Summary on constrained optimization, duality, consumer`s theory

```AAE 635 Discussions
12/11/2006
Summary on constrained optimization, duality, and consumer’s theory
--“This is not a substitute for your class notes; things that have been ignored in this notes
might appear on the exam.”-Solving constrained optimization problem in different approaches
f ( x1 ,....., xn ,1 ,......, n ) , subject to h( x1 ,....., xn , 1 ,......,  n )  0.
1. The Substitution Approach-Substituting for x1 from the budget constraint into the
objective function.
2. The Lagrangean Approach:
L = f ( x1 ,....., xn ,1 ,......, n ) + h( x1 ,....., xn , 1 ,......,  n )
The FONC:
Li = 0
L = 0
Checking SOSC by Bordered Hessian Approach:
To checking the SOSC, define the bordered Hessian as the (n+m)(n+m) matrix H 
 L11 L12 ... L1n L1 
L
L22 ... L2 n L2  
12
T

L x  L xx h x 
L
  
  , evaluated at (x*,
2L/(x, )2   xx
 
  

L x L    h x 0 m   L
 1n L2 n ... Lnn Ln 
 L1 L2  ... Ln
0 
*), where L(x, ) = f(x) + T h(x). Define the (m+k)(m+k) matrix Ck from H by
deleting from H the first (n-k) rows and (n-k) columns, k = m+1, …, n. Then, check
whether (1) k Ck  0 (for a maximum) or (1) m Ck  0 (for a minimum).
In a two-variable case with one constraint (m=1):
It reduces to checking whether C2  H is positive (for a maximum) or negative (for a
  2 L / x 12
 2 L / x 1x 2 h 1 / x 1 


 2 L / x 22
h 1 / x 2  . All expressions are
minimum), where H = det  2 L / x 1x 2
 h 1 / x 1
h 1 / x 2
0 

evaluated at (x*, *). (Note that this does not imply that 2L/xi2 &lt; 0, i = 1, 2).
3. Do we always need to test if we are at a maximum or min? No if we impose some
structures on the functions we deal with. What assumption do we use for constrained
optimizations to always yield interior solutions (for maximum or minimum) in a class of
differentiable functions?
Maximization Problem
Maximize f ( x1 ,....., xn ,1 ,......, n ) , subject to h( x1 ,....., xn , 1 ,......,  n )  0.
If f and h are quasi-concave functions, *  0, and either f is concave or
fx(x*)  0, then the FONC are sufficient to identify x* as a global interior
solution to the maximization problem (1). Note that CQ must hold for FONC
to be the necessary condition.
In consumption theory problem, (maximize utility subject to budget constraint) we use
this assumption to solve for FONC. Under non-satiation, * &gt;0 (the marginal utility of
income is positive) and Ux(x*) &gt; 0 for some xi.
Max u(x) s.t. y = px
L = u(x) + (y-px)
Example: Cobb-Douglas utility maximization: (HW 6): U  x1 x 2 : y  p1 x1  p2 x2
a
b
The Lagrangean: L  x1 x 2    y  p1 x1  p 2 x 2 
a
b
Take FONC and solve for the Marshallian demand functions.
ay
by
x1*  y, p  
and x 2*  y, p  
a  b  p1
a  b  p2
a
 a   b
  y, p     
 p1   p 2
*
b
  y 
 

 ab
a  b 1
&gt;0
The indirect utility function is given by: (substitute the optimal commodity demand
functions in to the utility function):
 ay


u ( x*)  V ( y, p)  
 a  b  p1 
a

by

 a  b  p 2
b

 a 
   

 p1 
a
b
 b   y 
  

 p2   a  b 
a b
The assumptions we need for interior max are fulfilled: The utility function is quasiconcave. The budget constraint is linear hence concave and quasi-concave. The marginal
utility of income is positive.
Another application of this is in a constrained profit maximization problem. * has an
interpretation of marginal profit of relaxing the constraint. (Because constraints usually
limit profit, relaxing constraint should increase profit.) (See HW5 problem question # 3
for constrained profit maximization)
Minimization problem
If CQ holds, f(x) and h(x) are quasi-convex functions, *  0, and either f(x) is convex or
fx(x*)  0, then x* is a global interior solution to the minimization problem.
