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Econ 604 Advanced Microeconomics Davis Spring 2005 16 February 2006 Reading. Chapter 5 (pp. 116-130) for today Chapter 5 (pp. 130-144) for next time Problems: To collect: Ch. 4: 4.2, 4.4, 4.6, 4.7 4.9 Next time: Ch. 5 5.1 5.2 5.4 Observe: 5.1 and 5.2 don’t fit into the standard Lagrangian setup, because, in each case, the products are perfect substitutes. Below a given price level, the consumer will devote all income to only one product 5.1 is a case of perfect substitutes. 5.2 pertains to perfect complements. Lecture #5 REVIEW IV. Utility Maximization and Choice A. An Introductory Illustration. The two good case. (This is largely a graphical representation) 1. The Budget Constraint 2. First Order Conditions for a Maximum Px/Py = Ux/Uy, which implies Ux/Px = Uy/Py Intuitively: Purchase goods until the MU per dollar spent on each good is the same. 3. Second Order Conditions for a Maximum. The intersection of first order conditions will be a maximum if the quasi-convexity conditions are satisfied (that is, for any order pairs on an indifference curve U, (xo, yo), (x1, y1), it is the case that U(xo + (1-)x1, yo + (1-)y1)> U This condition will be satisfied if the quasi-concavity condition in 2.107 is met. It will alternatively be satisfied if d2Y/dX2.>0 Example, Suppose U* = XY. Differentiating, dU = YdX + XdY. On a level curve, this expression equals zero, so dY/dX = -Y/X 1 Now suppose that you wanted to check the concavity of this level curve. What if we just took the derivative of the first order condition w.r.t. X? That is, d2Y/dX2 = Y/X2 > 0. Does this imply convexity in XY space? No! We must recognize that Y is implicitly a function of X. Rather, we have two (ultimately equivalent ways) to check for the concavity of the indifference curve. One is to totally differentiate U twice, subject to the constraint that utility is constant. We developed the sufficient condition for concavity of the constrained utility function in equation 2.107 [U22U11 -2 U1U2U12 + U12 U22]/ U22 <0 For U = XY, U1 = Y, U2 = X, U11 = 0, U22 = 0, U12 = 1 Inserting, -2YX/X3 = -2Y/X2 <0 (Note the sign is reversed relative to the standard second order condition. This is due to the way 2.107 is set up) The alternative approach is to solve the expression YX = U directly for Y and then differentiate twice. For example, let U = U* a constant. Then Y = U*/X dY/dX = -U*/X2 (with U* =XY, dY/dX = -Y/X) d2Y/dX2 = 2U*/X3 = 2Y/X2 <0 4. Corner Solutions B. The n-Good Case 1. First Order Conditions. For n goods, an optimizing consumer purchases all products so that the MU per dollar spent is equal for all goods. 2. Implications of First Order Conditions 3. Interpreting the LaGrangian Multiplier C. Indirect Utility Function We can express U(x1, x2,….xn) +(I –p x1 - …- p xn) as U(x1(p1…pn,I),……. xn(p1…pn,I) or V((p1…pn,I) The advantage of this is that indirect utility is expressed in terms of observables. Thus, given, of course our assumptions about the functional form of utility) we can talk about change in utilities in terms of changes in observable variables. We used the example of analyzing the effects of a lump sum tax to analyze this last time. D. Expenditure Minimization Associated with any utility maximization problem is a ‘dual’ problem where the agent minimizes expenditures needed to achieve a given level of utility. That is E = p1x1 + p2x2 + …pnxn Subject to U = U(x1,x2,….,xn) L = p1x1+p2x2 +…. + pnxn + ( U - U(x1, x2, …., xn)) This is important from a practical level because the dual yields exactly the same first order conditions as the original problem, and because it allows us to assess the price/income tradeoffs necessary to maintain a status quo level of utility. For example, given an assumption about the nature of utility, as well as prices and incomes, we can compute the change in, say, income necessary to offset the effects of a price increase. As we will see later, expenditures are directly observable, while utilities are not. PREVIEW 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 C. Changes in the Price of a Good. 1. Graphical Analysis – Price Reduction 2. Graphical Analysis – Price Increase 3. Effects of Price Changes for Inferior Goods. D. Individual’s Demand Curve 1. Shifts in the Demand Curve E. Compensated Demand 1. Relationship between Compensated and Uncompensated Demand 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 LECTURE_____________________________________________ V. Income and Substitution Effects. In this chapter we will use the utility maximization model to examine how a consumer responds optimally to the change in the price of a good. Our analysis will allow insights into the components underlying a quantity response (e.g., income and substitution effects) as well as insights into the ceteras paribus conditions that underlie the analysis of demand. A. Demand Functions. As we saw in the last chapter, we can write a constrained optimization problem involving n goods as a system of n+1 equations in n+1 unknowns. These expressions, at least in principle, can be solved for each of the individual goods xi* e.g., x1* = d1(p1, p2, …., pn, I) x2* = d2(p1, p2, …., pn, I) . . . xn* = dn(p1, p2, …., pn, I) These expressions d1 to dn are the demand functions for the individual. In this chapter, we consider the effects of a change in prices and income on the consumption of the good. Generally, these exercises are known as comparative statics analysis. 