MSc Economics: Economic Theory and Applications I. Consumer Theory

Transcription

1 MSc Economics: Economic Theory and Applications I Consumer Theory Dr Ken Hori Birkbeck College Autumn

2 1 Utility Max Problem Basic hypothesis: a rational consumer will always choose a most preferred bundle from the set of affordable alternatives x, given by the budget constraint p.x m. Using the utility function representation of preference orderings, the problem is max x 0 u(x) such that p.x m (UMP) Proposition The solution exists if u(.) is continuous, and the feasible consumption bundle set X is compact. The solution is unique if the preference is strictly convex. So given (p,m), the solution is a unique utility maximising consumption bundle x (p,m), which is the Marshallian demand function. 2

3 The maximised utility attained is then u(x (p,m)) v(p,m) the indirect utility function, which is an optimal value function. Properties v(p,m) is, a. Homogeneous of degree zero in (p,m). b. Non-increasing in p and non-decreasing in m. c. Continuous at all p À 0, m>0. d. Quasi-convex in p, i.e.{p : v(p,m) k} is a convex set k, or equivalently price indifference curves are convex. p 2 v(p,m) k lower contour set higher utility v(p,m) = k p 1 1.Price Indifference Curves 3

4 Remark Given (p,m), the optimal bundle is chosen such that the MRS of the goods equals the relative prices. Proof The Lagrangian, max x 0,λ L = u(x) λ(p.x m) FOCs u(x ) x i λp i =0 for i =1,..., k Thus u(x )/ x i u(x )/ x j = p i p j for i, j =1,...,k 4

5 Roy s Identity For p À 0 and m>0, x i (p,m)= v(p,m)/ v(p,m)/ m, i =1,..., k Proof Apply Envelope Theorem at x,i.e. where u(x ) v(p,m), v(p,m) v(p,m) m = L(x,λ) = λx i (p,m) x(p,m) constant = L(x,λ) = λ m x(p,m) constant Thus the ratio of the two gives x i (p,m). 5

6 2 Expenditure Min Problem Instead of finding the optimal consumption bundle given (p,m),canfind the minimum cost required to attain U given p, min x 0 p.x such that u(x) U (EMP) The optimal consumption bundle h (p,u) is the Hicksian demand function. The minimum cost required to attain U e(p,u) p.h (p,u) is the expenditure function. Properties e(p,u) is, a. H.d.1 in p,i.e.h (p,u) is h.g.d.0 in p. b. Non-decreasing in p and strictly increasing in U. c. Continuous at all p À 0. d. Concave in p. 6

7 Proof Prove e(p 00,U) te(p,u)+(1 t)e(p 0,U) for 0 t 1,where p 00 = tp +(1 t)p 0 As h (p,u) and h (p 0,U) are the expenditure minimising bundles at prices p and p 0 respectively, p.h (p 00,U) p.h (p,u) p 0.h (p 00,U) p 0.h (p 0,U) Multiply the former by t and the latter by 1 t, and then sum up, tp.h (p 00,U)+(1 t)p 0.h (p 00,U) te(p,u)+(1 t)e(p 0,U) But the left-hand side is {tp +(1 t)p 0 }.h (p 00,U)=p 00.h (p 00,U) = e(p 00,U) 7

8 Intuition: as p doubles, by retaining the same consumption bundle the expenditure will double; however one can possibly do better by choosing a more appropriate consumption bundle at the new p. Shephard s Lemma For p À 0, h i (p,u)= e(p,u), i =1,..., k Proof The Lagrangian optimisation for the EMP is, min h 0,λ L = p.h + λ(u u(h)) Applying Envelope Theorem at h,i.e.where p.h e(p,u), e(p,u) = L(h,λ) = h i (p,m) h(p,u) constant 8

9 3 DualityTheory Assuming unique solutions to the UMP and EMP, x(p,m) and h(p,u) C 1,and v(p,m) and e(p,u) C 2, 1. e(p,v(p,m)) m i.e. the minimum expenditure necessary to reach utility v(p,m), which is in turn the maximum utility attained at p and m,ism. 2. v(p,e(p,u)) U i.e. the maximum utility from income e(p,u) is U. 3. x i (p,m) h i (p,v(p,m)) the Marshallian demand at income m is the same as the Hicksian demand at utility v(p,m). 4. h i (p,u) x i (p,e(p,u)) the Hicksian demand at utility U is the same as the Marshallian demand at income e(p,u). 9

