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Lecture 4 Lecture 4 - Theory of Choice and Individual Demand David Autor 14.03 Fall 2004 Agenda 1. Utility maximization 2. Indirect Utility function 3. Application: Gift giving –Waldfogel paper 4. Expenditure function 5. Relationship between Expenditure function and Indirect utility function 6. Demand functions 7. Application: Food stamps –Whitmore paper 8. Income and substitution e¤ects 9. Normal and inferior goods 10. Compensated and uncompensated demand (Hicksian, Marshallian) 11. Application: Gi¤en goods –Jensen and Miller paper Roadmap: 1 Axioms of consumer preference Primal Dual Max U(x,y) Min pxx+ pyy s.t. pxx+ pyy < I s.t. U(x,y) > U Indirect Utility function Expenditure function E*= E(p , p , U) U*= V(px, py, I) x y Marshallian demand Hicksian demand X = d (p , p , I) = x x y X = hx(px, py, U) = (by Roy’s identity) (by Shepard’s lemma) ¶V / ¶p ¶ E - x - ¶V / ¶I Slutsky equation ¶p x 1 Theory of consumer choice 1.1 Utility maximization subject to budget constraint Ingredients: Utility function (preferences) Budget constraint Price vector Consumer’sproblem Maximize utility subjet to budget constraint Characteristics of solution: Budget exhaustion (non-satiation) For most solutions: psychic tradeo¤ = monetary payo¤ Psychic tradeo¤ is MRS Monetary tradeo¤ is the price ratio 2 From a visual point of view utility maximization corresponds to the following point: (Note that the slope of the budget set is equal to px ) py Graph 35 y IC3 IC2 IC1 B C A D x What’swrong with some of these points? We can see that A P B, A I D, C P A. Why should one choose A? The slope of the indi¤erence curves is given by the MRS 1.1.1 Interior and corner solutions [Optional] There are two types of solution to this problem. 1. Interior solution 2. Corner solution 3 Graph 36 y Typical case x The one below is an example of a corner solution. In this speci…c example the shape of the indi¤erence curves means that the consumer is indi¤erent to the consumption of good y. Utility increases only with consumption Graph 37 of x. y x 4 Graph 38 y x In the graph above preference for y is su¢ ciently strong relative to x that the the psychic tradeo¤ is always lower than the monetary tradeo¤. This must be the case for many products that we don’tbuy. Another type of “corner”solution results from indivisibility. Graph 39 y 10 I = 500 px= 450 py= 50 1 x Why can’twe draw this budget set, i.e. conect dots? This is because only 2 points can be drawn. This is a sort of “integer constraint”. We normally abstract from indivisibility. 5 Going back to the general case, how do we know a solution exists for consumer, i. e. how do we know the consumer can choose? We know because of the completeness axiom. Every bundle is on some indi¤erence curve and can therefore be ranked: A I B, A B, B A. 1.1.2 Mathematical solution to the Consumer’sProblem Mathematics: maxU(x; y) x;y s:t: pxx + pyy I L = U(x; y) + (I pxx pyy) @L 1: = Ux px = 0 @x @L 2: = Uy py = 0 @y @L 3: = I pxx pyy = 0 @ Rearranging 1: and 2: U p x = x Uy py This means that the psychic tradeo¤ is equal to the monetary tradeo¤ between the two goods. 3: states that budget is exhausted (non-satiation). Also notice that: U x = px U y = py What is the meaning of ? 1.1.3 Interpretation of , the Lagrange multiplier At the solution of the Consumer’sproblem (more speci…cally, an interior solution), the following conditions will hold: @U=@x @U=@x @U=@x 1 = 2 = ::: = n = p1 p2 pn 6 This expression says that at the utility-maximizing point, the next dollar spent on each good yields the same marginal utility. @U So what is @I ? Return to Lagrangian: L = U(x; y) + (I pxx pyy) @L = Ux px = 0 @x @L = Uy py = 0 @y @L = I pxx pyy = 0 @ @L @x @x @y @y = Ux px + Uy py + @I @I @I @I @I U By substituting = Ux and = y both expressions in parenthesis are zero. px py We conclude that: @L = @I equals the “shadow price” of the budget constraint, i.e. it expresses the quantity of utils that could be obtained with the next dollar of consumption. This shadow price is not uniquely de…ned. It is de…ned only up to a monotonic transformation. What does the shadow price mean? It’s essentially the ‘utility value’of relaxing