For the following three-person two-commodity pure exchange economy, the price of

Question

For the following three-person two-commodity pure exchange economy, the price of good y is normalized to $1 and px is written as p. The table below gives the utility functions, endowments, and demands for goods x and y, where

mi

denotes the value of consumer i's endowment. Calculate the Walras equilibrium price of good x, p, and the Walras allocation,

Check that the Walras allocation is Pareto efficient.

ui

i

xi

yi

xaya

(4, 0)

ma/(2p)

ma/2

(xb)3yb

(0, 24)

3mb/(4p)

mb/4

xc(yc)2

(6, 0)

mc/(3p)

2mc/3

Pick the market for good y. Set the total demand for y given by

p +

equal to the total supply of good y of  unit(s). Solving, we obtain

p = $  .

Therefore, the Walras prices are

(px, py) =

.

Then at the Walras allocation, consumer a consumes the bundle

,

consumer b consumes

,

and consumer c consumes

.

From the given utility functions, the marginal rates of substitution for each consumer is

MRSa =

,

MRSb =

,

and

MRSc =

.

At the Walras allocation,

MRSa =  ,

MRSb =  ,

and

MRSc =  .

Therefore, the Walras allocation  ---Select--- is is not Pareto efficient.

Person i

ui

i

xi

yi

a

xaya

(4, 0)

ma/(2p)

ma/2

b

(xb)3yb

(0, 24)

3mb/(4p)

mb/4

c

xc(yc)2

(6, 0)

mc/(3p)

2mc/3

Explanation / Answer

Pick the market for good y.

Since price of good y is normalized to 1, ma = 2p, mb = 12, and mc = 12p.

Substituting in the demand of y for each consumer and adding, the total demand is 9p + 3 which has to equal the total supply of good y of 12.

Solving, we obtain pˆ = $1.

Therefore, the Walras prices are (pˆx, pˆy)=(1, 1).

Then ma = 2, mb = 12 and mc = 12.

Hence, the Walras allocation E = ((1, 1),(9, 3),(4, 8)).

To check for Pareto efficiency, check the marginal rates of substitution at the Walras allocation:

MRSa = yˆa/xˆa = 1/1 = 1,

MRSb = 3yˆb/xˆb = 9/9 = 1, and

MRSc = yˆc/(2xˆc) = 8/8 = 1.

Since they are the same, the Walras allocation is Pareto efficient.

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