1. Suppose that an individual has a utility function U(c, h) = ln(c) + h (c repr

Question

1. Suppose that an individual has a utility function U(c, h) = ln(c) + h (c represents consumption of a necessity good such as shelter, and h represents health).

a. Find that optimal amount of c and h if the individual's income is 5, the price of c is1 and the price of h is1. (Hint use MRS= pc/ph ).

b. What happens to the consumption of c and h if the individual's income

increases to 10. Is health a normal or inferior good, or neither? What about c?

c. Suppose the price of c increases to 2, while income remains constant at 10. What happens to the consumption of c and h? Are c and h substitute goods,complementary goods, or neither?

Explanation / Answer

a). Let x be the optimal amount of c,

Since the price of both c and h is equal to 1 and total income is 5,

therefore optimal of h is equal to (5-x).

Uoptimal= ln(x) + (5-x),

For xoptimal, dU/dx=0 which gives, 1/x -1=0 or x=1.

So, coptimal=1 and hoptimal=4.

b).

Let x be the optimal amount of c,

Since the price of both c and h is equal to 1 and total income is 10,

therefore optimal of h is equal to (10-x).

Uoptimal= ln(x) + (10-x),

So, xoptimal, dU/dx=0 which gives, 1/x -1=0 or x=1.

coptimal=1 and hoptimal=9.

So an increase in the total income increases the consumption it shows it is Normal good.

c).

Let x be the optimal amount of c,

Since the price of both c is equal to 2 and h=1 and total income is 10.

Uoptimal= ln(x) + (10-2x),

So, xoptimal, dU/dx=0 which gives, x=0.5

coptimal=0.5 and hoptimal=9.

Since on increasing price of c, there is no change in optimal demand of h, therefore c and h are neither substitutes nor complementary.

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