The usual problems we work with are cost minimization and expenditure minimization.
Example1: Cost minimization with Cobb-Douglas production function: y  x1 x 2
a
b
Here h(x) = y  x1 x 2 is quasi-convex since x1 x 2 is quasi-concave.
a
b
a
b
L  w1 x1  w2 x 2   ( y  x1 x 2 )
a
b
Take and solve FONC to get the conditional input demand functions. They are functions
of w and y.
b
 aw  a b 1
 aw 
x *1   2  y a b and x *2   2 
 bw1 
 bw1 

a
a b
1
y a b
Substituting the optimal input demand functions in the definition of cost
C  w1 x1  w2 x2
b
a


 a
1
b
a b
a b
a
a




a b


a

b
a

b
we get C*      
w w2 y
 b 
 1
b


Example2: Expenditure minimization with Cobb-Douglas utility U  x1 x 2
a
b
L  p1 x1  p 2 x 2   (U  x1 x 2 )
a
b
Take FONC and solve for Hicksian Demand functions (these are functions of prices and
U)
The demand functions for the expenditure minimization is exactly the same as the cost
minimization solved above with w’s replaced by p’s and y replaced by U. The
expenditure function will also exactly look like the cost function with the necessary
replacements of the parameter names.
Again in the above examples the production function and utility function are both quasiconcave and hence h(x) is quasi-convex. (Note that h(x) is y-f(x) and U-u(x)). The
objective function is linear in both cases, hence is convex. So the FONC are necessary
and sufficient condition for interior solution.
Duality
Envelope theorem has wide variety of applications in economics.
V/  V = L(x*(), *(), ).
Applications of the Envelope theorem:
1. Shephard’s Lemma (The derivative of cost function with respect to input price is the
level of input demand)
c*(w1,w2, y) = min{ w1 x1 + w2x2: y = g(x1,x2))
L(x, , ) = {w1 x1 + w2x2 + ( y-g(x1,x2))
Solving this we have: C*(x1c( w1,w2, y) x2c( w1,w2, y)).
C*/w1 = x1c( w1,w2, y)
C*/w2 = x2c( w1,w2, y)
These are called conditional input demand functions (cost minimizing input demands)
C*/y = c( w1,w2, y) = marginal cost
Example: Given the following indirect cost function (actually called cost function in
many texts!)
b
a


 a
1
b
a b
a b
a
a




a


C*      
w a b w2 a b y b You can verify Shephard’s Lemma by taking
 b 
 1
b


derivative of C* w.r.t w’s. See the solution of input demand functions solved for the
Cobb-Douglas in previous section.
For other applications of Shephard’s Lemma to different kinds of cost functions, see
exam questions.
Another application of Shephard’s Lemma is in expenditure minimization models
E(p, U) = min pTx: U = u(x)
The Lagrangean: L(x, , ) = pTx + (U - u(x) )
Ep = x*(p, U) has similar properties with cost minimizing behavior in production.
3. Hotelling’s Lemma: the derivative of the indirect objective function with respect to
prices gives the profit maximizing net put decision rules x*(p) and y*(p).
V(p, w1,w2)=Max pg(x1, x2)-w1x1-w2x2 = *( x1*( w1,w2, p), x2*( w1,w2, p))
*/w1 =- x1*( w1,w2, p)
*/w2 = -x2*( w1,w2,p)
Example for profit maximization
L  px1 x 2  w1 x1  w1 x1
a
b
solving this using FONC will yield:
x  x1  p, w1 , w2   p
*
1
1
1 a  b
1 b
b
1 a
a
 a  1 a b  b  1 a b
 
 
 w1 
 w2 
 b  1 a b  a  1 a b
x 2*  x 2  p, w1 , w2   p
 
 
 w2 
 w1 
These are called unconditional input demand functions (profit maximizing input demand
functions)
Substitute the optimal choices in Profit definition above.
