Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consum

Question

Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consumer has $720 to spend, and the price of X, PX = 9, and the price of Y, PY = 9.

a) (4 points) How much X and Y should the consumer purchase in order to maximize her utility?

b) (4 points) How much total utility does the consumer receive?

c) (4 points) Now suppose PX decreases to 4. What is the new bundle of X and Y that the consumer will demand?

d) (6 points) How much money would the consumer need in order to have the same utility level after the price change as before the price change?

e) (6 points) Of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect?

Explanation / Answer

a.

U(x,y) = x*y   -----------------------------(1)

After differentiation of equation 1 w.r.t. x, we get MUx

MUx = dU/dx = Y

After differentiation of equation 1 w.r.t. Y, we get MUy

MUy = dU/dy = x

Also,

MUx/MUy = Px/Py

Y/x =9/9

y = x ------------------------------(2)

Now,

720 = y*Py + x*Px = 9Y + 9x

Y+x =80

Y = x                                                       (from equation 2)         

Thus,

2x = 80

X = 40 units

Y = 40 units

Thus, consumer should purchase 40 units of x and 40 units of y to maximize the utility.

b.

Total Utility = x*Y = 40*40

Total Utility = 1600

c.

If Px decrease to 4

MUx/MUy = Px/Py

Y/x =4/9

9Y = 4x    ------------------------ (3)

720 = 4*x + 9*y = 4x+4x = 8x

X = 90 units

Y = (720 - 360)/9 = 40 units

Thus, consumer will demand 90 units of x and 40 units of y.

d.

Similar utility is achieved when x = 40 an Y = 40

Funds required = 4*40 + 9*40 = 520

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