Econ152 – Wage Theory and Policy
Fall 2007
UC Berkeley Department of Economics
Andrea Weber
Problem set #1 - Solutions
1.
In the lecture we discussed the "neoclassical model of labor-leisure choice".
(a) Give the corresponding utility function that we used, specify the variables that enter
the utility function, and describe what the utility function intends to express.
(b) The utility function can be represented by a family of Indifference Curves. We
discussed four properties of Indifference Curves, one of which was that they are convex
to the origin. Give explanations of the three other properties of Indifference Curves.
(c) Suppose that the Indifference curves for consumption and leisure are concave to the
origin instead of convex. In general, how many hours will a person allocate to leisure
activities?
[Hint: Drawing some concave ICs and a budget line should guide you to the possible
solution(s)]
(a) and (b) see Borjas (2007), pages 27-29.
(c) If the indifference curves between consumption and leisure are concave, a worker will
either work all available time or will not work at all. It all depends on the relative slopes
of the indifference curves and the budget line.
As drawn in this figure
Econ 152 – Problem set #1 – Solutions
point B is preferred to point A, and the worker chooses not to enter the labor market.
Alternatively, as drawn in the following figure
the worker chooses not to consume any leisure and work all available time. Point B is
again preferred to point A.
2
Econ 152 – Problem set #1 – Solutions
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2.
Kurt owns a small farm near a big city and must decide whether to work on his farm or
take a job in the city. His utility depends on his income per day, Y, and on the number of
hours allocated to leisure activities, L.
Daily income from farm work is YF=20 * hF -h²F, where hF is hours of work on the
farm. Daily income from the job in the city is YC=15 * hC, where hC is hours of work in
the city.
(a) If Kurt can work on the farm or in the city, but not both, which sector would he
choose?
(b) If Kurt can work on the farm and in the city, how would he allocate his time?
To calculate the budget lines associated with each of the opportunities, it is easiest to
work through a numerical calculation of what Kurt’s earnings would be like if he
allocated 1 hour, 2 hours, 3 hours, etc. to each of the sectors and worked in that sector
exclusively. The calculation leads to
hours of work
1
2
3
4
5
6
7
8
Total Earnings
farm
city
19
36
51
64
75
84
91
96
Marginal Earnings
farm
city
15
30
45
60
75
90
105
120
19
17
15
13
11
9
7
5
15
15
15
15
15
15
15
15
(a) The table suggests that the budget line associated with working exclusively in the
city is given by CE and on the farm is the parabola FE. As a result, if Kurt can
only work in either the city or on the farm, it depends on his indifference curves
which option he chooses. He is going to work in the sector that maximizes his
utility. If his indifference curves are like A, he is better off working on the farm,
while if his indifference curves are like B, he is better off in the city.
Alternatively, if Kurt’s optimal choice is to work more than 5 hours, he’s going to
work in the city. Otherwise, he’s going to work in the farm.
Econ 152 – Problem set #1 – Solutions
4
Dollars of
Consumption
C
B
F
A
E
T-5
T
Hours of Leisure
(b) If Kurt can allocate his time to both the city and the farm, he is then better off
allocating the first few hours of work to the farm sector. As the table indicates, the
first hour allocated to the farm sector generates $19 worth of income, the second
hour generates $17, the third hour generates $15, and so on. So according to that
table, Kurt is thus best off by allocating the first 2 hours to the farm sector and
working any remaining hours he wishes in the city where each additional hour of
work generates constant $15.
Formally, Kurt is going to work in the sector that offers the greatest value for each
marginal hour (or even marginal minutes) worked. The marginal earnings in the
city are given by MYC=15, and the marginal earnings in the farm are given by
MYF=20-2hF. We know that as long as MYF>MYC, Kurt is going to choose to
work in the farm, and if MYF<MYC, then Kurt is going to choose to work in the
city. Then equalizing the marginal earnings:
MYF=MYC
20-2hF=15, and solving for hF
hF=5/2=2.5.
So, Kurt will work the first 2.5 hours in the farm and the rest of his working time
he’s going to spend on the city.
Econ 152 – Problem set #1 – Solutions
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3.
Suppose two workers Ben and Paul who have different preferences for consumption and
leisure:
Paul’s utility function is given by UP(C,L)= (C-100)*L and
Ben’s utility function is given by UB(C,L)= C*(L-5) (typo in problem set sheet!)
(a) What are Ben’s and Paul’s marginal utilities of leisure, respectively, if C=200? What
are their marginal utilities of consumption at a level of L=10?
(b) Use the results from (a) to sketch the shape Ben’s and Paul’s indifference curves?
How would you characterize each workers “taste” for leisure?
(a) The marginal utility of leisure is given by the change in utility that results from a one
hour increase in leisure, holding consumption fixed
MUL=U(C, L+1)-U(C,L)
Thus Paul’s marginal utility of leisure for C=$200 is MUL=(C-100)=200-100=$100,
and Ben’s marginal utility of leisure for C=$200 is MUL=C=$200.
