Download MEASURING PRODUCTION AND INCOME, Chapter 2

LABOR SUPPLY, based on Varian section 9.8
LABOR SUPPLY
REAL WAGE (W/P)
20
15
Serie1
10
Serie2
5
0
1
2
3 4
5 6
7
8 9 10 11 12 13 14 15 16
Hours worked
Labor Supply: 3 possibilities:
(1) Labor supply is unrelated to the real wage.
(2) Labor supply increases when the real wage increases.
(3) Labor supply decreases when the real wage increases: when the real wage
increases the individual can afford to take more leisure, which she likes.
Factors that increase aggregate labor supply at a given real wage:
1. Labor immigration. 2. Lower unemployment benefits should increase the
labor supply of the domestic population.
The Microeconomics behind labor supply
The individual or household faces the choice between consumption and leisure.
More consumption requires more hours worked and hence less leisure.
The problem of the individual is to maximize:
U = U(C,R)
where
C= consumption during a period of time, e.g. a day.
R = hours of leisure enjoyed during a day.
If C U(.), and if R U(.).
The two constraints the individual faces are:
(1) The time constraint:
LR  L
where L = labor supply in hours,
L is the time endowment which is 24 hours per day.
(2) P  C  W  L  M
where P= Price of the consumption good
W = Nominal Hourly Wage
M= non-labor income, e.g. government transfers
Let M  P  C
In other words, C is the quantity of goods that the individual receives that is not
related to hours worked.
 P  C W  L  P  C  P  C W  L  P  C
 P  C W  L  W  L  P  C  W  L
 P  C W  (L  L)  P  C W  L
 P  C W  R  P  C W  L
Now we have combined the two constraints that the individual faces, and the
result is similar to the usual budget constraint: px  x  p y  y  I
Thus, the goods that the individual derives utility from (C and R) are on the lefthand-side of the equation. And in front of the quantities of these goods are the
respective prices of these goods  W is the price of leisure: it is what the
individual gives up by taking one hour of leisure. P  C W  L is called full or
potential income. If R=0, then P  C  P  C W  L .
The constraint can be rewritten in real terms:
1 C  (W / P)  R  C  (W / P)  L
where 1 = real price of consumption, W/P is the real price of leisure = the
quantity of goods the individual gives up by consuming one more unit of leisure.
Graphical illustration of the choice possibilities of the individual:
Let C  0 ,  C  (W / P)  L  (W / P)  R
Slopecoefficient
Intercept
An Increase of the Real Wage
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
Leisure (0.0-1.0)
Note: The choice constraint cuts the x-axis where R= L .
In the figure we assume that L =1, and that W/P increases from 10 to 20.
The numbers on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
Note also that labor supply (L) = L - R: When R=0, then L= L .
If W/P  the intercept increases, and the slope becomes more negative.
If W/P , the individual can afford more of both C and R. On the other hand,
when W/P , R becomes more expensive in terms of the quantity of
consumption goods the individual gives up by consuming one more unit (hour)
of leisure.
3 hypothetical possibilities on demand for leisure and on labor supply (L= L -R)
when W/P :
1.No effect on the demand for leisure and on the labor supply if the substitution
(price) effect = income effect. The substitution effect is negative for the demand
of leisure when the price of leisure (that is, the real wage) increases. The
income effect for the demand of leisure is positive as a higher real wage means
that the individual can afford and wants more leisure when income increases.
2.Negative effect on the demand for leisure (= positive effect on labor supply) if
the substitution effect > income effect.
3. Positive effect on the demand for leisure (= negative effect on labor supply) if
the substitution effect < income effect.
The optimal choice with positive non-labor income ( C  0 )
C  (W / P)  R  C  (W / P)  L
 C  C  (W / P)  L  (W / P)  R
Slopecoefficient
Intercept
An Increase of Non-Labor Income
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
LEISURE (0.0-1.0)
In the figure we assume that L =1, W/P=10, and that C increases from 5 to 10.
The numbers on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
When C increases the individual wants more of both goods as they are assumed
to be so-called normal goods. You want more of normal goods when your
income increases.
An increase of C does not change the opportunity cost of enjoying leisure, and
constitutes therefore a pure income effect.
Summary:
The effect of changes in the exogenous variables on optimal demand for C and
R, and on optimal labor supply:
If C  C * , R * , L*  L  R* 
If W/P   C * , R * ?, L*  L  R* ?
MPL,W/P
Employment and taxes
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
LS
Eqb W/P:t=0
LD
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Employment (L)
Introducing a tax creates a wedge between what the firm pays (producer real wage) and what
the worker receives (consumer real wage). A tax decreases employment. L decreases in figure
from 5 to about 3.5. The producer real wage increases to 0.8 (from 0.7 when tax is zero) and
the consumer real wage drops to below 0.4. Taxes on labor include social insurance- and
income taxes.
A mathematical note on how to derive optimal demand-functions in case of
a Cobb-Douglas (or a logarithmic) utility function:
If the individual maximizes U ( x, y)  x  y 
subject to the budget constraint: px  x  p y  y  I
where x = quantity of good x, y= quantity of good y, px = price of good x, p y =
price of good y, and I= income.
The optimal demand for x and y are such that the consumer chooses to spend a
constant fraction of its income on these goods:
p x  x*


I

 x* 

I

   px
*
py  y

I


 y* 

   py
I

Note if   1  x*  (1  ) 
I ,
px
y*   
I
py
A mathematical example on the optimal choice of leisure (optimal labor supply):
Assume that the individual has the following utility function: U  C1/ 2  R1/ 2
The constraints of the individual are: (1) L  R  L  1
(2) C  W / P  L  C
Note: W, P and C can not be affected by the individual. Thus, they are
exogenous from the point of view of the individual.
Combining the constraints yields:
1 C  (W / P)  R  C  (W / P)  L
1 C  (W / P)  R  C  (W / P)
The result is similar to the usual budget constraint: px  x  p y  y  I
Optimal demands for C and R, and optimal labor supply are:
(W / P  C)  0.5  (W / P  C)
C*  0.5  I  0.5 
pc
1
(W / P  C )  0.5  0.5  C
R*  0.5  I  0.5 
pR
W /P
W /P
L*  1 R*  0.5  0.5  C
W /P
When C  0 :If C  C * , R * , L*  L  R* 
If W/P   C * , R * , L*  L  R* : More labor is supplied when W/P .
When C  0 :
If W/P   C * , R * =0.5 and L*  L  R* =0.5. That is, labor supply and
optimal leisure are unrelated to W/P.
Thus, the substitution effect equals the income effect.