1
1 a b
1
a
b
The profit function will be:  ( w1 , w2 , p)  p w1  w2 

 a b 
a b  


And you can verify Hotelling’s Lemma by taking derivatives of the profit function with
respect to w’s. You will get the optimal input demands solved above.
4. Roy’s identity
V(p, m) = max u(x): m = pTx
L(x, , ) = u(x) + (m- pTx )
Roy’s identity says that
The Marshallian demand x*(p,m)=-Vp/Vm
Proof: Using Envelope theorem: Vp= -*x* and Vm = * dividing the two will get us to
the desired result.
Example: Given the following indirect utility function, use Roy’s identity to find
Marshallian demand
 ay


u ( x*)  V ( y, p)  


a

b
p
1 

a

by

 a  b  p 2
b

 a 
   

 p1 
a
b
 b   y 
  

 p2   a  b 
a b
This indirect utility function is solved in one of the previous example above (assuming
ay
Cobb-Douglas utility function)…. Check if you get: x1*  y, p  
this comes from
a  b  p1
again the Cobb-Douglas utility function.
You might see envelope theorem in more places than listed here (e.g. see exam
questions). The main idea behind envelope theorem is, if you want to know how
parameters affect the indirect objective function, take derivative directly and evaluate the
derivative values at optimum choice.
Primal Dual results (Comparative statics):
Duality helps to get most of the comparative statics in economics so conveniently. You
can do comparative statics using primal approach, though most of them involve inverting
the bordered Hessian or the use of Cramer’s rule).
The concise way of doing comparative statics is using the following primal-dual result.
V - L = [Lx(x*, *), L(x*, *)]
 x * 
 *  , and is symmetric.
  
uT [V - L(x*, *, )] u = uT [Lx(x*, *), L(x*, *)]
 x * 
 * 
  
u0
for all (k1) vectors u satisfying h(x*, ) u = 0.
 x * 
,
* 
  
---This is equivalent to saying V - L = [Lx(x*, *), L(x*, *)] 
symmetric positive definite matrix if u is chosen to satisfy the constraint
For profit maximization this implies that, since L is zero, using Hotteling’s Lemma,
 *ww   [ x *w ] is symmetric positive semi-definite. This also implies [ x *w ] is
symmetric negative semi-definite. Hence input demand functions are down ward sloping.
Also output supply is upward sloping. NOTE: Profit maximizing input-demand is a
function of p and w.
Some further properties:
The profit function (indirect profit function) is convex in w and p as well.
Profit function is HD1 in w and P.
Hence, using Hotteling’s Lemma, input demand functions and output supply functions
are HD0 in p and w.
For cost minimization
The same applies with L being zero. Using Shephard’s Lemma , C * ww   [ x c w ] is
symmetric negative Semi-Definite, since w’s are not in the constraint..
This implies downward sloping demand curve. NOTE: Cost minimizing input demand is
a function of w and y.
Some further properties:
C* is concave in w’s.
C* are HD1 in w’s.
It follows that from Shepard’s Lemma, input demands are HD0 in w.
For expenditure minimization, the cost minimizing properties apply (with p replacing w
and u replacing y).
For Utility maximization: The Primal-Dual result says that the Slutsky matrix is
symmetric negative semi-definite.
Remember the equality between Slutsky matrix and the substitution matrix from
expenditure minimization.
 x c
 1
 w1
 c
 x 2
 w1
x1c   x1* x1* *

x1
 
w2   w1 k

*
*
c 
x 2   x 2  x 2 x *

1
w2   w1 k
x1* x1* * 

x2 
w2 k 
x 2* x 2* * 

x2 
w2 k 
They are connected via the identity xic(p,U) = xi*(p,E(p,U) ) and taking derivative wrt to
p. This is called instant Slutsky equation since it evaluates the input demands only at that
particular identity value.
Remember you can also do comparative statics if you know some identities. In the short
run profit max in the homework see the hint, and use the equality between the Hessian of
unconstrained profit function and the constrained profit function.
Cost functions and production technologies
Cost function for CRS (HD1) always has the form.
