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Person 1 and 2 are the only two individuals in an exchange economy. Each person drives utility from the consumption of two goods, x and y. Their utility functions are: 1. Person 1: U1 = xfyl-u Person 2: U2-x'' y," where (Xi,y) is consumption bundles of individual i E (1,2). The initial endowment bundles are: Person 1: (xgf.yt) Person 2: (x2,y2) Drive the utility maximizing consumption bundles of two individuals as functions of the price ratio and the quantities of their initial endowments. Obtain the excess demand functions for both goods. Obtain the competitive equilibrium price ratio in this exchange economy. Let(x,yf)= (40,80), (趷3D= (80,40), α=8,β=2 a. b. c. d. In separate diagrams, show indifference curves (with consumption bundle) and the initial endowments of each person. In an Edgeworth box diagram show the equilibrium allocations of good x and y. a. b.

Answer 1

Solution:

Denoting alpha by a and beta by b, for ease of writing.

Given: Utilities for two persons

Person 1: U1 = x1^ay1^1-a ; Person 2: U2 = x2^by2^1-b

a) Let the prices for two goods be Px and Py.

Then, given the endowments of the two persons, the budget lines become:

Person 1: Px*x1 + Py*y1 = Px*x1⁰ + Py*y1⁰

Person 2: Px*x2 + Py*y2 = Px*x2⁰ + Py*y2⁰

Since, we are more interested in the price-ratio than the absolute prices of the two goods, normalizing the price of good y, that is, Py = 1, we can denote price-ratio as p where p = Px/Py (with Py = 1, p = Px, so on normalizing price of good y, price-ratio is simply price of good x).

So, budget lines become:

Person 1: p*x1 + y1 = p*x1⁰ + y1⁰

Person 2: p*x2 + y2 = p*x2⁰ + y2⁰

We know that utility maximizing condition, that is in order to obtain optimal consumption bundles, we must have marginal rate of substitution (MRS) of each consumer equal the price ratio: MRS = p

MRSxy = Marginal Utility of good x (MUx)/Marginal utility of good y (MUy)

MUx = $\partial U/\partial x$ and MUy = $\partial U/\partial y$

Then, for person 1:

MU1x = $\partial U1/\partial x1$ = a*(y1/x1)^1-a

MU1y = $\partial U1/\partial y1$ = (1-a)*(x1/y1)^a

So, MRS1x1y1 = MU1x/MU1y = (a/(1-a))*(y1/x1)

Then with optimal condition of MRS1x1y1 = p

(a/(1-a))*(y1/x1) = p

y1 = x(1)p(1-a)/a

Substituting this in the budget line of person 1, we get

p*x1 + p(1-a)x1/a = p*x1⁰ + y1⁰

p*x1/a = p*x1⁰ + y1⁰

x1 = a*(p*x1⁰ + y1⁰)/p

And, then y1 = (p(1-a)/a)*(a*(p*x1⁰ + y1⁰)/p) = (1-a)*(p*x1⁰ + y1⁰)

Similarly, for person 2:

MU2x = $\partial U2/\partial x2$ = b*(y2/x2)^1-b

MU2y = $\partial U2/\partial y2$ = (1-b)*(x2/y2)^b

So, MRS2x2y2 = MU2x/MU2y = (b/(1-b))*(y2/x2)

Then with optimal condition of MRS2x2y2 = p

(b/(1-b))*(y2/x2) = p

y2 = (p(1-b)/b)*x2

Substituting this in the budget line of person 2, we get

p*x2 + p(1-b)x2/b = p*x2⁰ + y2⁰

p*x2/b = p*x2⁰ + y2⁰

x2 = b*(p*x2⁰ + y2⁰)/p

And, then y2 = (p(1-b)/b)*(b*(p*x2⁰ + y2⁰)/p) = (1-b)*(p*x2⁰ + y2⁰)

So, consumption bundle of person 1: (x1, y1) = (a(px1⁰+y1⁰)/p, (1-a)(px1⁰+y1⁰))

And consumption bundle of person 2: (x2, y2) = (b(px2⁰+y2⁰)/p, (1-b)(px2⁰+y2⁰))

Notice that consumption bundles for both persons are indeed functions of price-ratio, p and their respective initial endowments.

b) Excess demand functions for both goods:

Excess demand = total demand - total availability/supply

For good x:

Total demand for good x = x1 + x2

Total demand for good x = a(px1⁰+y1⁰)/p + b(px2⁰+y2⁰)/p

Total availability of good x = x1⁰ + x2⁰

Excess demand for good x = [a(px1⁰+y1⁰)/p + b(px2⁰+y2⁰)/p] - [x1⁰ + x2⁰]

Excess demand for good x = (a*y1⁰ + b*y2⁰)/p + (a-1)x1⁰ + (b-1)x2⁰ ... (1)

For good y:

Total demand for good y = y1 + y2

Total demand for good y = (1-a)(px1⁰+y1⁰) + (1-b)(px2⁰+y2⁰)

Total availability of good y = y1⁰ + y2⁰

Excess demand for good y = [(1-a)(px1⁰+y1⁰) + (1-b)(px2⁰+y2⁰)] - [y1⁰ + y2⁰]

Excess demand for good y = p((1-a)*x1⁰ + (1-b)*x2⁰) - ay1⁰ - by2⁰ ... (2)

c) Finding competitive equilibrium price ratio, say denoted by p'

For a competitive equilibrium, we require that both persons maximize their utility and feasibility condition holds. Feasibility condition means that for a good, demand equals supply, or in other words excess demand = 0.

In part (b), we have done most of the required work: we found the excess demand functions using the optimal (that is utility maximizing) consumption bundles.

Then setting any one of the two equations, (1) and (2), equal to 0, to get the competitive equilibrium price:

From (2) and by equating it to 0, we get

p'((1-a)*x1⁰ + (1-b)*x2⁰) - ay1⁰ - by2⁰ = 0

p' = (ay1⁰ + by2⁰)/((1-a)x1⁰ + (1-b)x2⁰)

d) With (x1⁰, y1⁰) = (40, 80) and (x2⁰, y2⁰) = (80, 40)

And utility functions as:

Person 1: U1 = x1^7/8y1^1/8 ; Person 2: U2 = x2^1/2y2^1/2

Competitive equilibrium price using part (c):

p' = (ay1⁰ + by2⁰)/((1-a)x1⁰ + (1-b)x2⁰)

p' = ((7/8)80 + (1/2)(40))/((1/8)(40) + (1/2)(80))

p' = (70 + 20)/(5 + 40)

p' = 90/45 = 2

Equilibrium allocation using part (a),

x1 = (7/8)(2*40 + 80)/2 = 70

y1 = (1/8)(2*40 + 80) = 20

x2 = (1/2)(2*80 + 40)/2 = 50

y2 = (1/2)(2*80 + 40) = 100

For person 1: (x1, y1) = (70, 20)

For person 2: (x2, y2) = (50, 100)