botes on consumer theory oand envelope theorm

7/31/2019 Botes on Consumer Theory Oand Envelope Theorm

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Economics 101ASection NotesGSI: David Albouy

Consumer Theory and the Envelope Theorem

1 Utility Maximization Problem

The consumer problem looked at here involves

Two goods: x and y with prices px and py.

Conusumers facing a budget constraint pxx + pyy I, where I is income.Consumers maximize utilityU(x, y) which is increasing in both arguments and quasi-concave in (x, y). Also the non-negativityconstraints x 0 and y 0 must hold as consumption cannot be negative.

Competition in this world is "perfect" or "pure" so consumers are price takers, i.e. px and py are fixedfor them. This may be justified by postulating a large number of consumers.

Since utility is increasing in both x and y we can safely assume that the budget constraint pxx +pyy Iis satisfied with equality. We will also assume that the non-negativity constraints are slack so that x > 0and y > 0. Setting the inequalities to equality, combining the two constraints, and rearranging a bit we getthe standard utility maximization problem (UMP for short)

maxx,y

U(x, y) s.t. pxx + pyy = I (UMP)

The solution to this problem is typically found by writing the Lagrangean

L (x,y,l) = U(x, y) + (Ipxxpyy)

and taking the first order conditions1

L

x =Uxd, yd

x dpx = 0L

y=

U

xd, yd

y dpy = 0

L

= Ipxxd pyyd = 0

The superscript "d" is used to refer to the fact that these solutions are the consumers demands.Solving yields the Lagrange multiplier d = d (px, py, I) and the demand functions

xd (px, py, I) yd (px, py, I)

To be more general we call these the uncompensated (or Marshallian or Walrasian) demand func-tions. These functions are "uncompensated" since price changes will cause utility changes: a situation thatdoes not occur with compensated demand curves. Substituting these solutions back into the utility function,

1 If the we did not assume the non-negativity constraints held the first two first order conditions would be

L

x=

UxD, yD

x

px < 0, with " = " ifxD > 0

L

y=

UxD, yD

y

py < 0, with " = " if yD > 0

Note that this is a simplification of the Kuhn-Tucker conditions described previously.

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the maximand, we get the actual utility achieved as a function of prices and income. This function is knownas the indirect utility function

V (px, py, I) U

xd (px, py, I) , yd (px, py, I)

(Indirect Utility Function)

This function says how much utility consumers are getting when they face prices (px, py) and have incomeI.

An interesting question is how much utility changes when either prices or income change. As shown inthe "Calculus and Optimization" hand-out we should expect that giving the person one dollar in incomeshould increase her utility by the Lagrange multiplier d

V

I= d

Taking the total derivative of Indirect Utility Function with respect to px yields

V

px=

U

x

xd

px+

U

y

yd

px(1)

This expression does not tell us much, however it can be simplified with the two useful substitutions. Firstthe the first two first order conditions tell us that

U

x= dpx and

U

y= dpy

so substituting these in to 1 we have

V

px= dpx

xd

px+ dpy

yd

px= d

px

xd

px+ py

yd

px

(2)

Second take the total derivative of the budget constraint pxxd + pyyd = I with respect to px to get

xd + pxxd

px+ py

yd

px= 0 px

xd

px+ py

yd

px= xd

Substituting this in to 2 we get a result known as Roys Identity

V

px= dxd (Roys Identity)

A basic (albeit somewhat flawed) intuition for this identity is straightforward: if px goes up by one dollarthen the consumer will lose xd number of dollars, which each have utility value d, so that utility dropsby dxd. The more correct intuition is that xd and yd are changing in response to the price increase, butbecause the consumer is optimizing the entire time and facing a budget constraint, these indirect effects sumup to zero. A similar result holds for y that

V

py= dyd

which you can prove for yourself. Note that if you were just given the indirect utility function you could solvefor xd, yd, and d, just be taking combining the derivatives of V appropriately, e.g. xd = (V/p) /d = (V/p) / (V/I).Example 1 Quasilinear utility takes a form where one of the goods consumed enters linearly into the

utility function, which in this case we take to be U(x, y) = x + h (y) where h0 (y) > 0 and h00 (y) < 0. (Thereader may like to try the case where h (x) =

x) The first order conditions for this problem simplify to

1 = dpx, h0

yd

= dpy, and pxxd + pyy

d = I . Solving yields

d = 1/px

yd = (h0)1

dpy

= (h0)