Min wTx : y = f(x) where f(x) is HD1.
C* = yAC(w)
Where AC represents the average cost, and is only a function of w. Cost function is
linear in y and therefore the marginal cost and average cost are always the same. Hence
the scale elasticity for HD1 production technology is 1. Also note the equality of MC and
AC.
 1
Example: Cobb-Douglass: f(x) = y = [ x1 x2 ] .
After taking and solving FONC, the corresponding cost function is:

w1 w12
C*= y 
The cost function corresponding to Cobb-Douglas is Cobb-Douglas.
 (1   )1


For simple CES: f(x) = y = [ x1  x 2 ]1 / 


Rewrite the y   [ x1  x 2 ]
Take the usual route (i.e. FONCs)
 1
w1  x1  0
w2  x 2


 1
0

y  [ x1  x 2 ]
Input demands:
1


1




1
1


1
2
x1  w1  1 [ w1  1  w2  1 ]1 /  y
x1  w2  1 [ w1  1  w2  1 ]1 /  y
 1

w
] Note that the cost function for CES
The cost function: C*  y[ w
production function is has the form CES itself.
See the class note (COST Minimization 2) for the general CES and how it is related to
Cobb-Douglas, linear, and Leontief technologies)
Cost function for Homothetic technology:
Homothetic function is a monotonic transformation of HD1 function.
y = F(f(x)) where F’(.) &gt; 0 and f(x) is HD1.
Again we can rewrite the production constraint as: F-1(y) = f(x) and use the cost function
for the HD1 form and replacing y by F-1(y) [call it G(y)].
The Cost function therefore will be: C*  G ( y ) K ( w)
Cost function for HD r production technology: see your class notes (COST MIN 2)
C*  y 1 / r K ( w)
To see this: Let y = F(x) is HD r
We can change this to HD1 by defining HD1 function. F(x) =[ f(x)]r where f(x) is HD 1.
Now rewrite the production function as: y1/r = f(x).
Then use the cost function for HD 1: replacing y1/r in place of y.  C*  y 1 / r K ( w)
Note that AC/MC = r. This is its scale elasticity. (AC is proportional to MC,
proportionality factor r)
Recall that all homogenous functions are homothetic (the converse is not true). As you
can see for HD1 therefore G(y) = y; for HD r function G(y) = y1/r
Allen Elasticity of Substitution
ij = (xic/wj) C/(xic xjc) = = [ln(xic)/ln(wj)]/Sj, where Sj = wj xjc/C is the j-th cost share
ij &gt;0 (&lt; 0 )  the two inputs are substitutes (complements)
For Cobb-Douglas: Allen elasticity is 1,
For CES Allen elasticity is 1/(1-)
For Leontief technology y = min{ax1, bx2}is 0 (no substitution possibility)
For Linear technology Allen elasticity is infinite (perfect substitution).
Scale elasticity: SE = i (g(x)/xi) (xi/g(x)) = AC/MC
SE &gt;1 Increasing RTS
SE =1 CRTS
SE &lt;1 Decreasing RTS.
\$
MC = C/y = marginal cost
AC = C/y = average revenue
IRTS
CRTS
DRTS
y
Back to Consumption Theory
See the first part of consumption theory class note for Adding-up restrictions:
j pj xj*(y, p) = y. (budget constraint)
Differentiating the budget constraint with respect to p gives
xi* + j [pj (xj*/pi)] = 0, i = 1, …, n, = &quot;Cournot aggregation restriction&quot;
In terms of elasticity:
(pi xi*/y) + j [(pj xj*/y)(ln(xj*)/ln(pi))] = 0
Differentiating the budget constraint with respect to y gives
j [pj (xj*/y)] = 1 = Engel aggregation restriction
In terms of elasticity:
j [(pj xj*/y) (ln(xj*)/ln(y))] = 1,
Please see in th notes other elasticity expressions that are implied by homogeneity of
degree zero of the Marshallian demand.
Household labor supply (time allocation).
 leisure (s)
 work within the household (lh)
 work as wage labor outside the household (lw) and wage rate is w.
