Econ 604 Advanced Microeconomics



Econ 604 Advanced Microeconomics

Davis Spring 2005, March 24

Lecture 8

Return Examinations

Reading. Chapter 7 (pp. 172-194)

Next time Chapter 8 (pp. 198-224 Problems: 6.9 7.1, 7.3; 7.5; 7.9

REVIEW

I. Shephard’s Lemma, Roy’s Identity and Price Indices. Recall that the one can get uncompensated demand from the maximization problem by Roy’s Identity. One can get compensated demand from the expenditure function via Shepard’s Lemma. Both are applications of the envelope theorem.

Shepard’s Lemma (E(Px,Py,V)/ ( Px = hx (Px,Py,V)

Roy’s Identity dx(Px, Py, I) (U/(Px /(U/(I

Finally, constructing price indices on the bases of compensated rather than uncompensated demand would result in a downward adjustment of cost of living indices.

VI. Chapter 6. Demand Relationships Among Goods. We considered the effects of increases in the price of one good on the demand for another. In the 2 good case, we showed that income and substitution effects again came into play.

A. The two good case. Goods are gross substitutes if (dX/(PY>0 and Gross Complements if (dX/(PY0 and net complements if (hX/(PY0 and Ui” Price Box | |Qty Box = Price Box | |Qty Box < Price Box |

| |Elastic Segment | |Unitary Elastic Segment | |Inelastic Segment |

In the leftmost panel, observe that when price changes, the effects on total revenue can be divided into a “price box” and a “quantity box”. In the case of a price reduction, for example, the price box is the revenues lost from units that would have sold at the higher price (Dimension (PQ). The quantity box denotes the extra revenues realized from lower the price (Dimension (QP). The left panel illustrates a situation where TR moves inversely with the price change. This is an elastic segment of the demand curve (recall |eQ,P| = |(Quantity box)/ (Price Box)| = |(QP /(QP| >1). People are price sensitive in the sense that total revenue increases when price falls.

The right most panel illustrates an inelastic segment (|eQ,P| = |(Quantity box)/ (Price Box)| = |(QP /(QP| 1 goods are luxury goods or cyclical normal goods (e.g., automobiles)

0 < eQ,I < 1 goods are normal goods (e.g., food)

eQ,I < 0 goods are inferior goods

Notice: Recall here the discussion at the outset of this chapter. We can only calculate income elasticities across individuals if we invoke some assumption about the way that a change in income is distributed across market participants. (We might, for example, and if reasonable, assume that all participants realize the same percentage change incomes. See, for example problem 7.1)

4. Cross Price Elasticity of Demand Another standard elasticity deals with the response of one good to the change in the price of a related good. This is termed cross price elasticity

eQ,P’ = percentage change in Q = (Q/Q = (Q(P’

percentage change in P’ (P’/P’ (P’(Q

As with income elasticities, cross price elasticities can be positive or negative. The sign is important.

eQ,P’>0 implies goods are substitutes

eQ,P’0.

f(tX1, tX2, …. , tXn) = tmf(X1, X2, …. , Xn).

Thus a function that is homogenous of degree zero implies

f(tX1, tX2, …. , tXn) = f(X1, X2, …. , Xn).

Euler’s theorem states that a function that is homogeneous of degree m, then

f1X1+ f2X2 +… fnXn = mf(X1, X2, …. , Xn)

Example: f(X,Y) = 10X + 20Y. Notice that increasing X and Y by k inflates the function by k.

k10X + k20Y = k(10X + 20Y) = kf(X,Y).

