Econ 463 R. Butler revised, spring 2013 Lecture 3
I. Different Functions for different Occasions, Or, “You’re Never Out of Style if You’re
Including Input Prices”
A. The Usual Suspects Diagram (of interrelated demand functions and their relatives):
max U(C,L)
s.t. PC=W(T-L)+V
min PC + WL
s.t. U(C,L)=U 0
solve
solve
Marshallian Demand
C=C(WT+V, W, P)
L=L(WT+V, W, P)
Hicksian Demand
C=C(U, W, P)
L=L(U, W, P)
substitute
Indirect Utility func.
~ ~
U  U (WT+V, W, P)
substitute
invert to get
Expenditure Function
E=E(U, W, P)
and Roy’s Identity (getting the demand function from the indirect utility function):
~
U
Pj
~   X j (Xj is the Marshallian demand functionfor the jth good, and Pj is its price).
U
E
Proof of Roys Identity--Using Cournot Aggregation Results (E=expenditure=money
income=M):
X i
E =  Pi X i differentiate w.r.t. Pj to get 0=Xj+  Pi
Pj
and differentiate w.r.t. E to get 1=  Pi
X i
.
M
Now for the proof, the numerator can be written as 
~
U X i
=
X i Pj
X i
 P P
i
=  (-Xj)
j
(since the marginal utility of the ith good on the left hand side =  Pi ), and the
~
X i
U X i
denominator can be written as 
=   Pi
=  . Now substitute and you
X i M
M
get the identity.
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B. Beyond Graphs 110: Labor Supply Functions from the Usual Suspects Diagram
B1.From Utility function to Labor Supply function: the Cobb-Douglas Case
1) U = C  L 
which is maximized subject to the budget constraint
2) WT + V=PC + WL
where the following symbols have been used
C= goods consumed
P= price of goods
E=”total expenditures” = P C + W L
W= wage rate
V= nonlabor income (Mom's check or your stock dividends)
T= total amount of time available (set=1 in Killingsworth)
L= amount of leisure time
H= amount of work (T=H+L, or H=T-L)
 = consumption parameter in the utility function
 =leisuree parameter in the utility function
 =LaGrange multiplier
This can be solved by setting up the following Lagrangian equation and maximizing with
respect to C,L and ë:
3) max L = C  L  +  (V + WT - WL -PC)
which yields the first order conditions
4)  WL=  PC
from which, along with the budget constraint, we can solve for the Marshallian Demand
functions (why do we call them "Marshallian" rather than "compensated" demand
functions):
 V + WT 
5) C = (1 - b) 

 P 
for consumption, and for leisure
 V + WT 
6) L = b 

 W 
where b =

+
2
From 6), we can substitute L=T-H, and solve for hours of work H, to get
H = (1-b) T - b(V/W)
If we think of this as a regression equation, then we would regress the number of hours
worked on an intercept (which coefficient would be estimate of "(1-b)T" if everyone had
the same potential works of hour, and if note, perhaps a control variable for health status
and age might be nice) and the slope variable (V/W). [[[[[ask: What would the slope
coefficient measure, and is there a test of the validity of this simple Cobb-Douglas utility
model? it has to do with the relationship between estimated coefficients. ]]]]]]]]
NOW FOR YOUR TURN: Try working out the labor supply function for the Leontiff
utility case: U = min(bLL, bCC).
B2. The General Relationship Between Hicksian and Marshallian Labor Supply
functions
The Slutsky equation (applied to labor supply). The Slutsky equation relates the
uncompensated Marshallian function (when a price falls, no adjustment is made for the
additional real income available since more things can be purchased, only non-wage
income, V, is held constant), to the compensated Hicksian function (now real income is
“held” constant in the sense of an adjustment in money income to keep the consumer on
the same indifference curve). Consider the demand for a good from both perspectives
(they will be initially equal, and a change in the price will induce a compensating
adjustment in income so that the equality will be maintained):
Let
E=expenditures=WT+V (which will equal full money income)=PC+WL
U=utility = U(C,L)
Demand:
Marshallian
L = Lm(P,W, E)
Hicksian
Lh= Lh(P,W, U)
Duality:
 =  (P,W,E) [indirect utility fn]
E=E(P,W,U) [expenditure fn]
m
Since H = T-L = T - Lm(P,W, E) = T - Lh(P,W, U)
With the Slutsky equations”
In terms of leisure (L):
Lh
W
In terms of hours of work (H)
H
W
U
Lm

