Econ 604 Advanced Microeconomics



Econ 604 Advanced Microeconomics

Davis

Spring 2006

2 March 2006

Reading. Chapter 6 (pp. 152-170) for today

Problems: To collect: Ch. 5. 5.6, 5.7, 5.8, 5.9

For next time: Ch. 6. 6.1, 6.3, 6.5

Test next time.

Lecture #7

REVIEW

E. Compensated Demand

1. Relationship between Compensated and Uncompensated Demand

F. A Mathematical Development of Price Change Responses

1. The Slutsky Equation

2. The Substitution Effect

3. The Income Effect

G. A short introduction to revealed preference

H. Consumer Surplus

1. Consumer Welfare and Expenditure Functions

2. A Graphical Approach

3. Consumer Surplus

4. Welfare Changes and Marshallian Demand Curve

PREVIEW

I. Shephard’s Lemma, Roy’s Identity and Price Indices.

VI. Chapter 6. Demand Relationships Among Goods

A. An Overview. The Two-Good Case

B. Substitutes and Complements

C. Net Substitutes and Complements

D. Composite Commodities

E. Home Production Attributes of Goods and Implicit Prices

Lecture________________________________________________

I. Shephard’s Lemma, Roy’s Identity and Price Indices. In closing chapter 5, it is instructive to review the relationships between primary problems, dual problems, direct utility and indirect utility. There are a variety of ways to derive product demand, and we offer this brief review to clarify the relationships between the concepts.

1) Maximizing utility subject to an income constraint allows for the expression of an indirect utility function Max U s.t., I = PxX + PyY.

2) An expenditure function E* may be derived by the dual problem, that is minimizing expenditures subject to a utility level Min PxX + PyY.subject to U(X,Y) = U*

Now the uncompensated demand functions developed by the maximization problems in (1) allow us also to derive indirect utility by inserting the demand functions (expressed in terms of prices and incomes) back into the utility function. Our standard method of finding the expenditure function is simply to solve the resulting expression for I (expressed in terms of U*, and prices). This is equivalent to solving the minimization problem in (2)

3) Given indirect utility and expenditure functions we can use the envelope theorem to express uncompensated and compensated demand, respectively.

In the case of an Expenditure function, we can derive compensated demand as the partial derivative of the expenditure function w.r.t. Px.

In the case of an indirect utility function, demand dx is the ratio of the partial derivative of indirect utility with respect to Px, divided by the partial derivative of indirect utilty w.r.t. I.

Consider these applications of the envelope theorem, in turn.

1. Deriving compensated demand with the Expenditure function. The expenditure function is derived by minimizing expenditures subject to a utility level, e.g.,

L = PXX + PYY + ((U* - U(X, Y))

The Envelope theorem states that at optimal values of X and Y,

(L/(PX = (E/(PX = X|U*

this application of the Envelope Theorem is known as Shephard’s Lemma.

2. Deriving uncompensated demand from the indirect utility function. In the standard maximization problem, the objective is

L = U(X, Y) + ((I - PXX - PYY)

Again, the Envelope theorem status that at optimal values of X and Y,

( L/(PX = (U/(PX = (X

and ( L/(I = (U/(I = (

Thus, (U/(PX = X

(U/(I

This application of the Envelope Theorem, is known as Roy’s Identity

Example. These relationships can be illustrated with our standard Cobb-Douglass function

U(X,Y) = X.5Y.5

Recall that indirect utility in this case is

V(PX, PY, I) = I/[2PX.5PY.5]

Thus, (U/(PX = I/[4PX1.5PY.5] = 2IPX.5PY.5 = I

(U/(I 1/[2PX.5PY.5] 4PX1.5PY.5 2PX

our well-known expression for uncompensated demand.

Similarly, we derived the expenditure function as

E = 2VPX.5PY.5

Applying Shephard’s lemma,

(E/(PX = V(PY/PX).5

again, the expression we have developed previously as our compensated utility function.

Importantly, these different concepts allow us alternative ways to estimate demand. Consider for example, the problem of constructing price indices. When specific components of a price index increase separately, we overstate the loss in well being if we compensate consumers with a simple increase in expenditures that offsets the effects of the single-good price increase on the total bundle (as would be suggested by an uncompensated demand function). This is the case because consumers may respond to the increase in the price of a single good by substituting away toward other goods. The appropriate way to develop the income increase necessary to compensate for an increase in the price of a single good is compensated demand.