The marginal utility of consumption is given by the change in utility that results from a
one Dollar increase in consumption, holding leisure fixed
MUL=U(C+1, L)-U(C,L)
Thus Paul’s marginal utility of consumption for L=10 hours is MUC=L=10,
and Ben’s marginal utility of consumption for L=10 hours is MUC=L-5=5.
(c) Ben’s MRS evaluated at C=$200, and L=10 is thus -40, and Paul’s MRS at the
same point is -10. This implies that Ben’s indifference curves are steeper than
Paul’s, or Ben’s marginal rate of substitution is larger than Paul’s. Compared to
Paul, Ben requires a sizeable monetary bribe to give up one hour of consumption.
On the other hand, Paul would give up more leisure in order to get one Dollar
worth of consumption than Ben. We can also say that Ben has a lower taste for
work than Paul (i.e. Ben places more value on leisure than Paul).
Econ 152 – Problem set #1 – Solutions
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Dollars of
Consumption
Dollars of
Consumption
Hours of Leisure
Ben's Indifference Curves
Hours of Leisure
Paul's Indifference Curves
4.
Karin’s preferences for consumption and leisure can be expressed as:
U(C,L)=(C-100)*(L-84). There are 168 hours in the week available to be split between
leisure and work. Karin earns $10 per hour after taxes. She also receives a weekly
stipend of $240 regardless of how much she works.
(a) Graph Karin’s budget line.
(b) What are Karin’s marginal utilities of consumption and leisure, respectively? What is
Karin’s marginal rate of substitution if L=120 and if she is on the budget line?
(c) Find Karin’s optimal amount of consumption and leisure.
(d) Which is the lowest wage so that Karin would be willing to enter the labor market?
(e) Now suppose Karin’s stipend is raised from $240 to $500 per week. How does this
affect her labor supply decision? Compute the optimal solution and show it in a sketched
graph.
(f) Now assume the stipend is fixed at $240, but her wage drops from $10 to $8. Will
Karin work more or less hours in response to this wage cut? Which two effects determine
her decision and which is the dominant effect? Compute the optimal solution and show it
in a sketched graph.
(a) If Karin does not work, she leisures for 168 hours and consumes $240. If she does
not leisure at all, she consumes $240+$10*(168) = $1,920.
Econ 152 – Problem set #1 – Solutions
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Dollars of
Consumption
$1,920
$240
168
Hours of Leisure
(b) If Karin leisures for 120 hours, she works for 48 hours and consumes
$240+$10*48 = $720. Thus her MRS when doing this is:
MRS = −
MU L
C − 100
720 − 100
=−
=−
= −$17.22
MU C
L − 84
120 − 84
(c) The optimal mix of consumption and leisure is found by setting the MRS equal to
the wage and solving for hours of leisure given the budget line: C=240+10(168L).
− w = MRS
C − 100
10 =
L − 84
240 + 10(168 − L) − 100
10 =
L − 84
L = 133
Thus, Karin will choose to leisure 133 hours, and consume $240+$10*35=$590
each week.
(d) The reservation wage is defined by the absolute value of the slope on the
indifference curve in the endowment point, or the MRS in absolute value when
working no hours. When working no hours, Karin’s leisure is 168 hours and
consumes $240. Thus,
wRES =
240 − 100
≈ $1.67
168 − 84
Econ 152 – Problem set #1 – Solutions
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(e) An increase in non-labor income results in a parallel outward shift of the budget
line (the green line in the graph). If leisure is a normal good, the new optimal
solution results in an increase in leisure and a reduction of hours work. This is
called the income effect.
Repeating the calculations from (c) Karin now chooses to leisure 146 hours and
work 22 hours. Her consumption increases to $ 720.
Dollars of
Consumption
$1,920
$500
$240
133
146
168
Hours o f Leisure
(f) A wage cut from $10 to $8, changes the slope of the budget line. In the graph the
new budget line is represented by the green line.
The new optimal solution is the point on the new budget line that is tangent to the
indifference curve.
The total effect of this wage change can be decomposed into two effects:
Because income is lower now, Karin will want to reduce the consumption of
leisure. This income effect is represented by the point on the new indifference
curve with slope equal to the old wage $10.
The wage change also changed the relative price of leisure and consumption.
Because one hour of leisure is cheaper now, Karin will want to trade some
consumption for leisure. This is called the substitution effect.
Repeating the calculations from (c) Karin now chooses to leisure 134.5 hours and
work 33.25 hours. Her consumption level drops to $506.
In this example the substitution effect dominates and the total effect results in a
reduction of working hours.
Econ 152 – Problem set #1 – Solutions
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Dollars of
Consumption
$1,920
slo pe -10
$ 1,584
slo pe -8
$240
133 134.75
168
Hours o f Leisure