W
h

U
V
H m
W
Lm

H
E

V
H m
H
E
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Derivation of the Slutsky equation:
Hicksian Demand for Leisure = Marshallian Demand for Leisure
Lh(P,W,U) = Lm(P,W,E(U, P,W))
Where, again, E(U, P,W) is the expenditure function, which equals money income.
Now take the derivative of both sides with respect to W (holding utility, U, constant, but
adjusting expenditures so as to maintain the equality on both sides of the equation) and
get:
Lh
W

U
Lm
W

E
Lm E
E W
E
 L by Shepherd’s lemma (the equivalent in production theory is: the derivative of a
W
cost function w.r.t. input price equals the conditional input demand function), and term
Lm
is just the income effect for leisure, positive if leisure is a normal good. Then
E
Lm
L is the amount of income necessary to return the consumer to the original
E
indifference curve. Now if we differentiated just the Marshallian function (holding V
constant in the expenditure constraint, E) we would have
Lm
Lm
Lm E Lm
Lm




T
W V W E E W W E E
since with V fixed, changing W changes total expenditure or income by T (the amount of
Lm
Lm
Lm
time available if you worked full time). Hence,


T which when
W E W V E
substituted into the first derivative equality in this section yields
Lh
Lm
Lm
Lm


T
L , or
W U W V E
E
Lh
Lm
Lm
Lm
Lm


( T  L) 

H
W U W V E
W V E
where the last expressions come from collecting terms and substituting H = T-L.
L
H
L
H


and
since whenever hours of work increases, leisure falls.
E
E
W
W
Hence the Slutsky equation for labor supply is
H h
W

U
H m
W

V
H m
H
E
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One of the implications of utility theory is that, holding real income (utility) constant, an
increase in the wage rate will increase hours of work (decrease leisure)—that is, that the
compensated derivative is non-negative. A regression of hours of work on wage and
H h
income, however, does not yield
, but the wage coefficient in the work hours
W U
regression yields
H m
W
, and the (non-wage) income coefficient in the work hours
V
H
H h
regression yields
, from which the compensated effect
can be calculated
E
W U
and used to test the theory. (Even if there is a backward bending labor supply function,
H m
H h
so that
is negative, the compensation effect
should be positive.)
E
W
m
II. Estimating Female Labor supply and the Sample Selection Problem
But sometimes getting unbiased estimates of these wage and income effects is tricky.
Return to the discussion after equation 6 above, where we have specified an hours of
work equation for the Cobb-Douglas utility function:
7) H = (1-b) T - b(V/W) where b =

+
Assume that everyone is equally healthy in the sample in the sense that they have the
same T; then we could regress hours of work, H, on an intercept and on (V/W). The labor
supply estimation model for a worker with a Cobb-Douglas utility function would be
8) H =  0   1
V

W
where  represents unmeasured tastes for work: if  > 0, the worker wants to work more
hours than average given their value of V/W; and if  <0 the worker wants to work fewer
hours than average given their value of V/W. Tastes for work average out to be zero
across the whole population of potential workers, E(  ) =0. The estimated coefficients
would measure ˆ0  (1  b)T and ˆ1   b .
The sample selection problem facing the 463 student (and econometricians in general) is
that we did not observe the desired hours of work function for everyone, only those
whose actual (market) wage exceeds their reservation wage. Hence, a full model of labor
supply for a population of workers with Cobb-Douglas utility preferences would look
like:
5
V