Households maximize utility from consumption of goods and leisure u(x, s).
Total time available is T.
The household production technology assumed to be represented by strictly concave
function production function f(lh).Denote the price of household production by ph.
The utility maximizing household’s problem is represented by
V(y, p, ph, w) = Max x, s, l {u(x, s): pT x  y + w lw + phf(lh): lw + lh + s =T and lh, lw, x, s
 0}
Substituting the time constraint assuming lw &gt;0
V(y, p, ph, w) = Max x, s, lh{u(x, s): pT x  y + w (T-lh -s) + phf(lh) : lh, x, s  0}
Which has a Marshallian demand functions: x+(y, p, w, ph ), s+( y, p, w, ph),
lh *(y, p, w,ph) and the corresponding output supply denoted by h*( y, p, w, ph) = f(lh *)
Two stage optimization:
First the household maximizes profit from its time endowment.
Second the household buys consumption good and leisure. (Strictly speaking leisure is
not bought in the market, but the consumer buys leisure through not earning money from
work. The price for leisure is the wage she could have earned by working).
Total income of a household if working full time (i.e. no leisure)
Y = y + w (T-lh ) + phf(lh) = y + wT + phf(lh)-w lh = y + wT +*(ph, w)
Given T, y and w are fixed, maximization of income implies maximization of profit from
home production by choosing the time allocation for home good.
Max  = phf(lh)-w lh
This looks like as if the home producer is actually facing the market wage rate for home
production. The FONC is phf’(lh) = w. That means the marginal value product of labor is
equal to wage rate. This gives us the profit maximizing labor supply.
The profit maximization problem yields the usual output supply h*(w, ph) and input
demand lh*( w, ph) functions.
Now given the total household income Y = y + wT + *(w).
In the second stage we buy goods (x) and leisure (s) to maximize utility.
V(p, Y) = Max x, s{u(x, s): pT x + ws = y + w T + *(ph, w)}
Again this yields the Marshallian demand for x#(Y, p, w ) and leisure s#(Y, p, w)
Notice the price of leisure being w.
Note that the single stage maximization and the two stages maximization yield the same
labor demand and output supply for home good.
lh *(y, p, w, ph) = lh *( w, ph)
h*( y, p, w, ph) = h*( w, ph)
The profit maximizing choices are independent of consumption parameters (p, y). That is
called separability of production and consumption decisions. NOTE that separability
works only when there is perfect labor market (that is when the household can get job
outside his farm). In many developing countries this assumption will not hold. Most of
microeconomics of development deals with the issue of non-separability of production
and consumption decisions in developing countries, and the inefficiency associated with
it.
Similarly the single stage and the two stage optimizations must yield the same
Marshallian demands for consumption and leisure.
x+(y, p, w, ph ) = x#(Y, p, w )
s+( y, p, w, ph) = s#(Y, p, w)
What is the impact of wage rate rise on labor supply (or on leisure)?
s+/w = s#/w + (s#/Y) (Y/w)
= s#/w + (s#/Y) [T - (*/w)]
= s#/w + (s#/Y) [T - lh*]
Notice that if leisure is normal good, s#/w is negative (recall the slutsky property).
s#/w is holding income constant. But we know that when w changes income changes,
the change being captured in the second term. (s#/Y) [T - lh*] is always positive for if
leisure is normal good, and the assumption of interior solution holds.
Therefore s+/w could be positive if the second term dominates the second term. This
would happen when the income elasticity of leisure is &quot;high&quot;.
One extension to this model: what is the impact of home good price on leisure
consumption?
s+/ph = (s#/Y) (Y/p)
= (s#/Y) h* &gt;0
You will spend less time working!! Since labor use must rise in home goods production,
(lh*/ph &gt;0) the increase must come from the reduction in outside employment
i.e. (lw* /ph &lt; 0). This is intuitive, right? This assumes that the farmer do not consume
the goods he produces. If the assumption is not true, then the rise in price has two
components. The first part is the substitution between leisure and the consumption of that
good. Second the income effect of the price rise, because ph shows up in two places.
Let’s see a special case where there is no home production.
