Lecture 5 - Choice, Demand. Normal and Inferior Goods 14.03 Spring 2003 Agenda 1. Expenditure function 2. Relationship between Expenditure function and Indirect utility function 3. Indirect Utility function 4. Demand functions 5. Changes in incomes and Engel’s law 6. Income and substitution effects 7. Normal, inferior and Giffen goods 8. Compensated and uncompensated demand (Hicksian, Marshallian) 9. Irish Potato famine Roadmap: [Chart 34] 1 Axioms of consumer preference Chart 34 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’s problem Maximize utility subjet to budget constraint Characteristics of solution: Budget exhaustion (non-satiation) • For most solutions: psychic tradeoff = monetary payoff • 2 Psychic tradeoff is MRS • Monetary tradeoff is the price ratio • From a visual point of view utility maximization corresponds to the following point: [Graph 35] Graph 35 y IC3 IC2 IC1 B C A D x What’s wrong with some of these points? We can see that A P B, A I D, C P A. Why should one choose A then? Theslopeoftheindifference curves is given by the MRS. 1.1.1 Interior and corner solutions There are two types of solution to this problem. 1. Interior solution 2. Corner solution [Graph 36] 3 Graph 36 y Typical case x The one below is an example of a corner solution. In this specificexample the shape of the indifference curves means that the consumer is indifferent to the consumption of good y. Utility increases only with consumption of x. [Graph 37] Graph 37 y x [Graph 38] 4 Graph 38 y x In the graph above preference for y is sufficiently strong relative to x that the the psychic tradeoff is always lower than the monetary tradeoff. This must be the case for many products that we don’t buy. Another type of ”corner” solution results from indivisibility. [Graph 39] 5 Graph 39 y 10 I = 500 px= 450 py= 50 1 x Why can’t we 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. Going back to the general case, how do we know a solution exists for con- sumer, i. e. how do we know the consumer can choose? We know because of the completeness axiom. Every bundle is on some indifference curve and can therefore be ranked: A I B, A B, B A. Â Â 1.1.2 Mathematical solution Mathematics: maxU(x, y) x,y s.t. pxx + pyy I ≤ L = U(x, y)+λ(I pxx pyy) − − ∂L 1. = Ux λpx =0 ∂x − ∂L 2. = Uy λpy =0 ∂y − ∂L 3. = I pxx pyy =0 ∂λ − − 6 Rearranging 1. and 2. U p x = x −Uy py This means that the psychic tradeoff is equal to the monetary tradeoff be- tween the two goods. 3. states that budget is exhausted. Also notice that: U x = λ px U y = λ py So what is the meaning of λ? 1.1.3 Interpretation of λ, the Lagrange multiplier ∂U/∂x ∂U/∂x ∂U/∂x 1 = 2 = ... = n = λ p1 p2 pn 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)+λ(I pxx pyy) − − ∂L = Ux λpx =0 ∂x − ∂L = Uy λpy =0 ∂y − ∂L = I pxx pyy =0 ∂λ − − ∂L ∂x ∂x ∂y ∂y = Ux λpx + Uy λpy + λ ∂I ∂I − ∂I ∂I − ∂I µ ¶ µ ¶ U By substituting λ = Ux and λ = y both expressions in parenthesis are zero. px py We conclude that: ∂L = λ ∂I We could have known this without calculations from the envelope theorem. λ 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. 7 This shadow price is not uniquely defined. It is defined only up to a monotonic transformation. 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? [Graph 40] Graph 40 y U U1 2 U0 x The problem is that a point of tangency doens’t exist 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’s≥ easy to≥ add these constraints to the maximization problem. 8 1.1.4 Lagrangian with Non-negativity Constraints max U(x, y) s.t. pxx + pyy I ≤ y 0 ≥ 2 L = U(x, y)+λ(I pxx pyy)+γ(y s ) − − − ∂L 1. = Ux λpx =0 ∂x − ∂L 2. = Uy λpy + γ =0 ∂y − ∂L 3. = 2sγ =0 ∂s − Point 3. implies that γ =0, s =0,orboth. 1. s =0,γ=0(since γ 0 then it must be the case that γ>0) 6 ≥ (a) Uy λpy + γ =0 Uy λpy < 0 − −→ − U y <λ py U x = λ 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,γ=0 6 Uy λpy + γ =0 Uy λpy =0 − −→ − U U y = x = λ py px Standard FOC, here the non-negativity constraint is not binding. 3. s =0,γ=0 Same FOC as before: p U x = x py Uy Here the non-negativity constraint is satisfied with equality so it doesn’t distort consumption. 9 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”. This is the value of maximized utility under given prices and income. So remember the distinction: Direct utility: utility from consumption of x1, ..., xn Indirect utility: utility obtained when facing p1, ..., pn,I Example: max U(x, y)=x.5y.5 s.t. pxx + pyy I ≤ .5 .5 L = x y + λ(I pxx pyy) − − ∂L .5 .5 = .5x− y λpx =0 ∂x − ∂L .5 .5 = .5x y− λpy =0 ∂y − ∂L = I pxx pyy =0 ∂λ − − We obtain the following: 10 .5x .5y.5 .5x.5y .5 λ = − = − px py 1 p x = p y = I x y 2 I I x∗ = ,y∗ = 2px 2py Half of the budget goes to each good. Let’s derive the indirect utility function in this case: I I I .5 I .5 U( , )= 2p 2p 2p 2p x y µ x ¶ µ y ¶ What is this indirect utility function good for? 1. It saves us time in recalculating utility for every set of prices and budget constraints. 2. It is used in theory as we will see when studying prices indices. 3. It allows us to calculate consumer demand as a function of prices and income. 1.3 Dual Problem of Utility Maximization Most economic problems have a dual, an inverse problem. For example, the dual of choosing output in order to maximize profits is minimizing costs at a given output level: cost minimization is the dual of profit maximization. 1.3.1 Expenditure function Consumer’s problem: maximize utility subject to a budget constraint. What is the dual of this problem? It’s minimizing expenditure subject to a utility constraint (i.e. a level of utility you must achieve) This yields the ”expenditure function”: the minimum expenditure required to attain a given utility level. Setup of the dual 1. Start with: max U(x, y) s.t. pxx + pyy I ≤ 11 2. Solve for x∗, y∗ z∗ = U(x∗,y∗) given px,py,I. ⇒ z is the indirect utility. 3. Now solve the following problem: min pxx + pyy s.t.U(x, y) z∗ ≥ 1.3.2 Graphical representation of dual problem [Graph 41] Graph 41 y U = z* x The dual problem consists in choosing the lowest budget set tangent to a given indifference curve. Example: min E = pxx + pyy .5 .5 s.t. x y Up ≥ where Up comes from the primal problem. .5 .5 L = pxx + pyy + λ Up x y − ¡ ¢ 12 ∂L .5 .5 = px λ.5x− y =0 ∂x − ∂L .5 .5 = py λ.5x y− =0 ∂y − ∂L .5 .5 = Up x y =0 ∂λ − p .5 p .5 x = y U ,y= x U ∗ p p ∗ p p µ x ¶ µ y ¶ .5 .5 E∗ =2Uppx py How do solutions to Dual and Primal compare? 1.3.3 Relation between Expenditure function and Indirect utility function Let’s look at the relation between expenditure function and indirect utility function. V (px,py,I0)=U0 E(px,py,U0)=I0 V (px,py,E(px,py,U0)) = U0 E(px,py,V(px,py,I0)) = I0 Expenditure function and Indirect Utility function are inverse one of the other.