Question

Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X2Y UB(X,Y) = X*Y

Therefore: For consumer A: MUX = 2XY; MUY = X2 For consumer B: MUX = Y; MUY = X

The initial endowments are: A: X = 75; Y = 15 B: X = 75; Y = 5

a) (20 points) Suppose the price of Y, PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium.

b) (8 points) How much of each good does each consumer demand in equilibrium? Consumer A's Demand for X: Consumer A's Demand for Y Consumer B's demand for X Consumer B's demand for Y

c) (4 points) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

Answer 1

Solution:

a) Given the information, as optimal consumption occurs for any consumer where MUx/MUy = Px/Py

So, for consumer A, optimal condition is: 2XY/X² = Px/1

2Y/X = Px or Y = Px*X/2 ... (1)

Similarly for consumer B, optimal condition is: Y/X = Px/1

So, Y = Px*X ... (2)

Then, with given endowments for each consumer, we can find the income, and use it to construct each budget line:

For consumer A: Px*X + Py*Y = Px*75 + Py*15

Px*X + 1*Px*X/2 = Px*75 + 15

1.5*Px*X = Px*75 + 15

X = (Px*75 + 15)/(1.5*Px)

X = 50 + 10/Px

Similarly for consumer B: Px*X + Py*Y = Px*75 + Py*5

Px*X + 1*Px*X = Px*75 + 5

2*Px*X = Px*75 + 5

X = (Px*75 + 5)/2*Px

X = 37.5 + 2.5/Px

Finally, as total endowment of X in the economy = endowment with A + endowment with B = 75 + 75 = 150

Then, we must have: demand of X by A + demand of X by B = total endowment

(50 + 10/Px) + (37.5 + 2.5/Px) = 150

(50 + 37.5) + (10 + 2.5)/Px = 150

87.5 + 12.5/Px = 150

Px = 12.5/(150 - 87.5) = 12.5/62.5 = 0.2

So value of price of X that is Px which will lead to competitive equilibrium is $0.2 per unit of X.

b) Then using the above part, demand of X by:

Consumer A = 50 + 10/0.2 = 100 units

Consumer B = 37.5 + 2.5/0.2 = 50 units (or total endowment of X - consumer A's demand for X = 150 - 100 = 50)

And demand of Y by:

Consumer A = 0.2*100/2 = 10 units

Consumer B = 0.2*50 = 10 units (or 20 - 10 = 10)

c) Marginal rate of substitution = MUx/MUy

For consumer A, in competitive equilibrium, this is = 2*100*10/(100)² = 2000/10000 = 0.2

Note that this MRS for A = Px/Py = 0.2/1 = 0.2, indicating the optimality (the condition is satisfied). This would be MRS for consumer B as well.