Practice 1 - Solutions - Seminar PDF

Preview

Summary

Seminar...

Description

PRACTICE 1 - Solutions Dr. Gemici, EC3337

1. Joseph has preferences over leisure (L) and consumption goods (C) given by U (C, L) = CL. His non-labor income is V = £0. Joseph earns w = £10 per hour. Suppose that the total number of hours available to him is 168. Write down the optimization problem that Joseph will solve together with the relevant budget constraint. What is his optimal consumption and leisure? Answer: The optimization problem is: U (C, L) = CL

max C,L

C = 10(168 − L)

s.t. The optimality condition is

M UL M UC

= w i.e.

Given the utility function, we have becomes

∂U ∂L

∂U ∂L ∂U ∂C

= w. ∂U ∂C

= L so that the above optimality condition

C = 10 L

(1)

= C and

Combining this with the budget constraint given by C = 10(168 − L), we get 10(168 − L) = 10 L

(2)

Solving for L, we get: L = 84. This is the optimal amount of leisure. Since H + L = 168, the optimal number of work hours for Joseph is H = 84 and optimal consumption is C = 10 ∗ 84 = 840.

1

2. Marni has preferences over leisure (L) and consumption goods (C) given by U (C, L) = CL. Marni’s non-labor income is V = £0. Suppose that the total number of hours available to her is 168. Derive the equation for her labor supply function. On a graph, draw Marni’s labor supply curve. Answer: The optimization problem is: max C,L

s.t.

U (C, L) = CL C = w(168 − L)

The tangency condition is the usual

MUL MUC

= w i.e.

Given the utility function, we have becomes

∂U ∂L

= C and

∂U ∂L ∂U ∂C

∂U ∂C

= w. = L so that the above tangency condition

C = 10 L

(3)

C = w(168 − L)

(4)

And the budget constraint is Now combining the tangency condition (3) and the budget constraint (4), we get w(168 − L) =w (5) L Solving the above for L, we get L = 84 so that H = 84. Notice that the w cancels out on both sides so that in this case we have that the optimal leisure and optimal labor supply do not depend on wages at all. The labor supply curve is: H(w) = 84 This labor supply curve is just a vertical line. See attached figure. Note that this question is the same as the first question since the utility function is the same and the non-labor income is 0 in both. The only difference is that in question 1 you are given a specific number for the wage rate and in question 2 you are not. In question 2, you are asked to solve for the individual’s labor supply curve i.e. you are asked to solve for the equation that represents the individual’s optimal labor supply for ALL wage rates (this labor supply function is written as H(w)). For this particular question, it turns out that the optimal labor supply is 84 regardless of the wage rate (and hence is just a vertical line), but this will not always be the case. You can see in the next question a more common situation where the optimal labor supply and leisure are different for different wage rates.

2

3. Selma has preferences over leisure (L) and consumption goods (C) given by √ U (C, L) = CL. Selma’s non-labor income is V = £100. Suppose that the total number of hours available to her is 168. Write down the optimization problem that Selma will solve together with the relevant budget constraint. Derive the equation for her labor supply as a function of hourly wages denoted with w. Answer: The optimization problem is: max C,L

Given the utility function, we have becomes

MUL MUC

∂U ∂L

√

CL

C = w(168 − L) + 100

s.t. The tangency condition is the usual

U (C, L) =

=

= w i.e. √ 1 √C 2 L

and

∂U ∂L ∂U ∂C

= w.

∂U ∂C

=

√ 1 √L 2 C

so that the above tangency condition

C =w L

(6)

C = w(168 − L) + 100

(7)

And the budget constraint is Now combining the tangency condition (6) and the budget constraint (7), we get w(168 − L) + 100 =w L

(8)

+100 Solving for L, we get: L = 168w 2w Since H + L = 168, labor supply H is 168 − L. So the labor supply function H(w) will be

H(w) = 168 −

168w + 100 2w

You can see that we will have a different optimal labor supply H for each different wage rate w (unlike the case in question 2 where H(w) = 84 i.e. H was 84 regardless of the wage rate).

3

4. An individual with 100 hours a week to allocate between work and leisure has the utility function represented by U (C, L) = CL. Suppose this person currently has a weekly total income (i.e. labor income plus non-labor income) of £900 and chooses to work 40 hours per week. What is this person’s non-labor income per week (V) and hourly wage (w)? Answer: The optimization problem is: max

U (C, L) = CL

C,L

C = w(100 − L) + V

s.t. The optimality condition is the usual

M UL MUC

Given the utility function, we have becomes

∂U ∂L

= w i.e.

= C and

∂U ∂L ∂U ∂C

∂U ∂C

= w. = L so that the above tangency condition

C =w (9) L Since we are already given that H = 40, we know that L = 60 (since H + L = 100). Then we can rewrite the tangency condition (9) as: C = w ⇒ C = 60w 60

(10)

Also,the budget constraint is: C = 40w + V

(11)

Now combining the tangency condition (10) and the budget constraint (11), we get 60w = 40w + V ⇒ V = 20w

(12)

The question also states that the total income [i.e. labor income (40w) plus non-labor income V (which we just derived to be 20w)] is £900. Using this, we get: 40w + 20w = 900 ⇒ w = 15 which means that non-labor income V is: V = 20w = 20 ∗ 15 = 300

4...