Question

An individual has the utility function: u(c,h)= ln(c) - a/H
where C represents consumer spending. H is the amount spent on insurance disease. The parameter α indicates whether the individual is sick or not, such that α = 0 when the person is in good health and α = 1 when the person is sick. The probability of getting sick is equal to k. The individual has an income m, and has the budget constraint C + H = m.

The individual chooses C and H to maximize the expected utility.
a) Write this person's maximization problem, so that the objective function does not
depends only on C.
b) Derive the first order conditions.
c) Find the equilibrium choices of C and H.
d) How does H vary with income?

Answer 1

Solution:-

Given that

Utility function

C = dollars spent on consumption

H = amount spent on Health insurance.

Expected utility of the individual:

.......(1)

Budget constraint : C + H = m .........(2)

a)

Maximization problem:

subject to budget constraint

C +H = m

Put H = m - c into expected utility function:

objective function in terms of consumption C is represented by the above equation.

b)

FOC:

......(3)

(3) represents the FOC

c)

Equilibrium choices of C and H

FOC

and budget constraint m - c = H

put the value of H into budget constraints

Assume:  

and

  

H can not be negative

Therefore, neglect the value of

for

and

Equilibrium choices of C and H.

d)

To capture the effect of income on H, differentiable G with respect to m,

k > 0 and m > 0

with the increase in income, amount spent of health insurance will increase.