Question

Consider a consumer whose utility function is

u(x, y) = x + y (perfect substitutes)

a. Assume the consumer has income $120 and initially faces the prices px = $1 and py = $2. How much x and y would they buy?

b. Next, suppose the price of x were to increase to $4. How much would they buy now?

  

c. Decompose the total effect of the price change on demand for x into the substitution effect and the income effect. That is, determine how much of the change is due to each of the component effects. (Hint: Recall from class the two properties that determine the location of “z”, the reference point for distinguishing the income and substitution effects.

d. if the case in which the consumer’s utility function is u(x, y) = min{x, y} (perfect complements)，how does it change for question a,b and c above.

Answer 1

a) Since two goods are perfectly substitute then MRS = 1

Here P_x < P_y , Therefore, the consumer will consume all x. and the demand for x will be

whereas demand for y = 0.

that is they will buy 120 unit of x and 0 unit of y.

b) If Price of x increases to $4, then price of Y will be less than that of X. Since two goods are perfectly substitute to each other and P_x > P_y , then they will consume only Y, the demand for Y is

and they consume nothing of X.

d) If two goods are perfect complements, then, x=y

put x = y, in the budget equation, M = x.p_x +yp_y

we get,

I.e,

Now for (a) P_x = 1 and P_y = 2 and M=120

therefore,

i.e, x=40 and since x = y, then y also = 40

For (b), p_x = 4 and p_y = 2

then , and x =y=20.