10 4 Slutsky Decomposition x j (p,m) = h j(p,u) x j(p,m) m x i(p,m) Proof Use identity 4 for good j and differentiate both sides by p i, h j (p,u) = x j(p,m) + x j(p,m) e(p,u) m But by Shephard s Lemma e = h i (p,u), which at the optimal point equals x i (p,m). Another result of the Duality Theory. It holds as the optimal outcome of the UMP is the same as that of the EMP. This equation holds only for constant m. For the case m = p.w see example later. 10

11 Intuition: An increase in the price p i has two effects, Substitution effect: changes the relative price between various commodities. Income effect: decreases the overall level of real income for any consumer who purchases a positive quantity of commodity i. p 2 total effect = substitution effect + income effect substitution effect p 1 2.Substitution and Income Effects 11

12 h i hj The matrix of substitution terms is symmetric since, using Shephard s Lemma, h j = 2 e p j = 2 e p j = h i p j Thematrixisinfactnegativesemi-definite as e(p,u) is concave the compensated own-price effect is non-positive, h i = 2 e p 2 i 0 These are properties of unobservable Hicksian demands. But h from Slutsky i the xj observable matrix + x j m x i is also symmetric and negative semi-definite. This is now a testable prediction. 12

13 This matrix is in fact useful in considering the integrability problem: Given observed demand functions, can we find the original utility function or the expenditure function (i.e. reversing the Roy s Identity or Shepherd s Lemma)? The integrability condition that ensures the existence of an expenditure function that is consistent with the observed demand functions, is that this matrix is symmetric and negative semi-definite. 13

14 Also from h i Slutsky, 0 we can state using m > 0 x i < 0, i.e.normal good must be an ordinary good. x i For an inferior good (i.e. x i m sign of x i is ambiguous. < 0), the However x i > 0 x i m < 0, and hence a Giffen good must be an inferior good. 14

15 5 Welfare Measurement Question What is the point of Duality Theory? Answer For welfare analysis. For welfare measurements an estimation of e(p,u) is required, but this is unobserved. However UMP is, and one can estimate the unobservable Hicksian demand from observable Marshallian demands. 15

16 5.1 Money Metric Utility Fns A consumer has a choice between receiving a goods bundle x or some income m. Given current prices p, howmuchincome m(p, x) would he need to be indifferent between the two? Answer: the minimum cost required to buy a bundle z that is on the same IC as x,i.e. m(p, x) e(p,u(x)) Gives a monetary value to the utility of holding x, and is thus called the money metric utility function. Alternatively, how much income μ(p; q,m) does one need at p to be as well off as having income m at prices q? The solution is the money metric indirect utility function, μ(p; q,m) e(p,v(q,m)) 16

17 Friday, 5 November, 1999, 17:09 GMT Can't buy me love? Marriage can bring you as much joy as 60,000 a year, claim economists using a mathematical formula which takes into account income, personal traits and happiness levels. It's a Sunday morning, and you are just surfacing from sleep. You turn over in bed, and put your arm around your loving, faithful partner. Life is good. Rewind. It's a Sunday morning, and you are just surfacing from sleep. You turn over in bed, and put your arm around your pile of 50 notes. There are 1,200 of them. Life - apparently - is just as good. A study by two economists claims to have found that, contrary to generations of wisdom, money can actually make you happy. A lasting marriage brings as much happiness as having an extra 60,000 added to your pay packet, Professor Andrew Oswald of the University of Warwick and David Blanchflower of Dartmouth College in the US say. Similarly, losing a job causes 40,000-worth of unhappiness. The study of 100,000 people randomly sampled across the UK and US also compared satisfaction and mental well-being rates in other countries. It found there had been a decline in the number of people married (72% in the early 1970s, 55% by the late 90s). But married people said they were much happier than the unmarrieds. Getting divorced, separated, or widowed made people much more unhappy than losing their jobs. The happiest people were women, the highly educated, married couples, and those whose parents have not divorced, the report says. Women who co-habit are happier than those who live alone, but are not as happy as those who are married. Happiness and satisfaction with life tend to be shaped like a U or J, it says, with high levels in youth and old age, but a drop in the 30s. Andrew Oswald: Happily married, but not a non-financial millionaire 17

