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Fall 2011 Macroeconomics 73253

Homework 1

The following questions are designed to get you to work through several Robinson Crusoe

examples. Please read Chapter 2 of my notes carefully, especially the worked example. Please make

sure you answer each part of each question!

Question 1: Suppose that Robinson Crusoe has preferences over consumption c and

leisure e given by the utility function U(c, e) = ln c + ln e. Assume that he has a total of

T = 12 hours a day which he can divide between leisure and labor supply. He has access

to the following technology for converting labor supply l into output y, y = 4l1/2.

Combining a)-c) on your problem set: How much does Robinson consume? work?

How much leisure does he enjoy?

Answer: Rewrite Robinsons problem as:

maxl[0,12]

ln(4l1/2) + ln(12 l)

Or, ignoring the constant ln 4 term:

maxl[0,12]

1

2ln(l) + ln(12 l)

The first order condition is:1

2l

1

12 l= 0

Or l = 4. Hence, e = 12 4 = 8 and c = 4(4)1/2 = 8.

Question 2: Let Robinson Crusoes preferences over consumption-labor supply pairsbe given by:

c1 exp{(1 )f(l)}

1 , (1)

where c 0 is consumption, l 0 is labor, > 0 and f is a twice differentiable, increasing

and convex function. (A convex function f is one such that f is concave). Remark: f is

not the production function, it is part of Robinsons preferences.

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Part a) Verify that these preferences: 1) are increasing in c and decreasing in l, 2) exhibit

diminishing marginal utility with respect to consumption. Give a sufficient condition to

ensure that they exhibit diminishing marginal utility with respect to labor.

Answer: Take first and second derivatives. Let:

U(c, l) =c1 exp{(1 )f(l)}

1 ,

So:

Uc(c, l) = c exp{(1 )f(l)} > 0

and

Ul(c, l) = c1 exp{(1 )f(l)}f(l) < 0.

Also,

Ucc(c, l) = c1 exp{(1 )f(l)} < 0

and

Ull (c, l) = c1 exp{(1 )f(l)}[(1 )f(l)2 f

(l)].

Ull is negative (diminishing returns to labor) if(1 )f(l)2 f

(l) < 0.

Now add in technology. Assume that output is given by:

y = A0 + A1l,

where A0 0 and A1 > 0. Consider an individual solving:

maxc1 exp{(1 )f(l)}

1 subject to : c A0 + A1l.

Part b) Assume that A0 = 0 and show that the individuals optimal choice of l is

independent ofA1.

Answer: Rewrite optimization as:

maxl0

(A1l)1 exp{(1 )f(l)}

1 ,

In fact you can see that A11 just acts as a multiplicative constant. It blows the utility up

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or down, but does not affect the tradeoff between consumption and labor. But we will

proceed and solve problem. The FOC is:

(A1)1l exp{(1 )f(l)} (A1)

1l1 exp{(1 )f(l)}f(l) = 0

Lots of things cancel to give:

l f(l) = 1.

We cannot go further without knowing f, but we can say that the optimal l is independent

of A1.

Part c) Now, assume that A0 > 0 and show that the individuals optimal choice of l

increases in A1 and decreases in A0. (Hint: Derive the first order condition. Assume that

the optimal l satisfying this condition is a differentiable function of A1. Totally differenti-

ate the first order condition with respect to A1. Hence, recover dl/dA1. What is its sign?

Repeat for A0.)

Answer: The FOC now is:

A1(A0 + A1l) exp{(1 )f(l)} (A0 + A1l)

1 exp{(1 )f(l)}f(l) = 0

We can simplify this a bit:

A1 (A0 + A1l)f(l) = 0

This equation holds at each (A0, A1), but its solution will be different at each parametervalue. Lets start with A1, to emphasize the dependence of the optimal l on A1, I will

write l(A1) and re-express the FOC as:

A1 (A0 + A1l(A1))f

(l(A1)) = 0

Let us totally differentiate with respect to A1. This gives:

1 l(A1)f(l(A1)) A1

dl

dA1

f(l(A1)) (A0 + A1l(A1))f

(l(A1))

dl

dA1

= 0

Hence,dl

dA1=

1 l(A1)f(l(A1))

A1f(l(A1)) + (A0 + A1l(A1))f (l(A1))

The denominator is positive, but what about the numerator? We can use the first order

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conditions. This tells us that:

A1 A1l(A1)f

(l(A1)) = A0f(l(A1))

So combining expressions:

dl

dA1=

A0A1

f(l(A1))

A1f(l(A1)) + (A0 + A1l(A1))f (l(A1))

> 0.

The analysis for the A0 case is similar.

Part d) What implications do your answers in b) and c) have for the relative size of

income and substitution effects.

Answer: Substitution effects dominate income effects.

Part e) A careful examination of hours and wage data suggests that while the hours of

men have declined as real wages (or the marginal product of labor) have risen, household

hours have not changed much. Comment on the validity of a model with preferences of

the form (1) for the analysis of household labor supply.

Answer: The model with A0 seems most relevant for modelling households.

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