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In: Economics

Each day, Luke must decide his leisure hours, L, and his consumption, C. His utility function...

Each day, Luke must decide his leisure hours, L, and his consumption, C. His utility function is given by the following equation ?(?, ?) = (? − 30)(? − 12). Luke pays 10% of his wage income as tax. Show all the steps, with definition of every new notation used in the steps.

a) Suppose that Luke’s pre-tax wage is $5/hour. Find Luke’s daily budget constraint equation and graph it. (5 pts.)

b) If Luke’s pre-tax wage is $5/hour, what is Luke’s optimal consumption, (? ∗ , ? ∗ )? (5 pts.)

c) Assume that the pre-tax wage is an unknown variable, ?. What is Luke’s leisure demand ?(?)? (5 pts.) d) What is Luke’s reservation wage, the lowest wage Luke is willing to work for? (3 pts.)

Solutions

Expert Solution

*Answer:

Given that,

U(L,C) = (C-30) (L-12)

*a)

Suppose that Luke's pre-tax wage is $5/hour. Find Luke's daily budget constraint equation and graph it.

Luke can work for 24 hours in a day, in that case he will earn $50 + $5(24) = $170, or he can leisure the whole day, in which case he consumes only $50.

Budget constraint

C = 50 + 5(24 - L)

*b)

If Luke's pre-tax wage is $5/hour, what is Luke's optimal consumption, (L*, C)?

When utility function U(L,C) = C*L. This implies that the marginal rate of substitution is C / L.

Therefore, Luke's marginal rate of substitution is:

MRS = (C-30) / (L-12).

Luke's optimal mix of consumption and leisure is found by setting MRS equal to wage ans solving for hours of leisure from given the budget constrain.

w = MRS

Putting the budget constrain equation in place of C

Thus Luke will choose 20 hours of leisure and consume 50 + 5*4 = $70 per day

*c)

Assume that the pre-tax wage is an unknown variable, w. What is Luke's leisure demand L(w)?

*d)

What is Luke's reservation wage, the lowest wage Luke is willing to work for?

The reservation wage is defined as the MRS when Luke is not working. When he is not working he leisures for 24 hours and consumes $50.

The reservation wage = $1.67

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Rahul Sunny answered 2 years ago
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