Question

13. (14pts) An exchange economy with two consumers (A and B) and two goods (1 and 2): Endowments wA= (2/3, 1/3) and w= (1/3, 2/3) Utility functions: u (xf,x4) = xfx4 and u(xf,x) = xfx 1 for normalization. Setting p1 Calculate the competitive equilibrium price, allocations, and utilities for each consumer 1) (2pts) Derive the contract curve and the core path (express x 2) as a function of xf). (2pts) 3) Suppose that we add the third consumer in this economy: wC- (2/3, 1/3), uc(xf,x£) = xfx%. Calculate the competitive equilibrium price, allocations, and utilities for each consumer. (3pts) 4) Compare 1) and 3) and explain intuitively why we have such changes in prices, allocations, and utilities. (2pts) V are fair? (2pts) 5) What are two properties for the fair allocations? The allocations in 3) 6) We learned in the class how to achieve the fair allocation. Could you suggest the way in this three consumers' economy? And show that your way is correct by the calculations of allocations and utilities on the equilibrium in your way. (3pts) 198#

Answer 1

1)

Given:

uX1y1 U2= X2y2. W1- ( 2/3, 1/3), W2 = (1/3,2/3)

Consumer 1's problem:

2p+1 max u X1y1 s.t. x1p+ yi 3

At equilibrium, MRS is equal to the price ratio:

Ут %3D р+ ут - Xір X1

Substituting the value in the budget constraint:

2p+1 X1p+X1p 2p/3+1/3x1 6p

Similarly, for consumer 2,

P+2 max u2 X2y2 s.t. x2p+ y2

py2 X2Pp X2

2p+xp- P2 3 p+2 X2 6p

Hence, both consumers' demands are:

(X1. yi) 2+1 2p+, (x2. 2) ( P+ 2, - ( 6p 6 6p 6

Total demand is equal to total endowment:

2p 1 p+2 1-3p +3-6pp 1 + 6p 6p

Equilibrium allocations are:

(1, y)- 2. (2. y) = )

Utilities derived at these allocations by individuals 1 and 2 are 1/4 and 1/4 respectively

2)

At the pareto optimal allocations, slope of both consumers' indifference curve are equal:

$MRS_1=MRS_2 \rightarrow \frac{y_1}{x_1}=\frac{y_2}{x_2}$

Also,

X1 + X2 1, y1+y2-1

Using this,

$\frac{y_1}{x_1}=\frac{1-y_1}{1-x_1} \rightarrow y_1-y_1x_1=x_1-x_1y_1 \rightarrow x_1=y_1$

Is the equation for the contract curve.

3)

Third consumer's problem:
$max \ x_3y_3 \ s.t. \ x_3p+y_3= \frac{2p+1}{3}$

At equilibrium,

2p+1 y3 py3 X3p X3 X3 6p

Individual 3's demand is given by:

$(x_3,y_3)=(\frac{2p+1}{6p},\frac{2p+1}{6})$

At equilibrium, total demand is equal to total endowment:

$\frac{2p+1}{6p}+\frac{p+2}{6p}+\frac{2p+1}{6p}=\frac{5}{3} \rightarrow 5p+4=10p \rightarrow p=\frac45$

Competitive equilibrium allocations are

$(x_1,y_1)=(\frac{13}{24},\frac{13}{30}), (x_2,y_2)=(\frac{14}{24},\frac{14}{30}), (x_3,y_3)=(\frac{13}{24},\frac{13}{30})$

Utilities at equilibrium are 169/720, 216/720, 169/720 for individuals 1, 2, and 3 respectively.

4)

These changes in utilities, prices, and allocations are because of the addition of the additional consumer. Due to the addition of this consumer, price of good x has fallen and equilibrium allocations for both consumers 1 and 2 have declined, along with their utilities. This is because the endowments have changed in a ratio lower to the increase in number of consumers.