Econ 604 Advanced Microeconomics
Davis
Spring 2005
23 February 2006
Reading. Chapter 5 (pp. 128-150) for today
Chapter 6 (pp. 152-170) for next time
Problems: To collect: Ch. 5. 5.1 5.2 5.4
Next time: Ch. 5. 5.6, 5.7, 5.8, 5.9
Lecture #6
REVIEW
Comments on Homework: Many of you had difficulties with problem 4.6 (The problem where good Z was not optimally consumed with a budget of $2. Observe that in this problem, if you take first order conditions, the optimal quantity of z to consume is negative. Some of you switched the negative sign to positive. This is, of course, incorrect. The point is that you should be care to attend to such details when you work with a system. The fact that the optimal amount of Z is negative implies that some problem exists with the system.
Chapter 4:
V. Income and Substitution Effects. Analysis of demand.
A. Demand Functions
1. Homogeniety
B. Changes in Income
1 .Normal and Inferior Goods.
2. Engel’s Law (A statement about income effects on the food SHARE rather than quantity of food demanded)
C. Changes in the Price of a Good (with indifference curves).
1. Graphical Analysis – Price Reduction
2. Graphical Analysis – Price Increase
3. Effects of Price Changes for Inferior Goods.
D. Individual’s Demand Curve (Derive from Indifference Curves)
1. Shifts in the Demand Curve
E. Compensated Demand (Reconstruct from Indifference Curves)t
Example #5. Compensated demand functions. (I repeat this, because we will use the notion of indirect demand further in today’s lecture.) Consider the Cobb-Douglas utility function U(X,Y) = X.5Y.5. The demand functions for X and Y are given by
X* = I/2Px and Y* = I/2Py
The indirect utility function can be solved by inserting X* and Y* back into the utility function. This yields
Utility = V(I, Px, Py) = I/(2Px.5Py.5)
Solving this expression for I and substituting in to X* and Y* yields the compensated demand functions
X = VPy.5/Px.5 and Y = VPx.5/Py.5
Notice that even though Py did not enter into the uncompensated demand function for X it does enter into the compensated demand function. This example makes clear what is being held constant with the two demand forms. With uncompensated demand, expenditures are held constant, so a rise in the price of X causes a reduction in utility. With compensated demand utility V is held constant. When the price of X increases, expenditures must also be raised to keep utility constant. Of course, the price of Y will affect how this expenditure increase is spent.
PREVIEW
F. A Mathematical Development of Price Change Responses
1. Direct Approach
2. Indirect Approach
3. The substitution Effect
4. The income Effect
5. The Slutsky Equation
G. Revealed Preference and the Substitution Effect
1. Graphical Approach
2. Negativity of the Substitution Effect
3. Mathematical Generalization
H. Consumer Surplus
1. Consumer Welfare and Expenditure Functions
2. A Graphical Approach
3. Consumer Surplus
4. Welfare Changes and Marshallian Demand Curve
Lecture________________________________________________
F. A Mathematical Development of Price Change Responses. Now we go back to the income and substitution effects developed graphically last lecture, and develop these results analytically.
1. Direct Approach One way to separate out analytically income and substitution effects would be to start with our standard constrained optimization problem and solve for ¶dx/¶Px and then separate out income and substitution effects. That is, we would start with the problem
L = dx(Px, Py, I) + l(I - PxX - PyY)
and take FONC w. r. t. Px and I. Solving, we could develop expression for ¶dx/¶Px and ¶dx/¶I that we could use to develop a compensated demand function hx which incorporates the substation effect, but abstracts out the income effect.
In general, however this solution is rather cumbersome, and not very informative. It is instructive to take an indirect approach. This indirect approach has the advantage of allowing us to see how working with the dual to a problem can provide important insights.
2. Indirect Approach. Consider an expenditure function. Recall that the expenditure function reflects the minimum amount that must be spent by an individual in order to achieve a given level of utility U*.
minimum expenditure = E(Px, Py, U*)
Then, by definition at reference prices Px and Py, compensated demand function hx equals the normal (uncompensated) demand given a budget I equal to expenditures.
hx(Px, Py, U*) = dx(Px, Py, E(Px, Py, U*))
Now, we can isolate the (compensated) substitution effect simply by taking the partial derivative of hx with respect to Px.
¶hx/¶Px = ¶dx/¶Px + ¶dx/¶E · ¶E/¶Px
Rearranging
¶dx/¶Px = ¶hx/¶Px - ¶dx/¶E · ¶E/¶Px
The expression to the right of the equality reflects a combination of substitution and income effects associated with a price change on uncompensated demand. Further, the left of the two terms is the substitution effect. The sign on this term is negative.