1. Homogeneity One of the most straight-forward of these comparative statics exercises regards the effects of an equal percentage increase in all prices, as well as the income. For such increases to affect preferences, consumers would have to suffer “money illusion” (or perhaps money disillusion”). We presume that consumers are not affected by such offsetting changes. Formally, we assume that demand functions are homogeneous of degree 0 in all prices and income. That is, for any constant t, we assume that x1* = d1(p1, p2, …., pn, I) = d1(tp1, tp2, …., tpn, tI) Example #1: Consider the Cobb-Douglas utility function U(X,Y) = X.3Y.7. Following our analysis from the last chapter, we can show that X* = .3I/Px and Y* = .7I/Py Clearly, doubling I and each of the prices would leave unaffected both X* and Y*. Thus, Cobb-Douglas functions are homogeneous of degree 0. Example #2: The CES utility function. Suppose that the demand function was given by U(X,Y) = X.5 + Y.5. Following our analysis from the last chapter, the demand functions are given by X* = (I/Px)[1 + Px/Py] -1 Y* = (I/Py)[1 + Py/Px] -1 Again, doubling y, x and I do not affect X* and Y*, making these functions also homogeneous of degree 0. Y Y B. Changes in Income. Now let’s move on to more general comparative statics effects. We start with an analysis of a change in income. In terms of an indifference map, income increases will lead to a utility expansion. The question is whether the consumption of a good X will increase (as shown in the left panel) or decrease, (as shown in the right panel). U3 U3 U2 U2 U1 X1 X2 X3 X I1 I2 A Normal Good I3 U1 X3 X2 X1 X I1 I2 I3 An Inferior Good Notice in the above chart that there’s nothing unusual about the indifference curves in the panel on the right. Either relationship is possible. For some goods (such as automobiles, education and housing) consumption usually increases with income. For other goods (macaroni and cheese in a box, second hand clothing and generic beer) consumption diminishes with income increases. We define these relationships as follows 1 .Normal and Inferior Goods. When consumption of a good increases with income (x/I >0) the good is said to be a normal good. When (x/I <0) the good is said to be an inferior good. 2. Engel’s Law: One of the most robust findings regarding consumer expenditures regards the relationship between food expenditures and income. As first studied by Ernst Engle (1857) the percentage of income devoted to food tends to fall as income increases. This relationship is known as Engel’s Law. It has been established over time and across cultures. For this reason, the percentage of income spent on food is sometimes taken as a poverty measure. Question: Does a conclusion that the share of income going to food falls as income increases imply that food is an inferior good? Answer: Not necessarily. Engel’s law implies that (PxX/I)/ I = PxI[(X/I) – PxX]/I2 <0. Obviously, if X/I <0, the law holds. However, if PxX is large enough, the relationship could be negative, even with X/I >0. (Thus, Engel’s law does not imply that food is necessarily an inferior good. C. Changes in the Price of a Good. Now let’s consider a third comparative static effect: the effects of a price change. As will be obvious momentarily, the effects of price adjustments are a bit more involved than income effects. Y 1. A Price Reduction– A Graphical Analysis. Consider a demand function for just two goods, X and Y. If the price of a good falls, then the Budget Constraint Y = I/Py + (Px/Py)X will “flatten out. The new optimal consumption bundle is the point of tangency between the highest attainable indifference curve consistent with the new budget constraint. I1=Px 1X+Py Y I1=Px 2X+Py Y X1 X2 U2 U1 X Income Effect Y Substitution Effect 2. Graphical Analysis – Price Increase 2 I1=Px X+Py Y I1=Px 1X+Py Y X2 X1 U2 U1 Income Effect The increase in the consumption of good X can be divided into two parts, a substitution effect attributable to the change in relative prices along the original indifference curve, and an income effect which reflects the increase in effective income available as a result of the price reduction. Observe in the chart on the left that these two effects are reinforcing in this case. This need not always be true. Substitution Effect effective income due to the price increase. X The quantity reduction due to an increase in the price of a good X may similarly be decomposed into income and substitution effects. Starting at original income I1, the substitution effect associated with increasing the price of a good is found by rotating the budget constraint back along the original indifference curve. The income effect then is the reduction in Y 3. Effects of Price Changes for Inferior Goods. For inferior goods, substitution and income effects work in the opposite directions. Most of the time, the net result will still be an inverse relationship between price and quantity. The panel on the left illustrates (or I1=Px 1X+Py Y tries to illustrate the effects of a price reduction on an inferior good. Rotating the budget constraint along the original indifference curve generates a fairly sizable 2 substitution effect. However, the I2=Px X+Py Y parallel upward shift of the budget line results in negative income X1 X2 effect. Still, in general, we would U2 expect that, as shown here, the U1 effects of a price reduction would X be positive, on