the budget constraint by one unit (e.g., one dollar). [Q: What’sthe sign of @2U=@I2, and why?] We could also have determined that @L=@I = without calculations by applying the envelope theorem. At the utility maximizing solution to this problem, x and y are already optimized and so an in…nitesimal change in I does not alter these choices. Hence, the e¤ect of I on U depends only on its direct e¤ect on the budget constraint and does not depend on its indirect e¤ect (due to reoptimization) on the choices of x and y. This ‘envelope’result is only true in a small neighborood around the solution to the original problem. Corner Solution: unusual case When at a corner solution, consumer buys zero of some good and spends the entire budget on the rest. What problem does this create for the Lagrangian? 7 Graph 40 y U U1 2 U0 x The problem is that a point of tangency doens’texist for positive values of y. Hence we also need to impose “non-negativity constraints”: x 0, y 0. This will not be important for problems in this class, but it’seasy to add these constraints to the maximiza- tion problem. 1.1.4 An Example Problem Consider the following example problem: 1 3 U(x; y) = 4 ln x + 4 ln y Notice that this utility function satisifes all axioms: 1. Completeness, transitivity, continuity [these are pretty obvious] 1 3 2. Non-satiation: Ux = 4x > 0 for all x > 0. Uy = 4y > 0 for all y > 0. In other words, utility rises continually with greater consumption of either good, though the rate at which it rises declines (diminishing marginal utility of consumption). 3. Diminishing marginal rate of substitution: 1 3 Along an indi¤erence curve of this utility function: U = 4 ln x0 + 4 ln y0. Totally di¤erentiate: 0 = 1 dx + 3 dy. 4x0 4y0 dy Ux 4y0 Which provides the marginal rate of substitution U = = . dx j Uy 12x0 The marginal rate of substitution of y for x is increasing in the amount of y consumed and decreasing in the amount of x consumed; holding utility constant, the more y the consumer has, the more y he would give up for one additional unit of x. 8 Example values: px = 1 py = 2 I = 12 Write the Lagrangian for this utility function given prices and income: maxU(x; y) x;y s:t: pxx + pyy I 1 3 L = ln x + ln y + (12 x 2y) 4 4 @L 1 1: = = 0 @x 4x @L 3 2: = 2 = 0 @y 4y @L 3: = 12 x 2y = 0 @ Rearranging (1) and (2), we have: U p x = x Uy py 1=4x 1 = 3=4y 2 The interpretation of this expression is that the MRS (psychic trade-o¤) is equal to the market trade-o¤ (price-ratio). @L What’s @I ? As before, this is equal to , which from (1) and (2) is equal to: 1 3 = = : 4x 8y The next dollar of income could buy one additional x, which has marginal utility 1 or it could buy 1 additional 4x 2 3 1 3 y0s, which provide marginal utility (so, the marginal utility increment is ). It’simportant that @L=@I 4y 2 4y = is de…ned in terms of the optimally chosen x; y . Unless we are at these optima, the envelope theorem @x @x @y @y does not apply So, @L=@I would also depend on the cross-partial terms: Ux px + Uy py . @I @I @I @I 9 1.1.5 Lagrangian with Non-negativity Constraints [Optional] max U(x; y) s:t: pxx + pyy I y 0 2 L = U(x; y) + (I pxx pyy) + (y s ) @L 1: = Ux px = 0 @x @L 2: = Uy py + = 0 @y @L 3: = 2s = 0 @s Point 3: implies that = 0, s = 0, or both. 1. s = 0; = 0 (since 0 then it must be the case that > 0) 6 (a) Uy py + = 0 Uy py < 0 ! U y < py U x = px Combining the last two expressions: U p x > x Uy py This consumer would like to consume even more x and less y, but she cannot. 2. s = 0; = 0 6 Uy py + = 0 Uy py = 0 ! U U y = x = py px Standard FOC, here the non-negativity constraint is not binding. 3. s = 0; = 0 Same FOC as before: p U x = x py Uy Here the non-negativity constraint is satis…ed with equality so it doesn’tdistort consumption. 10 1.2 Indirect Utility Function For any: Budget constraint Utility function Set of prices We obtain a set of optimally chosen quantities. x1 = x1(p1; p2; :::; pn;I) ::: xn = xn(p1; p2; :::; pn;I) So when we say max U(x1; :::; xn) s:t: P X I we get as a result: max U(x1(p1; :::; pn;I); :::; xn (p1; :::; pn;I)) U (p1; :::; pn;I) V (p1; :::; pn;I) ) which we call the “Indirect Utility Function”.
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