W
V
V
  iff  0   1
  >0
9.b) H =  0   1
W
W
9.a) H* =  0   1
9.c) H = 0
iff  0   1
V
  0
W
and H* is the desired hours of work, given the parameters of the Cobb-Douglas function
and the value of V/W of each individual worker. Suppose we fix V, then the labor supply
function would look something like (including, of course those “off” the labor supply
function because of positive and negative values of their errors,  ):
desired hrs
of work, H*
true slope= -  1
0
V
W2
wage, W
true reservation wage
The problem with estimating the labor supply equation for the population as given in the
above graph is that not everyone (each individual is given by a dot) is observed in the
sample of workers, but only those for whom H>0. That is, we ignore observations for
those whom H  0, this truncating the observed sample that we use to estimate the model.
This truncation, or sample selection, is nonrandom and related the values of V/W (since
V/W determines the likelihood that the desired hours of work will be H  0 or not). The
observed sample, with the resulting estimated regression line, is given below for the case
of this sort of sample selection:
6
desired hrs
of work, H*
true slope= -  1
V
W2
estimated slope using OLS
0
wage, W
true reservation wage
estimated reservation using OLS
Mathematically, regression functions of hours of work are conditional on the value of the
independent variables, V/W, as follows:
10) E( H |
V
V
V
 E ( | )
) =  0  1
W
W
W
V
) =0, as illustrated in the first graph. But with
W
V
the truncation at zero hours of work, E ( | )  0 and OLS estimates of the labor supply
W
function may be biased. The simplest solution is to add another term to the equation that
V
captures the “ E ( | )  0” effect. This can be done “approximately” by predicting the
W
likelihood of working Prob(H>0) (call this “predtd_p”), and then including this variable
and its square in the labor supply equation. This is want I suggested in lecture one, which
you should now refer to. The other way, more exact if the regression errors are normally
V
distributed, is to include a  -correction term in the equation, where  = E ( | ) . This
W
is a procedure first pioneered by Jim Heckman, so it is sometimes referred to as a
Heckman sample selection correction or even as a Heckit.
For the population as a whole, the E ( |
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Appendix to Lecture 3: Indirect Utility and Expenditure Functions using the CobbDouglas
As an alternative to the utility function for getting a specific estimating form for the labor
supply equation, you can begin with an indirect utility or expenditure function instead and
see what they imply about the estimation form. They are actually easier to work with
than the utility function because you don't need to maximize these functions (they are
already the result of a maximization process), but only calculate partial derivatives. The
indirect utility function is just expressing total utility as a function of income and prices.
One important property of a correctly specified indirect utility function is that it is
homogeneous of degree zero in V,W and P. Recall that this means if you double all
income and prices, then the utility won’t change at all What is the intuitive explanation?
Also, the indirect utility function must be nondecreasing in V and nonincreasing in P.
Why?
It is achieved by substituting the demand functions into the utility function (maximizing
behavior implicitly assumed in such a function, can you tell why?):


~
 V + WT  
a1) U = C(.)  L(.)  =  (1 - b) 
 
 P 

  V + WT  
 b 
 
  W 

Now Roy's identity just says that
~
U(P,W,V)
a2) -L = ~ W
= -(T-H)
U(P,W,V)
 (V  WT )
This leads to a rather lengthy expression, which can be simplified into that of equation 6
above.
We can get the expenditure function for the Cobb Douglas utility function by returning to
equation (7) and solving for "(V+WT)" (which are total expenditures) on the left hand
side of the equation, with U (the level of utility), P and W on the right hand side. This
function tells me the minimum expenditure it takes to achieve a given level of utility with
prices P and W. So again, this function assumes utility maximization has already taken
place:


~
a3) E = V + WT = [ U (P(1  ))  (W (1  ))  ]  
1


8
The derivative of the expenditure function with respect to W yields the demand for
leisure function, which is just T minus the labor supply function (since H = T - L). This
function should always be homogeneous of degree one in P, W, whatever the form of the
original utility function. This labor supply function derived from the expenditure
function is the "compensated" one since utility is being held fixed. Hence the second
partial derivative of the expenditure function with respect to wages, or the derivative of
the compensated labor supply equation with respect to W, will be positive as the increase
in wages will cause a worker to move around the indifference curve and substitute more
goods for the now more expensive leisure. Will the Marshallian labor supply equation
also always slope upwards? i.e., will the derivative of that equation with respect to W
also be positive? Why?
Hicksian Demand from First Principles:
a4) min L = PC + WL +  (U0 - C  L  )
get
a5)  WL=  PC
as before (see equation 4), which along with the utility function, allows you to solve for L
and C as a function of U0, P, and W:
1
L=
1
U 0  

 W  
[
]
 P
and
C=
U 0  

 P  
[
]
 W
These Hicksian Demand functions could be substituted into the cost constraint to get the
expenditure function derived earlier (by other means).
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