Download Example of computing a competitive equilibrium in an

Simon Fraser University
Department of Economics
Prof. Karaivanov
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 ) = b2 f where b is the amount of
b
bananas she eats and f is the amount of sh she eats. Annie has an endowment of wA
=7
f
bananas and wA = 3 kilos of sh. Ben has a utility function uB (b; f ) = 2f + 3 log(b) and
f
b
endowments of wB
= 0 bananas and wB
= 4 kilos of sh. Assume the price of bananas is 1.
(a) Write down the de nition of a competitive equilibrium for this economy.
(b) Solve for the competitive equilibrium sh price pf and the competitive equilibrium allocation of sh and bananas between Annie and Ben.
Solution:
(a) The de nition was given twice in class.
A competitive equilibrium is an allocation (four quantities) (fA ; fB ; bA ; bB ) and prices (pb ; pf )
(we will use pb = 1 later) such that:
(i) given the prices, consumers maximize their utility at the allocation (fA ; fB ; bA ; bB )
(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:
f
f
fA + fB = wA
+ wB
b
b
bA + bB = wA
+ wB
(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 b2 f
b;f
s.t. b + pf f = 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 2 ln(mA
f
pf f ) + ln f
Take the derivative and set to zero:
1
2pf
+ =0
(mA pf f ) f
1
cross-multiply and solve for the demand for sh by A:
mA
3pf
fA =
(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:
bA =
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. B 0 s consumer problem
is:
max 2f + 3 ln b
f;b
s.t. b + pf f = mB
plug in from the constraint into the utility function:
max
b
2(mB
pf
b)
+ 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).
2
3
+ =0
pf
b
3p
or, bB = minfmB ; 2f g.
3
From the budget constraint: fB = maxf mpfB
; 0g: Given the prices and the endowments
2
we also know that:
mB = 0 + 4pf
Notice that this is always larger than
3pf
2
so there always will be an interior optimum, i.e.
bB =
and
fB =
mB
pf
3pf
2
3
4pf
=
2
pf
3
= 2:5
2
Now, use the de nition for competitive equilibrium (CE) from part (a) and the demands you
obtained above to solve for the CE price of sh pf 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 bA +bB to equal supply of bananas in the economy:
b
b
wA
+ wB
: That is:
2(7 + 3pf ) 3pf
b
b
bA + bB =
+
= wA
+ wB
=7
3
2
from where we nd (solving the above equation for pf ) that the CE price is
pf =
2
3
f
f
You may check that the sh market also clears at this price, i.e., that fA + fB = wA
+ wB
(as
Walras' law claims). Finally (do not forget!) the CE allocation is obtained by plugging pf
into the demands b and f obtained above. You should obtain:
bA = 6; bB = 1; fA = 4:5 and fB = 2:5
3