Question

A consumer has preferences represented by the utility function u(x, y) = x^(1/2)*y^(1/2). (This means that MUx=(1/2)x^(−1/2)*y^(1/2) and MUy =1/2x^(1/2)*y^(−1/2)

a. What is the marginal rate of substitution?

b. Suppose that the price of good x is 2, and the price of good y is 1. The consumer’s income is 20. What is the optimal quantity of x and y the consumer will choose?

c. Suppose the price of good x decreases to 1. The price of good y and the consumer’s income are unchanged. How much will the consumer increase his consumption of x after the price change?

d. Based on your calculations in part b and c, what is the own price elasticity of demand for good x?

Answer 1

a) Marginal rate of substitution = MUx/MUy

MRS = 0.5 x^{​​​​​​-0.5}y^0.5/ (0.5x^0.5y^-0.5)

MRS = y^{​​​​​​y0.5}​​​​​​ * y^{​​​​​​0.5}/(x^{​​​​​​0.5} * x^{​​​​​​0.5})

MRS = y/x

b) MUx/ MUy = Px/Py

(y/x) = (2/1)

y = 2x

I = Px * X + Py * Y

I = 20

20 = 2x + y

20 = y +y

2y = 20

y = 10

x = y/2 = (10/2) = 5

c) If Px = 1

MUx / MUy = (Px/Py)

y/x = 1

y = x

I = Px X + Py Y

20 = X +Y

20 = X + X

2X = 20

X = 10

y = 10

d) For good x

q = 5

p = 2

Q = 10

P = 1

Average of quantity = (5+10)/2 = 7.5

Average price = (2+1)/2 = 1.5

Change in price = 1-2 = -1

Change in quantity = 10-5 =5

Price elasticity of demand = % change in quantity/ % change in price

% change in quantity =( Change in quantity/ average of quantity)*100

% change in quantity = (5/7.5)*100 = 66.67%

% change in price = (change in price/ average of price)*100

% change in price = (-1/1.5)*100 = -66.67%

Price EOD = (66.67%/-66.67%) = -1