net. It is not a Income Effect Substitution Effect logical impossibility that the income effect might dominate a substation effect. Such goods are termed “Giffen Goods. However, it is doubtfully the case that such goods exist. To see this intuitively, reflect for a moment on what it would imply for a good to be a Giffen Good: As a consequence of a price reduction, income increased enough that consumers no longer wanted the good! D. Individual’s Demand Curve. It often convenient to express quantity as a function of price, holding other things constant. This price/quantity relationship, for example, represents the underpinnings of much of the graphical analysis in elementary economics. Given the demand function for a good x, x* = d(px, py, I), we can write a demand curve by holding all variables other than px constant. Formally, Individual Demand Curve: An individual demand curve shows the relationship between the price of a good and the quantity of that good purchase by an individual assuming that all other determinants of demand are held constant. A demand curve can be readily derived from an indifference map, simply by considering the quantities of X optimally chosen as the price of X, px, changes. P Y I1 =Px 1X + Py Y I2 =Px 2X + Py Y P1 I3 =Px 3X + Py Y P2 P3 U3 U2 U1 X1 X2 X3 X X1 X2 X3 Notice in the figure to the right, that as px falls, reflected by the progressively flatter indifference curves, the quantity of x increases. This gives rise to the standard inverse relationship between price and quantities, as shown, for example, in the graph below. 1. Shifts in the Demand Curve. A change in either the price of good Y or income I will cause a new demand curve to be constructed. It is important to keep in mind that a demand curve is simply a two dimensional representation of an n dimensional relationship. Example #3: Consider the Cobb-Douglas utility function U(X,Y) = X.3Y.7. Recall that the demand functions for X and Y are given by X* = .3I/Px and Y* = .7I/Py = 70/Py Setting I = 100 generates the demand curves X* = 30/Px and Y* Notice that a new I would cause an outward shift in each demand curve. Notice also, however, that in this case, Py does not affect X and vice versa. Example #4: Consider again the CES utility function given by U(X,Y) = X + Y . The demand functions are given by .5 .5 X* = (I/Px)[1 + Px/Py] -1 Y* = (I/Py)[1 + Py/Px] -1 Suppose we set I = 100 and Py = 1. Then the demand curve for X becomes X X = (100/Px)[1 + Px] -1= 100/[ Px + Px2] Notice that a new higher I would shift the demand curve out, and a new higher Py would shift the curve in (indicating that the goods are substitutes) E. Compensated Demand Curves. Notice in the development of the above demand curve, that the actual level of utility varies as price changes. This occurs, of course, because income effects impact utility as well as substitution effects. Thus only nominal income is held constant as the price falls, for example. Although this is the most conventional way to construct a demand curve, it is not the only way. Sometimes it is useful to construct a demand curve holding real income constant. The idea here would be to “compensate” individuals with income increases or reductions as prices change, so that they stay on the same indifference curve. These compensated demand curves thus, illustrate pure substitution effects.. P Y Compensated Demand Curve: A compensated (or Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased on the assumption that the other prices and utility are held constant. The curve therefore illustrates only substitution effects. Mathematically, the curve is a two-dimensional representation of the Px 1/Py Px 2/Py Px 3/ Py P1 P2 P3 U X1 X2 X3 X X1 X2 X3 compensated demand function. The panel above to the left illustrates the development of a compensated demand curve from an indifference map. Notice that as the price of x falls, income is adjusted (implicitly) so that the budget line remains tangent to the initial indifference curve. The inverse relationship between price and quantity generate the standard demand curve, shown above on the right. The use of compensated or uncompensated demand is a matter of choice. In most empirical work uncompensated demand curves are estimated, because they are the curves for which data are available. However, compensated demand curves are useful in theoretical work of welfare analyses, because it is often desirable to evaluate the effects of changes that hold utility constant. X Example #5. Compensated demand functions. 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 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. 1. Relationship between Compensated and Uncompensated Demand The same reference price vector, income level and initial level of utility, compensated and uncompensated demand functions must, by construction intersect at one point. However, at all other prices they differ. Intuitively the compensated demand is less sensitive than uncompensated demand to price increases above the point of intersection, because income supplements enable the consumer to purchase more of the good. On the other hand at prices below the point of intersection, compensated demand is steeper, since income must be taken away from compensated demand as price falls in order to keep utility constant. This relationship is illustrated graphically below. Price h(x) - compensated demand dx - uncompensated demand hx > dx Quantity hx < dx