Some economists estimate that the failure to appropriately adjust for substitution effects results that the inflation rate is .2 points higher than it actually is each year.

VI. Chapter 6. Demand Relationships Among Goods. In chapter 5 we considered the effects of a change in the price of a particular good on consumption. In this short chapter we consider the effects of changes in the price of related goods.

A. An Overview. The Two-Good Case. It is convenient to start with a two good case, as we have in other topics. The two panels below illustrate some possible consequences on the change in the consumption of a good X brought about by a reduction in the price of a good Y,

[pic]

Notice in both panels that the amount of Y consumed increases with the price reduction (from Yo to Y1). However, the consequences for good X differ. In the panel shown on the left, there is relatively little substitutability between the goods. As a consequence the consumer took advantage of the price reduction to increase consumption of good X. In contrast, as suggested by the very flat indifference curves in the panel on the right, when the price of Y fell, the consumer increased consumption of Y for two reasons. First, because of an income effect, and second, because the goods are close substitutes. As a consequence of the price reduction, consumers shifted away from X and toward the now-cheaper substitute Y.

The ambiguous effects of a price reduction in a related good can be illustrated with a Slutsky-type decomposition of uncompensated demand.

(X/(PY = (dX/(PY = (X/(PY|U* - Y((X/(I)

Notice that the first term on the right is a substitution effect, and reflects the tradeoffs between the related good and X. The second term on the right is an income term. It is the product of the (now cheaper) units of Y purchased, multiplied by the extent to which X moves with an increase in Income.

In a two good world, these terms have opposite signs. Assuming that indifference curves are convex, the goods are substitutes, and so (X/(PY >0. However, if the good is a normal good, the second term is negative, since both Y>0 and ((X/(I)>0. Thus the net effect is ambiguous. The effects of an increase in PY are positive or negative depending on whether or not the substitution effect dominates the income effect.

Example 6.1. Consider again the Cobb-Douglas function

U(X,Y) = X.5Y.5

We learned previously that

dx(PX, PY, I) = I/2PX

and that

hx(PX, PY, I) = VPY.5/ PX.5

It is obvious from the uncompensated demand function that

(dx/(PY = 0.

Let us decompose this net effect into substitution and income effects. Note first that

(hx/(PY = V/[2(PYPX).5]

But, recall our indirect utility is given by

V - I/[2PX.5PY.5]

Thus

(hx/(PY = (X/(PY|U* = I/[4PXPY]

To derive the income effect observe that

(dx(PX, PY, I)/ (I = 1/2PX

Thus -Y(dx/ (I = -I/[4PYPX]

Combining terms

(X/(PY = (dX/(PY = (X/(PY|U* - Y((X/(I)

0 = I/[4PXPY] - I/[4PXPY]

the desired result. Interestingly, observe that PY is left out of the uncompensated demand function not because PY exerts no influence on X, but rather because the income and substitution effects just cancel each other out in this case.

B. Substitutes and Complements. For the case of many goods, the Slutsky analysis for cross-price effects mentioned above generalizes to the following expression, for an too goods i and j

(Xi/(Pj = (di/(Pj = (Xi/(Pj|U* - Xj((Xi/(I)

Generally speaking, goods are substitutes if a consumer responds to an increase in the price of a good j by increasing consumption of good i. Goods are complements if a consumer responds to an increase in the price of a good j by decreasing consumption of good i.

There are two ways to make this intuitive relationship precise. One is to combine substitution and income effects together. Goods are said to be Gross Substitutes if

(Xi/(Pj = (di/(Pj > 0

They are Gross Complements if

(Xi/(Pj = (di/(Pj < 0.

The notion of gross substitutes and gross complements have some undesirable features. The most important of these is that they are asymmetric. That is, X may be a gross substitute for Y and, at the same time Y may be a gross complement for X. This would arise if the substitution effect of an increase in PY dominated the income effect on X, and, if the income effect of an increase in PX dominated the substitution effect on good Y.

The following example illustrates

Example 6.2

Let U(X,Y) = ln X + Y

Then the Lagangrian becomes

L = ln X + Y + ((I – PxX – PyY)

From FONC we have

1/X = (PX ; 1 = (PY.

