β)(β α)x1 A
(3) From the optimization formula for the cobb-douglas utility function, we get
(x1 A,x2 A) = (4α, 4(1α)p1
p2
) and (x1 B,x2 B) = (4β p2
p1
,4(1β)). We then apply the market clearing condition to get 4α +4β p2
p1
= 4. So p1
p2
=
β
1α . Plug the price ratio back
to the optimal bundles to get the following optimal allocation: (x1 A,x2 A) = (4α,4β)
and (x1 B,x2 B) = (4(1α),4(1β))
2
Simon Fraser University Prof. Karaivanov
Department of Economics Econ 301
Example of computing a competitive equilibrium in an exchange
economy
Problem:
Suppose there are only two goods (bananas and sh) and 2 consumers (Annie and Ben) in
an exchange economy. Annie has a utility function uA(b; f) = b2f where b is the amount of
bananas she eats and f is the amount of sh she eats. Annie has an endowment of wA b = 7
bananas and wf
A = 3 kilos of sh. Ben has a utility function uB(b; f) = 2f + 3 log(b) and
endowments of wB b = 0 bananas and wB f = 4 kilos of sh. Assume the price of bananas is 1.
(a) Write down the denition of a competitive equilibrium for this economy.
(b) Solve for the competitive equilibrium sh price p f and the competitive equilibrium allocation of sh and bananas between Annie and Ben.
Solution:
(a) The denition was given twice in class.
A competitive equilibrium is an allocation (four quantities) (fA ; fB ; b A; b B) and prices (p b; p f)
(we will use p b = 1 later) such that:
(i) given the prices, consumers maximize their utility at the allocation (fA ; fB ; b A; b B)
(i.e., these quantities are their consumer demands given the prices and the endowments)
(ii) markets clear, i.e. demand equals supply for each good:
fA + fB = wA f + wB f
b
A + b B = wA b + wB b
(b) Set pb = 1. Then proceed in the usual way { solve each consumer’s problem of maximizing utility subject to her budget constraint. That will give us his demand for each good as
a function of her income and the prices. For person A we must solve:
max
b;f
b2f
s.t. b + pff = mA
Proceed as usual (remember ch. 5-6). Take a monotonic transformation of the utility (log) and
substitute b from the BC to get a problem of one variable:
max
f
2 ln(mA pff) + ln f
Take the derivative and set to zero:

2pf
(mA pff) +
1 f
= 0
1
cross-multiply and solve for the demand for sh by A:
fA = mA
3pf
(notice this is the usual formula for Cobb-Douglas demand that I used directly in class, however
I wanted the full derivation here!). From the budget constraint, nd A’s demand for bananas:
b
A =
2mA
3
Also, given the prices and the endowments we also know that:
mA = 7 + 3pf
Proceed the same way for person B who has quasi-linear preferences. B0s consumer problem
is:
max
f;b
2f + 3 ln b
s.t. b + pff = mB
plug in from the constraint into the utility function:
max
b
2(mB b)
pf
+ 3 ln b
Take the derivative and set to zero (be careful here because this is quasi-linear preferences, so
if income is low enough the consumer will spend his all income on b { a corner solution).

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2
pf
+
3 b
= 0
or, b B = minfmB; 3p 2f g.
From the budget constraint: fB = maxfmB
pf

32
; 0g: Given the prices and the endowments
we also know that:
mB = 0 + 4pf
Notice that this is always larger than 3p 2f so there always will be an interior optimum, i.e.
b
B =
3pf
2
and
fB = mB
pf

3 2
=
4pf
pf

3 2
= 2:5
Now, use the denition for competitive equilibrium (CE) from part (a) and the demands you
obtained above to solve for the CE price of sh p f and then plug this price in the demand
functions to obtain the CE allocation.
2
By Walras’ Law you can use only the market clearing condition for one of the markets (the
other will automatically clear at the same prices). For example, take the market for bananas.
Market clearing requires demand for bananas b A+b B to equal supply of bananas in the economy:
wb
A + wB b : That is:
b
A + b B = 2(7 + 3pf)
3
+
3pf
2
= wb
A + wB b = 7
from where we nd (solving the above equation for pf) that the CE price is
p f = 2
3
You may check that the sh market also clears at this price, i.e., that fA + fB = wA f + wB f (as
Walras’ law claims). Finally (do not forget!) the CE allocation is obtained by plugging p f
into the demands b and f obtained above. You should obtain:
b
A = 6; b B = 1; fA = 4:5 and fB = 2:5
3

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