1

pypx

xd =Ipyyd

px=

I

px

pypx

(h0)

1

pypx

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Notice that yd does not depend on income I, only on relative prices py/px. Indirect utility is given by

V (px, py, I) = xd + h

yd

=I

px

pypx

(h0)

1

pypx

+ h

(h0)

1

pypx

It is fairly easy to see that V/I= 1/px = d

. Roys identity is more difficult to check

V

py= 1

px (h0)

1

pypx

pypx

1

h00 (py/px)

1

px+ h0

(h0)

1

pypx

1

h00 (py/px)

1

px

= 1px

(h0)1

pypx

pypx

1

h00 (py/px)

1

px+

pypx

1

h00 (py/px)

1

px

= 1px

(h0)1

pypx

= dyd

The reader is invited to check Roys Identity for themselves in the case of x.

2 Expenditure Minimization Problem

The consumer problem can be approached in a different way which produces some useful tools. Instead ofmaximizing utility given a certain income, imagine how much income it would take to achieve a certain levelof utility. In other words consider the following expenditure minimization problem (EMP for short),which as always take prices as given

minx,y

pxx + pyy s.t. U(x, y) u (EMP)

This problem looks very much like the UMP above except that the objective function and constraint havebeen switched around. We wish to minimize the income I = pxx +pyy needed achieve a fixed level of incomeu, for given prices (px, py). Our third parameter in parameter in this problem (after px and py) is no longerI, but u. This problem can typically be solved by writing the Lagrangean

L (x, y, ) = pxx + pyy + [u U(x, y)]

Assuming the constraint holds with equality, the first order conditions are

L

x= px c

U (xc, yc)

xc= 0

L

y= py c

U (xc, yc)

yc= 0

L

= u U(xc, yc) = 0

The first two FOC are quite similar to above replacing c with 1/d, but the third constraint corresponding

to the constraint is very different. Solving these three equations in the three unknowns yields the Lagrangemultiplier c = c (px, py, u), the shadow price in dollars of having to provide an extra unit of utility to thisconsumer, as well as the compensated demands

xc (px, py, u) yc (px, py, u)

which are a function now of the required utility u, not income I. Note here that even though utility staysthe same, quantities demanded will change as px and py change since the individual is trying to minimizeher expenditures on consumption. Levels of required income I are assumed to automatically adjust to letmake sure that individual can still achieve utility u, although not necessarily the bundle of goods previously

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consumed. The individual is fully compensated for changes in price which could otherwise affect her utilityif I were held fixed. Substituting in the solutions back into the objective function, the minimand, we getthe expenditure function

E(px, py, u) pxxc (px, py, u) + pyyc (px, py, u) (Expenditure Function)

which is precisely the amount I needed to maintain utility level u, for given prices px and py.As before we can differentiate E with respect to u to get E/u = . Taking the total derivative of E

with respect to px we getE

px= xc + px

xc

px+ py

yc

px(3)

We can simplify this expression with two substitutions. First we substitute in the first order conditionswhich imply, px =

cU/x and py = cU/y into 3 to get

E

px= xc + c

U

x

xc

px+ c

U

y

yc

px= xc + c

U

x

xc

px+

U

y

yc

px

(4)

Second we differentiate the constraint U(xc, yc) = u totally with respect to px to get

U

x

xc

px +U

y

yc

px

which implies that the second term in 4 is zero. This implies the result known as Shepards Lemma (theanalogue to Roys Identity) that

E

px= xc (Shepards Lemma)

Again the (somewhat misleading) intuition for this is clear. If px changes by a small amount then xc will

not change by very much and so the increased cost of consuming these units is precisely xc. The betterintuition is that there are changes in xc and yc, but because of optimizing behavior, the consumer avoidsspending any more than xc, although since she was optimizing before she cannot avoid spending any less.The case for y is identical

E

py= yc

Thus, given the expenditure function you can derive the compensated demands just be taking the partialderivatives of E.

