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.

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