Thus, this function is homogenous of degree 1. Trivially,

fxX + fyY = 10X + 20Y = f(X,Y)

Example: Demand relationships are typically homogeneous of degree zero. Recall, for example, in the case of a Cobb-Douglas utility function in X and Y

dx = X = I/2Px,

dy = Y = I/2Py

Increase I and Px by k will recover not affect demand. So this function is homogeneous of degree 0. (Note, you can take the quantity-weighted derivatives of dx or dy and you will find that they sum to zero)

When m=0, then the Euler’s theorem states that the sum of the quantity weighted first derivatives equals zero. Consider a demand function X = dx(Px, Py, I)

By Euler’s Theorem+

((X/(PX) PX + ((X/(PY)PY +((X/(I)I = 0

Convert to elasticities by dividing by X,

((X/(PX) PX /X+ ((X/(PY)PY/X +((X/(I)I/X = 0/X

eX,Px + e X,Py + e X,,I = 0

This is another way to state the homogeniety of degree zero property of demand functions. An equal percentage change in all prices and incomes will leave the quantity demanded of X unchanged.

Example: Cobb-Douglas Elasticities Consider the Cobb Douglas demand function

U(X,Y) = X(Y( where ( + ( = 1.

Demand functions are

X = (I/PX Y = (I/PY

The elasticities are easy to calculate. For example,

eX,Px = ((X/(PX) PX /X = (-(I/PX2)( PX /X)

= (-(I/PX2)( PX2 /(I )

= -1

Similarly

eX,I = 1

eX,Py = 0

eY,Py = -1

eY,I = 1

eY,Px = 0

Hence these demand functions have elementary elasticity values. Further

sX = PXX/I = PX(I/PXI = (

sY = PYY/I = (

The constancy of income shares provides another way of shown the unitary elasticity of demand.

Homogeneity holds trivially,

eX,Px + e X,Py + e X,,I = 0

-1 + 0 + -1 = 0

Finally, consider the elasticity version of the Slutsky equation.

eXP = eSXP - sx eXI

-1 = eSXP - ((1)

Thus

eSXP = -(1 - () = -(

In words, the compensated price elasticity of demand for one good is the income share for the other good. This is a special case of the more general result that

eSXP = -(1 - sx)(

where ( is the elasticity of substitution (between two goods) in chapter 3 (note 6). (To see this, note that in the case of a Cobb Douglas function (=1.)

D. Types of Demand Curves. Economists consider various types of demand forms. Here in closing we consider some of the problems associated with two of these functions.

1. Linear Demand Consider a demand function of the form

Q = a + bP + cI + dP’

Where a, b, c and d are demand parameters, and .

b0 (the good is a normal good)

d< > 0 if the related good is a gross substitute or a gross complement.

Holding I and P’ constant

Q = a’ + bP

Where a’ = a + cI + dP’. Clearly this describes a linear demand curve. Further, changes in a’ will shift demand. Despite the simplicity of this demand statement, linear demand has the deficiency that elasticity changes as one moves along the demand function. To see this notice that

eX,Px = ((X/(PX) PX /X = bP/Q

Obviously as P rises Q falls, and demand becomes more elastic.

Example: Linear Demand. Consider a demand function

Q = 36 – 3P.

Price elasticity of demand is

eX,Px = -3P/Q = -3P/(36 – 3P)

Notice demand is unit elastic when P = 6. For P>6 demand is elastic. For P0, b0 (a normal good) and d0, as for the linear good. Notice that one can easily “linearize” such a function by taking natural logarithms (ln)

lnQ = lna + blnP + clnI + dlnP’.

Notice that one can estimate the parameters of such a function with ordinary least squares. Notice also that

eQ,P = ((Q/(P) P /Q = b aPb-1IcP’d P/(aPbIcP’d)

= b

Thus, the price elasticity of demand is constant.

Example: Elasticities, Exponents and Logarithms. Notice in the above example that income and cross price elasticities are also directly read from the exponents of the demand functions

eQ,I = c ; eQ,P’ = d

Therefore, from a linear regression, one can read elasticities without having to make any mathematical computations. For example, if one estimated

lnQ = 4.61 - 1.5lnP + .5ln(I) + ln(P’)

We know that

eQ,P = -1.5 ; eQ,I = .5

and eQ,P’ = 1

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