The problem becomes:
V(p, w, y) = Max x, s{u(x, s): pT x = y + w( T-s) )}
Marshallian demands are: s+(p,w, y) and x+(p, w, y)
First the full (maximum)income is Y = y + wT
Second, purchase goods and leisure using the full income.
V(p, w, Y) = Max x, s{u(x, s): pT x + ws = y + wT}
Now the Marshallian demand will be: x#(Y, p, w ) and s#(Y, p, w)
Given the equivalence of the two approaches:
s+(p, w, y) = s#(Y, p, w)
s+/w = s#/w + (s#/Y) (Y/w)
= s#/w + (s#/Y)T
The second term (the endowment effect) is positive and theoretically could dominate the
first term ( which is negative). Therefore, it is possible that when wage increases workers
reduce hours spent on working, and increase their leisure time.
Note that materials below are not on the exam.
______________________________________________________________________
Duality between Expenditure minimization and Utility Maximization
Properties of indirect utility function:
V(p, y) = Maxx {u(x) : pT x  y, x  0},
Solution: Marshallian demand x(p, y):
Properties of V(p, y)
 Increasing in y,
 non-increasing in p,
 Homogeneous of degree zero in (p, y)
 Quasi-convex in p.
Properties of the expenditure function:
E(p, U) = Minx {pT x: u(x)  U, x  0},
Solution: Hicksian demand x(p, U)
Properties of E(p, U)
 Non-decreasing in U and p,
 Homogeneous of degree one in p,
 Concave in p.
Duality between V and E: (identities)
1. xc(p, U) = x*(p, E(p, U)) Goods are goods
2. x*(p, y) = xc(p, V(p, y))
1 and 2 says goods are goods two.
3. E(p, V(p, y)) = y Money=money
4. V(p, E(p, U)) = U. Happy = happy
Example: Start from the indirect utility function for Cobb-Douglas:
a
b
a b
 a   b   y 
V ( y, p)      

 p1   p 2   a  b 
Using identity 4, we can solve for the expenditure function.
a
b
 a   b   E ( p, U ) 
V ( y, E ( p, U ))      

 p1   p 2   a  b 
a b
U
b
a


 a
1
b
a b
a b
a
a





 p a b p a b U a b
Solving for E ( p,U )      
2
 b 
 1
b


Using Roy’s identity we can solve for Marshallian demands
Using Shephard’s Lemma, we can solve for Hicksian demands.
This is an illustration of the use of the identities.
Theoretically, if you have any of the four functions (expenditure, Indirect utility,
Hicksian demand, or Marshallian demand) you can always recover the rest of the
three functions using the identities. Some of them will involve taking integrals
though, which is not fun to do!!
Equivalent Variation, Compensating Variation
I want to touch on these welfare measures of price changes. Assume price of good 1
increases and price of good 2 remain fixed (a two good model). Assume price of good 2
is set to 1.
The price change obviously lowers the utility of an individual for a fixed income. So
what is the welfare loss from this price change, from U1 to U2?
Then equivalent variations (EV) asks “ how much is the consumer willing to pay to
prevent this price change, given she doesn’t have the right to the old low price?” This is
the difference between the income she already has and the income she needs to attain the
lower utility level with old price” So this is all about attaining the new utility level at old
price set; and finding the difference between old income and the income needed to attain
U2 using old prices.
The Compensating Variation (CV) asks: “ Given the consumer has the right to the old
price, how much money can we offer the consumer to compensate for the loss in utility
due to the price rise. This is comparing the old income and the income needed to attain
old utility with the new prices. In short we use new prices to compare the income needed
to attain U1 and U2.
Given we are using different sets of prices to compare income measure of the utility
changes, EV and CV will be different. They will be equal if and only if there is no
income effect (in which case Hicksian demand and Marshallian demand coincide).
Figure 1: The welfare effects of an increase in p1
x2
c
U1
b
C
a
A
U2
B
Old price line
x1
CV=c-b
EV= b-a
For a price rise CV &gt; EV !!!
For a price fall EV becomes CV and CV becomes EV.
I really enjoyed teaching you guys. GOOD LUCK!
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