Taking the ratio of these terms

X = PY/PX

Inserting into the budget constraint renders the demand functions

I = PY + PYY

= (1 + Y)PY

or

PYY = I- PY

Thus, if PY increases the budget share spent on Y (that is, PYY) must decrease. Thus, expenditures on X increase, implying that X and Y are gross substitutes. However, solving the above for Y, note that

Y = [I- PY ]/ PY

So

(Y/(PX = 0, implying that the goods are unrelated.

C. Net Substitutes and Complements. An alternative definition circumvents this problem with asymmetries. The definition relies on compensated demand effects. Specifically, if

(Xi/(Pj|U* > 0 goods i and j are said to be net substitutes, and if

(Xi/(Pj|U* < 0 goods i and j are said to be net complements.

Where U* is a constant.

These terms are sometimes called “Hicksian” substitutes and complements. Symmetry exists in these definitions by simple application of Young’s Theorem, that is

(Xi/(Pj|U* = (Xj/(Pi|U*

[Recall from Shephard’s Lemma, Xi = (E/(Pi. Thus, (Xi/(Pj = (2E/[(Pi(Pj], which would render the same result were the order of differentiation reversed]

D. Composite Commodities. To this point, our discussion of price effects has been confined to the effects of a price change on a single related good. More generally, a price change affects all other goods available to a consumer, and the analysis quickly becomes overwhelming. (If there are n goods, there are (n+1)n/2 possible substitution effects. To simplify the analysis, it is conventional to aggregate goods, and consider the effects of price changes using some good X and a composite “other goods” Y. This section considers the assumptions implicit in such an analysis.

Consider an economy that consists of n goods, X1, and n-1 other goods, X2, to Xn If X1 is available for price P, and the other goods are available for prices Po2 …Pon . Write the total expenditure on other goods as Y

Y = Po2X2+ …+Pon Xn .

Thus, the budget constraint is

I = P1X1 + Y

Now consider the effect of a constant percentage markup in all other goods Y. Denote this markup as t.

I = P1X1 + tY

We can analyze the effects of changes in t just as we did any of the other “price effects” previously. Note the following

a) The analysis does require that all other goods increase by the same relative amount

b) This analysis does not require that the change in consumption of other goods X2 … Xn be equivalent. What is necessary is that the MUi/Pi be equivalent for all goods.

This leads to the following definition

Composite Commodity: A composite commodity is a group of goods for which all prices move together. These goods can be treated as a single “commodity” in that the individual behaves as if he or she were choosing between other goods and total spending on the entire composite group.

This definition and the Composite Commodity Theorem, this result, which makes it possible to aggregate goods, allows the simplification of a number of very complex problems into a two dimensional space.

Importantly, however, the general result that price changes for n-1 other goods may be analyzed as a single price change requires strict proportionality in price increase. Even slight departures from strict proportionality may negate the composite commodity theorem if cross- substitution effects are large.

Example: Housing Costs as a Composite Commodity. Suppose that an individual receives utility from 3 goods; food (X), housing services (square feet Y) and household services (energy use Z). Suppose utility takes the CES form

Utility = U(X,Y,Z) = -1/X -1/Y -1/Z

Using the Lagrangian Technique demand functions for the three goods are

X = I/[PX+ ( PXPY).5 + ( PXPZ).5]

Y = I/[PY+ ( PYPX).5 + ( PYPZ).5]

Z = I/[PZ+ ( PZPX).5 + ( PZPY).5]

Letl I = 100, PX = 1, PY = 4 and PZ = 1 then optimal levels of the goods become

X* = 25

Y* = 12.5

Z* = 25

Hence 25 (1) = 25 is spent on food, and 12.5*4 + 25*1 = 75 is spent on housing-related needs.

Let’s assume that housing services and housing operations move together,

Define a composite commodity H = 4Y+1Z. Here define the initial PH =1 (arbitrarily) The initial quantity of housing is simply the total dollars spent on housing

H = 4(12.5) + 1(25) = 75

Further, since all price increases are by assumption proportional, the following price relationship holds

PH = PZ = .25PY

Now we can rewrite the demand function for X as

X = I/[PX+ ( 4PXPH).5 + ( PXPH).5]

= I/[PX+ 3(PXPH).5]

As before, I = 100, PX = 1 and PH = 1 so X* = 25. Housing is most easily calculated from the budget constraint as everything but food. H = 75.

An increase in housing costs. Suppose the prices of Y and Z increase proportionally to PY = 16 and PZ = 4. Then PH also rises proportionally to PH =4.

