Labor Demand and Supply
Prof. Eric Sims
University of Notre Dame
Fall 2010
Sims (ND)
Labor Demand and Supply
Fall 2010
1 / 16
Labor
We now go all the way to endogenizing production
People have to work in order to produce anything
Labor is different than capital in that it is used up in the production
process. Unused labor today cannot be used in the future
Sims (ND)
Labor Demand and Supply
Fall 2010
2 / 16
Production Function
Let production function now be:
y = zf (k, n )
Properties:
Constant returns to scale
Increasing and concave
Still think of output as fruit. z represents exogenous factors like
weather. k is “stock” of trees. n is amount of time spent harvesting.
Sims (ND)
Labor Demand and Supply
Fall 2010
3 / 16
Labor Demand
Firms pay a wage rate, w , to hire labor
Firms behave competitively
Choice of how much labor to hire is intratemporal – there is no
dynamic dimension
Sims (ND)
Labor Demand and Supply
Fall 2010
4 / 16
Firm Objective Function
Similar to last time, just incorporating these changes:
max
0
0
n,n ,I ,k
zf (k, n ) − I − wn +
1
z 0 f (k 0 , n 0 ) − w 0 n 0 + (1 − d )k 0
1+r
s.t.
0
k = I + (1 − d )k
Sims (ND)
Labor Demand and Supply
Fall 2010
5 / 16
Solution
Looks nastier than it really is
Solution is to equate marginal product of labor with the real wage
each period:
w = zfn (k, n )
This implicitly defines a labor demand function: n = n (w , z ).
Derivative signs?
Investment demand function is same
Sims (ND)
Labor Demand and Supply
Fall 2010
6 / 16
Labor supply
Labor is supplied by households, who also take real wage as given
Households are endowed with time
Households derive utility from leisure, which is the time endowment
that is not spent working
Notation:
h: time endowment (say 16 hours a day)
l: leisure time
n: labor time
h = n+l
Sims (ND)
Labor Demand and Supply
Fall 2010
7 / 16
Household utility function
Household lifetime utility is now:
U = u (c ) + v (l ) + β u (c 0 ) + v (l 0 )
Just think of leisure as another “good”: v (·) has the same properties
as u (·) – increasing and concave
Sims (ND)
Labor Demand and Supply
Fall 2010
8 / 16
Budget constraint
Income is now endogenous: equal to wn + Π, where Π denotes
profits returned to households via dividends or capital gains.
Because firm is distinct decision making unit, households take Π as
given and out of their control
Savings component is same as before
Sims (ND)
Labor Demand and Supply
Fall 2010
9 / 16
Household problem
Choose consumption, labor, and saving to maximize utility subject to
constraints, taking prices as given:
max
u (c ) + v (l ) + β u (c 0 ) + v (l 0 )
0
0
c,c ,l,l
s.t.
w 0 n0 + Π0
c0
c+
= wn + Π +
1+r
1+r
h = n+l
h = n0 + l 0
Sims (ND)
Labor Demand and Supply
Fall 2010
10 / 16
How to Solve
Plug constraints into objective function; take derivatives; set to zero
Solution is the same intertemporal consumption Euler equation and
two static intratemporal labor supply conditions
New FOC:
v 0 (h − n ) = u 0 (c )w
v 0 (h − n 0 ) = u 0 (c 0 )w 0
Sims (ND)
Labor Demand and Supply
Fall 2010
11 / 16
Graphical Representation of Solution
Indifference curve / budget line diagram for consumption today and
tomorrow same
New indifference curve budget line diagram for consumption today
and leisure today
Relative price of leisure is the real wage. Why?
Sims (ND)
Labor Demand and Supply
Fall 2010
12 / 16
The labor supply function
FOC is: v 0 (h − n ) = u 0 (c )w
Holding consumption fixed, an increase in w necessitates an increase
in n / reduction in l
i.e. labor supply slopes up
Compensated vs. uncompensated and backward-bending supply
What else affects labor supply? Anything which affects consumption.
What affects consumption?
Interest rates and expectations about future income
Labor supply function: n = n (w , r , z 0 )
Signs of derivatives?
Sims (ND)
Labor Demand and Supply
Fall 2010
13 / 16
Equilibrium in the Labor Market
ns = n(w,r,z’)
w
w*
nd= n(w,z,k)
n*
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Labor Demand and Supply
n
Fall 2010
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Comparative Statics
Go through comparative statics graphically
Sims (ND)
Labor Demand and Supply
Fall 2010
15 / 16
Mathematical Example
Use log-log utility
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Labor Demand and Supply
Fall 2010
16 / 16