Intermediate Microeconomics

Midterm 2: Review problem

Exercise 1

Consider a two-agent economy with two goods, *x *and *y*. Agents’ utility functions are

identical: *u**A*(*x, y*) = *u**B*(*x, y*) = min*{**x, *2*y**}*. Initial endowments are *e**A *= (1*, *1)*, e**B *=

(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 (*x**A**, y**A*), (*x**B**, y**B*) 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, *x**i *= 2*y**i *must always hold for agent i’s optimal consumption bundle, *i *= *A, B*.

Budget constraints for the two agents are:

1 *· *1 + *p **· *1 = 1*x**A *+ *py**A*

1 *· *3 + *p **· *1 = 1*x**B *+ *py**B*

Combine this with optimality conditions to get

1 + *p *= (2 + *p*)*y**A **⇒ **y**A *= 1 + *p*

2 + *p*

*, x**A *= 2*y**A *= 2(1 + *p*)

2 + *p*

3 + *p *= (2 + *p*)*y**B **⇒ **y**B *= 3 + *p*

2 + *p*

*, x**B *= 2*y**B *= 2(3 + *p*)

2 + *p*

Market clearing

Total endowment in the economy *e *= *e**A *+ *e**B *= (4*, *2).

Check that market clearing holds for allocations we found above:

*x**A *+ *x**B *= 2(1 + *p*)

2 + *p*

+

2(3 + *p*)

2 + *p*

=

8 + 4*p*

2 + *p*

= 4

*y**A *+ *y**B *= 1 + *p*

2 + *p*

+

3 + *p*

2 + *p*

=

4 + 2*p*

2 + *p*

= 2

Hence, set of competitive equilibria is

(*x**A**, y**A*) = (2(1 + *p*)

2 + *p*

*,*

1 + *p*

2 + *p*

)*, *(*x**B**, y**B*) = (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:

*x**A *= 2*y**A*. In this case, as long as no endowment is wasted, agent B’s bundles

are also located on the kinks of her indifference curves: *x**B *= 4 *− **x**A *= 4 *− *2*y**A *=

2(2*−**y**A*) = 2*y**B*. 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 (*x**A**, y**A*) = (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 *x**A *= 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 *e**A *= (1 + *a, *1 *−*

*a*)*, e**B *= (3 *− **a, *1 + *a*) before bilateral trade happens.

Rewriting agents’ budget constraints and keeping the optimality condition *x**i *=

2*y**i *we can obtain:

(*x**A**, y**A*) = (2(1 + *a *+ *p*(1 *− **a*))

2 + *p*

*,*

1 + *a *+ *p*(1 *− **a*)

2 + *p*

)

(*x**B**, y**B*) = (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)

*x**A *= 4

2 + *p*

*, y**A *= 2

2 + *p*

*x**A *= 1*/*2 = 4

2 + *p*

*⇒ **p *= 6

at this price level *y**A *= 1*/*4*, x**B *= 7*/*2*, y**B *= 7*/*4, so market clearing holds,

and (*x**A**, y**A*) = (1*/*2*, *1*/*4) constitutes competitive equilibrium for *a *= 1*, p *= 6

(equivalently, for endowment *e**A *= (2*, *0)*, e**B *= (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 *c**t *and income *I**t*. Lastly, we know that the consumer can borrow or save

at an interest rate of *δ *and his utility function is *U*(*c*1*,**c*2*,**c*3) = *u*(*c*1) + *δ**u*(*c*2),

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 (*c*1*,**c*2*,**c*3) must be less than

or equal to the discounted value of his income stream. In other words,

*pc*1 + *δ **pc*2 *≤ **I*1 + *δ**I*2

(2) Set up the Lagrangian:

*L *= *u*(*c*1)+ *δ**u*(*c*2)+ *λ*(*I*1 + *δ**I*2 *− **pc*1 *− **δ **pc*2)

Then the first order condition is:

*∂**L*

*∂**c*1 = *u**0*(*c*1)*− **λ **p *= 0 (1)

*∂**L*

*∂**c*2 = *δ**u**0*(*c*2)*− **λδ **p *= 0 (2)

*∂**L*

*∂ λ *= *I*1 + *δ**I*2 *− **pc*1 *− **δ **pc*2 = 0 (3)

By equations (1) and (2), we get *u**0*(*c*1) = *u**0*(*c*2) = *λ **p*. Therefore, *c*1 = *c*2. Plug

this into equation (3), we get *c*1 = *c*2 = *I*1+*δ**I*2

(1+*δ*)*p*.

Exercise 2

Consider an exchange economy where consumers A and B have endowments *e**A *=

(4*,*0) and *e**B *= (0*,*4), where *x **> *0 is a constant. The two consumers have CobbDouglas utility functions, *u**A*(*x*1 *A**,**x*2 *A*) = (*x*1 *A*)*α*(*x*2 *A*)1*−**α *and *u**B*(*x*1 *B**,**x*2 *B*) = (*x*1 *B*)*β*(*x**B *2)1*−**β*

(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 *MRS**A *= *MRS**B*. So we have 1*− **αα **x x**A *2 *A*

1

=

*β*

1*−**β*

*x**B*

2

*x**B*

1

=

*β*

1*−**β*

4*−**x**A*

2

4*−**x**A*

1

. Rearrange the equation and we get the equation of the contract

curve, which is

*x**A*

2 =

4(1*−**α*)*β**x*1 *A*

4*α*(1*−**β*)*−*(*β **−**α*)*x*1 *A*

(3) From the optimization formula for the cobb-douglas utility function, we get

(*x*1 *A**,**x*2 *A*) = (4*α**, *4(1*−**α*)*p*1

*p*2

) and (*x*1 *B**,**x*2 *B*) = (4*β **p*2

*p*1

*,*4(1*−**β*)). We then apply the market clearing condition to get 4*α *+4*β **p*2

*p*1

= 4. So *p*1

*p*2

=

*β*

1*−**α *. Plug the price ratio back

to the optimal bundles to get the following optimal allocation: (*x*1 *A**,**x*2 *A*) = (4*α**,*4*β*)

and (*x*1 *B**,**x*2 *B*) = (4(1*−**α*)*,*4(1*−**β*))

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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