Econ 604 Advanced Microeconomics



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) + ((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 ................
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