The rightmost term reflects an income effect. Consider the sign of this term. ¶E/¶Px is clearly greater than zero, since a consumer must be compensated for a price increase in order to be indifferent between the new higher price and an old lower one. If a good is a normal good, ¶dx/¶E>0 as well. Thus, the sign on the expression is negative only because of the ‘– ‘sign appearing to the left of it. On reflection that should be appealing. For a normal good, a price increase reduces income which will cause the consumer to purchase less of the good.
5. The Slutsky Equation. The above decomposition can be stated a bit more clearly with some notational changes. First, rewrite the substitution effect
¶hx/¶Px = ¶X/¶Px|U* = constant
Next rewrite the income effect. First, observe that E and I refer to the same thing, and that compensated demand dx = an amount X consumed. Thus, ¶dx/¶E =¶X/¶I. Next, E/¶Px =X. (recall E(U*) = PxX+PyY). Thus,
-¶dx/¶E· ¶E/¶Px = -¶X/¶I· X
Combining, we write what is termed the Slutsky equation as
¶dx/¶Px = ¶X/¶Px|U* = constant - ¶X/¶I· X
Intuitively, this equation says that the effects of a $1 increase in the price of a good X can be divided into a substitution effect prompted by the lower relative price of other goods, and an income effect that arises because the consumer is poorer as a consequence of the price reduction.
Notice further, as we illustrated with indifference curves for the 2 good case, that when a good is normal, ¶X/¶I>0, so the income effect reinforces the substitution effect. On the other hand if a good is inferior ¶X/¶I<0 and the income effect damps the effect of substitution.
In the extreme, it is possible (at least as an analytical matter) that the income effect term dominates the substitution effect. In this case, the overall slope of the demand curve would be positive. Such goods are called Giffen Goods.
Example. Slutsky Decomposition with a Cobb Douglas Utility function
Perhaps the ideas of this section would be made more concrete with an example. Consider again the Utility Function U(X,Y) = X.5Y.5. Suppose further that PX = .25, PY = 1 and I = 2.
We have, in several past examples, derived the uncompensated demand functions for this function under these conditions as
X = dx(Px, Py, I) = I/(2Px)
and
Y = dy(Px, Py, I) = I/(2Py)
By the Slutsky equation
¶dx/¶Px = ¶X/¶Px|U* = constant - ¶X/¶I· X
Let us verify that the left and right sides are equivalent. On the left side,
¶dx/¶Px = -I/ 2Px2
To develop the first term on the right side, we need to find the indirect demand function for X. Given X and Y,
V = (I/(2Px)).5(I/(2Py)).5 = I/(2Px.5Py.5)
Thus
I = 2VPx.5Py.5
Substituting for I in the income constraint, and solving yields the expenditure function
2VPx.5Py.5 = I
Substituting I into the direct demand functions yields
X = hx(Px, Py, I) = 2VPx.5Py.5/(2Px) = VPy.5/Px.5
Thus,
¶hx(Px, Py, I)/ ¶Px = -VPy.5/(2Px15)
Using the indirect utility function again, we can express compensated demand in terms of income
-IPy.5/[(4Px15) Px.5Py.5] = -I/(4Px2)
To calculate the income effect, - ¶X/¶I· X, simply take the deriviative of dx w.r.t.
¶X/¶I = ¶dx(Px, Py, I)/ ¶I = 1/(2Px)
Thus, - ¶X/¶I· X = - X/2Px = -I/(4Px2)
Combining terms, we have
¶dx/¶Px = ¶X/¶Px|U* = constant - ¶X/¶I· X
-I/ 2Px2 = -I/(4Px2) -I/(4Px2) = -I/(2Px2)
Thus, for example, given the above utility function, I = 2, Py=1 and Px = .25, an increase in Px from .25 to 1 changes quantity demanded from 2/2(.25) = 4 to 2/2(1) = 1. The change from 4 to 2 is driven by the substitution effect. A change from 2 to 1 is the income effect (Note: How do you get these changes? .. By inserting parameter values into the indirect demand function.)
But observe, the fact that income shares are constant in this example, implies that income and substitution effects are equal. The effects of the price change in X on consumption and income are equal. For good Y these effects are also equal, but opposite in sign. Thus the amount of Y consumed does not change.
