Question

1) Suppose a consumer's choice set consists of 2 goods. Can they both be superior goods at the same level of income? Why or why not?

2) Suppose a consumer's utility function is U=(x^.5)(y^.5) and the consumer has $100 to spend. Let Px=10 and Py=5. How much of x and y will be purchased?

Answer 1

(1)

A superior good is a good whose consumption increases more than proportionately than increase in income.

Let budget constraint be: M = X.Px + Y.Py, with usual notations.

When income increases, M will increase. If both X and Y are superior goods, quantity of X and Y will both incraese more than proportionately compared to increase in M, assuming constant Px and Py. This will lead to an imbalance, since right side of the equation will be higher than the left side of the equation, and the budget will be exceeded. Therefore both goods cannot be superior.

(2)

U = x^0.5y^0.5

Budget line: 100 = 10x + 5y, or 20 = 2x + y

Utility is maximized when MUx/MUy = Px/Py = $10/$5 = 2

MUx = $\dpi{80} \partial$ U/$\dpi{80} \partial$x = 0.5 x (y/x)^0.5

MUy = $\dpi{80} \partial$ U/$\dpi{80} \partial$y = 0.5 x (x/y)^0.5

MUx/MUy = y/x = 2

y = 2x

Substituting in budget line,

20 = 2x + 2x = 4x

x = 5

y = 2 x 5 = 10