Econ 604 Advanced Microeconomics



Econ 604 Advanced Microeconomics

Davis Spring 2004, March 23

Lecture 9

Reading. Chaper 7 (pp. 172-194)

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

REVIEW

E. Home Production Attributes of Goods and Implicit Prices. We outline briefly some models that economists have developed to gain insight into the question of why goods are substitutes or complements

1. Household Production Model. “Inputs” generate utility when combined with other household resources. This allows establishment of implicit prices for nonmarketed goods.

2. The linear attributes model A way to convert many heterogeneous products into a narrower dimensionality. For example, steak, eggs, tunafish, salmon and milk, all provide proteins and vitamins. We could identify demand for proteins in this way, as well as an implicit price for proteins. One interesting insight of this analysis is that consumers may be expected to frequently make “lumpy” changes in consumption bundles.

Appendix: Separable Utility and the Grouping of Goods. Finally, we considered the theoretically predicted effects of some stronger assumptions about the nature of substitution between products. In particular we considered Separable utility. Simple separable utility allows us to impose a conditions that all goods are gross substitutes or gross complements. Also group separability allowed, among other things for two stage budgeting, that is consumers might first decide on their expenditures on various classes of goods, and then they could consider allocations within classes.

PREVIEW

Tonight we consider two final components of standard demand analysis; (a) Converting individual demand curves into a market demand curve and (b) elasticities. Specifically, we proceed as follows

VIII. Chapter 8 Market Demand and Elasticitity

A. Market Demand Curves

1. The two consumer case

2. The n consumer case

B. Elasticity

1. Motivation and a general definition

2. Price Elasticity of Demand

3. Income Elasticity of Demand

4. Cross Price Elasticity of Demand

C. Relationships Between Elasticities

1. Sum of income Elasticities for all Goods

2. Slutsky Equation in Elasticities

3. Homogeneity

D. Types of Demand Curves

1. Linear Demand

2. Constant Demand Elasticty

Lecture________________________________________________

VIII. Chapter 8 Market Demand and Elasticitity. We have considered in some detail price and quantity effects for a particular consumer. Suppose now we consider the effects of aggregating across consumers. We also devote some attention to the elasticity measures very widely used in empirical work

A. Market Demand Curves

1. The two consumer case. Consider an economy that consists of two consumers (person 1 and person 2). Their individual (uncompensated) demand curves for a good X may be written as

X1 = d1x(PX, PY, I1) and

X2 = d2x(PX, PY, I2)

Market demand is simply the sum of individual demands for good X. Thus

Total X = X1 + X2 = d1x(PX, PY, I1) + d2x(PX, PY, I2)

= D(PX, PY, I1,I2)

Observations:

- If each individual’s demand curve for good X (holding PY,I1 and I2 fixed) is downsloping, market demand for good X will be downsloping as well.

- Market demand here is created from uncompensated individual demands. Compensated Market demand could be constructed in the same way.

Graphically, market demand is simply the horizontal summation of individual demands (e.g., a

|P | |P | |P | |

| | | | | | |

| | | | | | |

|P1 | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | X1 | | X2 | | X1 + X2 |

| |Individual 1 | |Individual 2 | |Market |

Factors that shift individual demands would generally shift market demand in a similar manner.

- A change in the price of a related good can affect all individual demands uniformly.

- Income effects, however, are a bit more complicated, since incomes can change differently for different individuals. The way the income changes can affect demand. (This is often overlooked)

Example:

Consider two consumers with the following simple linear demand curves for Oranges

X1 = 10 - 2PX + .1I1 + .5PY

X2 = 17 - PX + .05I1 + .5PY

Where

PX = Price of oranges

Ii = Individual i’s income (in thousands of dollars)

PY = Price of Grapefruit (a gross substitute for Oranges)

Market Demand becomes

DX = X1 + X2 = 27 - 3PX + .1I1 + .05I2 + PY

(Notice that we can sum across PX and PY, assuming the law of one price. On the other hand, we cannot sum across individual incomes.)

To graph DX, we nee values for the variables other than own price. Let I1 = 40, I2 = 20 and PY = 4. Then

DX = 27 - 3PX + .1(40) + .05(20) + 4

= 27 + 9 - 3PX

= 36 - 3 PX

If the price of grapefruit were to rise to $6, then demand would shift out to

DX = 27 - 3PX + .1(40) + .05(20) + 6

= 27 + 11 - 3PX

= 39 - 3 PX

On the other hand, setting PY = 4 again, impose a redistributive income tax that takes 10 from 1 and transferrs it to 2. The following results:

DX = 27 - 3PX + .1(30) + .05(30) + 4

= 27 + 8.5 - 3PX

= 35.5 - 3 PX

Notice that none of these changes affects the market coefficient on own prices.

2. The n consumer case This simple analysis with two consumers extends readily to the case of n consumers. Given a representative consumer j with demand for good i

Xij = dij(P1, … , Pn, Ij)

Then the market demand for m consumers would be

Xi = ( Xij = Ddi(P1, … , Pn, I1, I2,…,In )

B. Elasticity

1. Motivation and a general definition As we have seen, economists are often interested in the way that one variable A affects another variable B. Economists, for example, are often interested in the way that changes in various prices affect the quantity demanded of a good. An important problem with such comparisons is that the variables are not measured in readily comparable terms (for example, steak is measured in pounds, the price is in dollars. Oranges may be sold by the dozen, and price changes may be on the order of dimes.

One coherent way to address these different units of measure is to denominate all these changes in percentage terms. This of elasticity

eB,A = percentage change in B = (B/B = (B(A

percentage change in A (A/A (A(B

Notice from the definition that elasticity is a(n inverse) slope coefficient, weighted by a location. We know, for example, that apple consumption will decrease with an increase in the price of apples. However, the location allows us to speak more meaningfully of the magnitude of the response.

2. Price Elasticity of Demand Perhaps the most important elasticity measure is own price elasticity, or the responsiveness of changes in a price on the quantity of that good consumed.

a. Definition

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

percentage change in P (P/P (P(Q

Barring a Giffen Good relationship, own price elasticity measures are always negative numbers (since (Q/(P 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

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)

When m=0, then the Euler’s theorem states that the sum of the quantity weighted first derivatives equals zero.

Now, 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 elastcity 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 special case of the more general result that

eSXP = -(1 - sx)(

where ( is the elasticity of substitution in chapter 3 (note 6).

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 normal)

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 l 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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