The UMP and EMP are mathematically known as "dual" problems. What is a constraint in one isthe objective in the other and vice-versa. Because of this a useful identities relating compensated anduncompensated demands holds, namely

xd (px, py, I) = xc (px, py, V (px, py, I)) (ID1)

xc (px, py, u) = xd (px, py, E(px, py, u)) (ID2)

Similarly there are two identities relating the indirect utility function with the expenditure function

V (px, py, E(px, py, u)) = u (ID3)

E(px, py, V (px, py, I)) = I (ID4)

These identities can be proven formally (the proof is a little bit beyond the scope of this course), but a slowreading of them should show them to be quite intuitive. These identities can be used to simplify problemsolving. For instance, solving the UMP one gets xd, yd, d and V. Setting V (px, py, I) = u, and solvingfor I yields the expenditure function E which can then be differentiated to get xc, yc and c using ShepardsLemma. Similarly one can proceed from the EMP, and solve the equation E(px, py, u) = I for u to get V,which can then be used to get xd, yd, and d, using Roys Identity.

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Example 2 Continuing with quasilinear utility, the FOC imply that px = c, py =

ch0 (yc), and xc +h (yc) = u. Solving the system we get

c = px

yc = (h0)1

pypx

xc = u h

(h0)1

pypx

Notice that yc = yd, as neither involves either u or I, so that the identity holds trivially. The expenditurefunction is given by

E(px, py, u) = pxxc + pyy

c

= pxupxh

(h0)1

pypx

+ py (h

0)1

pypx

Solving for u in this equation will yield the indirect utility function derived above. Shepards Lemma canalso be verified rather similarly to how Roys Identity was verified.

3 The Envelope Theorem

The derivations of Roys Identity and Shepards Lemma, as well as the interpretation of the Lagrangemultipliers are all special cases of what is known as the envelope theorem. Stated generally, say we wish tosolve the maximization problem

maxx,y

f(x, y, ) s.t. g (x, y, ) c

where x and y are control variables and is a given parameter which effects f and/or g, but over which wedo not maximize over, i.e. it is given exogenously. The Lagrangean is then written as

L (x,y,,) = f(x,y,) + [c

g (x,y,)]

for which the first two first order conditions are given by2

L

x=

f (x, y, )

x g (x

, y, )

x= 0

L

y=

f (x, y, )

y g (x

, y, )

y= 0

Do not take the derivative with respect to , as is given exogenously. The solution to this problem, givenby x () and y (), depends on as different values of will imply different solutions.

Substituting the solutions into the function f gives the value function

F() = f [x () , y () , ]

2 Equivalent is the minimization problem

minx,y

f(x,y,) s.t. g (x,y,) c

as this is equivalent to the maximization problem

maxx,y

f(x,y,) s.t.g (x,y,) c

Writing out the Lagrangean one getsL (x,y,) = f(x,y,) + [g (x,y,) c]

which gives equivalent first order conditions as the negative signs cancel.

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which is the maximized value of f, which ultimately depends on . Taking the total derivative of F() weget

dF()

d=

f

x

dx

d+

f

y

dy

d+

f

(5)

The last term represents the direct effect of on f, while the first two terms represent the indirect effect of on f by changing x and y. This expression can be simplified in two steps. First substituting in the

first order conditions f/x = g/x and f/y = g/y into 5 yields

dF()

d=

g

x

dx

d+

g

y

dy

d+

f

=

g

x

dx

d+

g

y

dy

d

+

f

(6)

Second, differentiating the constraint g (x, y, ) = c totally with respect to yields

g

x

dx

+

g

y

dy

+

g

= 0 g

x

dx

+

g

y

dy

= g

which substituting into 6 and rearranging yields the envelope theorem

dF()

d=

f

g

=

L

(Envelope Theorem)

Therefore the change in the value function is given by the partial derivative of the Lagrangean with respectto - a helpful simplification.3 This covers Roys Identity, Shepards Lemma, and the interpretation of theLagrange multiple (check!).

3 In the unconstrained case or in the case where the constraint is slack ( = 0) is just a special case with

dF()

d=

f

The total effect is given by just the partial, direct effect seen in f.6

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