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

Question

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

a. Find that optimal amount of c and h if the individuals income is 5, the
price of c is 1 and the price of h is 1.

b. What happens to the consumption of c and h if the individuals 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?

David · Accepted Answer

Solution to 1: 
U(c, h) =  ln (c) + h 
a. 
With pc = 1, ph =1 and income (m) = 5 
Budget set: c + h = 5  ……….. (1) 
Marginal utility of consumption (MU
C
) = (1/c) 
Marginal utility of consumption (MU
H
) = 1 
At optimum point, MU
C
/ MU
H
 = Pc/Ph i.e. 
(1/c) = 1/1, implies optimal consumption of necessity good (c*) = 1 …… (2) 
Substitute this in (1), we get optimal amount of h as: 
h* = 5-1 = 4 
b. 
With pc = 1, ph =1 and new income (m) = 10     
Budget set: c + h = 10  …………… (3) 
At optimum point, MU
C
/ MU
H
 = Pc/Ph i.e. 
(1/c) = 1/1, implies optimal consumption of necessity good (c*) = 1 …… (4) 
Substitute this in (3), we get new optimal amount of h as: 
h* = 10-1 = 9 
We note that with rise in income the consumption of h has increased, so h is a normal good.  But 
the increase in income has no effect on consumption good (c), so it can be taken as normal good. 
c. 
With price of c rising to 2, price of h =1 and income = 10, new budget line is given as: 
2c + h = 10 ……….. (5) 
At optimum point, MU
C
/ MU
H
 = Pc/Ph i.e. 
(1/c) = 2/1,

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

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