Then demand for X would fall to

X = 100/[1+ 3(4).5]

= 100/7

Housing purchases are given by

PHH = 100 - 100/7 = 600/7

Or, with PH = 4

H* = 150/7

This is exactly the level of housing purchases that arises from inserting PX = 4 and PY = 16 into the original demand equations

X* = 100/7

Y* = 100/28

Z* = 100/14

So the total amount of “housing” consumed is

H* = 4Y* + 1Z* = 150/7

Hence, we obtain the same responses regardless of whether we use the composite approach, or look at proportional changes in the housing related goods separately.

E. Home Production Attributes of Goods and Implicit Prices.

To this point, we have skirted the issue of why goods are complements and substitutes. In closing this chapter, we outline briefly some models that economists have developed to gain insight into these issues.

1. Household Production Model. Assume that individuals do not derive utility directly from goods they purchase in the market, but rather, only from those goods, used as inputs, along with time. Raw hamburger, doesn’t provide utility, for example, rather raw hamburger, along with cooking and cleanup time causes utility.

This problem can be analyzed by considering these goods as a household production function. Specifically. Consider three goods, X, Y and Z, each of which can be purchased in the market. Utility comes from a combination of these goods. For example, suppose

U(a1, a2) where

a1 = f1(X, Y, Z)

and

a2 = f2(X, Y, Z)

As before, the individual maximizes utility subject to the standard budget constraint

PXX + PYY + PZZ = I

We won’t derive results here in detail, but two interesting observations are possible from the structure of the problem.

First, such a model clarifies the relationship between market goods. Essentially, the production relationships a1 and a2 are, at least in principle observable. Thus, this approach allows us to treat households as multi-product firms

Second, and more importantly, this setup provides a framework for establishing “implicit” or “shadow” prices of home-produced goods a1 and a2. We illustrate this with a very simple variant of the home production model, that of linear attributes.

2. The linear attributes model K. J. Lancaster developed a particularly simple variant of the household production model to examine the attributes of goods that generate utility for a consumer. Consider, for example food consumption. Rather than considering separately the utlity of a very wide array of foods, consider the attributes of the different foods, for examples, calories (a1) and vitamins (a2) that the foods provide.

In simple form, suppose that aX1 denotes the caloric content of food X. Then we may write.

a1 = aX1X+ aY1Y+ aZ1Z

a2 = aX2X+ aY2Y+ aZ2Z

Now consider the amounts of Calories and vitamins available from each of the goods. If a person consumed only X, then their budget would be

I = pXX so X = I/pX.

Thus, caloric intake becomes a1*= aX1(I/pX) and vitamins becomes a2*= aX2(I/pX), as illustrated at point X* along ray X in the figure below. Similarly for points Y* and Z* along the respective rays Y and Z.

Now the ray X* to Z* illustrates combinations of a1 and a2 that may be purchased with budget I and combinations of X and Z. Similarly for the cords X and Y and Y and Z.

One thing immediately obvious from the figure is that rational consumers would never consume all three goods. Rather, individual with a preference for a1 would consume Y and Z, while consumers with a preference for a2 would consume X and Y. Thus the attributes model predicts the following

a. Consumers will purchase fewer goods than the maximum available (particularly if there are fewer attributes than goods)

b. Second if preferences or prices change, consumption patterns may shift abruptly. This is interesting in that, as is observed with households, discontinuous shifts in consumption bundles occur in response to small price changes. A small increase in wages for females, for example may cause women with children to leave the house and work full time (rather than part time) because the attributes of work include may include social relations and the satisfaction associated with maintaining out of the house responsibilities in addition to income.

Appendix: Separable Utility and the Grouping of Goods. One shortcoming of general utility theory is that it says relatively little about demand relationships between goods. Other than the result that net substitution effects are symmetric, virtually any type of relationship is possible. Stronger results can be obtained only by placing more restrictions on the utility function. One particularly useful restriction is the assumption of separability, or an assumption that consumption decisions about one good (or group of goods) do not affect the utility associated with consuming other goods. For example, in the public goods literature it is typically assumed that the amount consumers decide to give to charity does not affect directly the utility of other goods derive from other goods.

Simple Separability. Consider the following simple case. Suppose an individual consumes three goods, X1, X2 and X3, and that his or her utility is additively separable

U(X1, X2, X3) = U(X1) + U(X2) + U(X3)

Where Ui’>0 and Ui” ................
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