Prior to proceeding, consider again our assertion that ¶E/¶Px =X. Intuitively, this says that when the price of X increases by a dollar, $X extra dollars of expenditures are needed to maintain utility at the same level. We can demonstrate this formally by solving the dual problem to utility maximization
L = PxX + PYY +l(-U(X,Y))
Applying the envelope theorem to this problem
¶L/¶Px =¶E/¶Px = X
Notice, we are recovering in this way the demand function. This result called Shepard’s Lemma, is important in empirical work because it implies that one can find the demand function for a good simply by taking the derivative of the expenditure function. Since, by construction, the expenditure function holds Utility constant, this is a compensated demand curve.
To illustrate, recall that in the above example,
E = V(2Px.5Py.5),
Thus
¶E/¶Px = VPy.5 Px-.5
We’ll return to this in section I, below.
G. Revealed Preference and the Substitution Effect. The principal unambiguous prediction that can be derived from the utility maximization model is that the demand curve has a negative slope. The proof of this assertion requires a diminishing MRS (which makes necessary conditions for a maximum also sufficient). Some economists consider basing demand on unobservable utility considerations is undesirable. This section outlines an alterative approach.
1. Graphical Approach. In the panel to the left, suppose that with the budget line I1 allocation A is selected over B. We will say that A is revealed preferred to B, and the fact that A is preferred to B implies that B will only be selected when A is unaffordable (e.g, with budget line I3.) Similarly given more is preferred to less, bundle C must be revealed preferred A. This implies that B will be selected over C if C is unaffordable. (That is, at relative prices of Y less than those implied by I2.) Given an observed preference over alternative bundles A and B, we are able to rank a series of alternative bundles as well. This turns out to be a fairly powerful observation.
2. Negativity of the Substitution Effect. Using the principle of rationality, we can show why the substitution effect must be negative (or zero). Suppose an individual is indifferent between two bundles C (XC and YC) and D (XD and YD). Let PxC and PyC denote the prices at which C is chosen and PxD and PyD the prices at which D is chosen. Then when C is chosen, if the individual is indifferent, D must cost at least as much as C (and probably more). That is
PxC XC + PyC YC < PxC XD + PyC YD
Similarly, when D is chosen, C must cost at least as much as D (or probably more)
PxD XD + PyD YD < PxD XC + PyD YC
Rewriting
Pxc(Xc -XD) + PyC (YC - YD)<0 and
PxD(XD -XC) + PyD (YD- YC) <0
Adding together yields
(PxC- PxD)(XC -XD) + (PyC - PyD)(YC- YD)<0
Now, suppose that only the price of X changes; PyC = PyD. Thus
(PxC- PxD)(XC -XD) <0
But this expression states only that price and quantity must move in the opposite directions. (If the first term above is negative, the other must be positive, and vice versa). This is precisely a statement about the non-positive nature of the substitution effect.
¶X/¶Px|U* = constant<0
Notice, that we needed neither a utility function nor an assumption of diminishing MRS. In terms of the above graph, this simply says that if there exists a pair of bundles between which a consumer is indifferent, then it must be the case that prices which make one unaffordable must imply lower prices for some goods in the preferred bundle. Another bundle will be preferred only if the relative prices of those previously preferred goods increases.
3. Mathematical Generalization. Generalizing is straightforward. If at prices Pi0 bundle Xi0 is preferred to bundle Xi1 and bundle Xi1 is affordable, then
S Pi0 Xi0<S Pi0 Xi1
But prices when bundle Xi1 is bought, say Pi1, bundle Xi0 must be more expensive
S Pi1 Xi0>S Pi1 Xi1
We summarize this as follows:
Strong Axiom of Revealed Preference. If a commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is reveled preferred to bundle 2, and if bundle 2 is revealed preferred to bundle 3, …., and if bundle K-1 is reveled preferred to bundle K, then bundle K cannot be revealed preferred to bundle 0.
Most of the other properties we have developed can also be developed from the revealed preference axiom. For example, it is easy to show that demand functions are homogenous in prices and Income. As shown by Houthakker (1950) revealed preference and utility theory are equivalent conditions. The revealed preference approach is widely used in the construction of price indices.
H. Consumer Surplus. One important applied area in economics involves developing a monetary measure of the gains or losses individual experiences as a result of price changes. In order to make such calculations, we need a measure of the welfare consequences of a price change. This is the notion of consumer surplus
1. Consumer Welfare and Expenditure Functions. An easy way to develop the notion of consumer surplus is to consider again the notion of an expenditure function
expenditure = E(Px, Py, Uo)
where Uo is a target level of utility