18 5.2 CV and EV Policy makers are interested in the welfare changes from (p 0,m 0 ) to (p 1,m 1 ). Use compensating and equivalent variations, or v(p 1,m 1 CV )=v(p 0,m 0 ) v(p 0,m 0 + EV )=v(p 1,m 1 ) CV = m 1 e(p 1,v(p 0,m 0 )) EV = e(p 0,v(p 1,m 1 )) m 0 Thesignsassumethatoneisalwaysbetter off in (p 1,m 1 ). E.g.1: If a change in agricultural policy leads to a fall in (p, m) for the farmers, the amount of compensation required can be estimated using CV at new prices. E.g.2: Checking which of the possible policies make the consumers better-off can be better analysed using EV at current prices. 18

19 5.3 CS as an Approximation The Marshallian consumer surplus for a price move p 0 p 1 is the area to the left of the market demand curve, CS = Z p 1 p 0 x(p)dp Let m 0 = m 1 = m, and the price of good 1 change from p 0 to p 1.Then, CV = EV = Z p 0 p 1 Z p 0 p 1 h(p, v(p 0,m))dp h(p, v(p 1,m))dp Hence the correct measure of welfare is an integral of the Hicksian demand curve rather than the Marshallian. However Marshallian CS can still be used for approximation. 19

20 Slutsky equation for own-price change h i (p, U) = x i(p, m) + x i(p, m) m x i(p, m) Thus for a normal good h i is less negative than x i (or h i is steeper than x i ): p h(p,u 1 ) p 0 p 1 h(p,u 0 ) Consumer surplus x(p,m) x It follows then EV > CS > CV The order reverses for an inferior good. For a quasilinear utility function, as h(p, u 0 )=x(p, m) =h(p, u 1 ), EV = CS = CV 20

21 6 Aggregation Issue H consumers with income m =(m 1,..., m H ). k consumption goods with prices p. Demand vector for consumer h is, x h (p,m h )=(x h 1(p,m h ),..., x h k (p,mh )) Define the aggregate demand function as X(p, m) = HX x h (p,m h ) h=1 Question: Do any of the properties for individual workers, such as Roy s Identity or Slutsky s equation, carry through this aggregation? If that is the case then the aggregate behaviour can be treated as it were generated by a single representative consumer. It turns out that the aggregate demand function possesses no interesting properties other than homogeneity and continuity. 21

22 Implications: Problem for having micro underpinning on macro aggregate theories. Hard to test consumer theories. Consider Roy s Identity. What we want is, X i (p, m) = V/ V/ M, i =1,...,k where V (p, m) = P H h=1 vh (p,m h ) and M = P H h=1 mh.but P H H h=1 vh / P H h=1 vh / M 6= X v h / v h / m h h=1 and hence X i (p, m) 6= P H h=1 xh i (p,mh ). The necessary and sufficient condition for successful aggregation is that the indirect utility function is of the Gorman form, v h (p,m h )=a h (p)+b(p)m h Then V (p,m)= X H h=1 ah (p)+b(p)m 22

23 Try Roy s Identity again, V/ p P H i V/ M = h=1 a + b M b(p) HX a p = i + b m h = b(p) h=1 HX h=1 v h / v h / m h Hence this time X i (p, m) = P H h=1 xh i (p,mh ). The point is that in x h i (p,m h )= 1 a + 1 b m h b(p) b(p) consumers have the same marginal propensity to consume xh i m = 1 b h b, independent of m h. Hence the aggregate demand function depends only on the total income, and not on the distribution of income. Two examples of Gorman form utility functions are quasilinear v(p,m)=v(p)+ m, and homothetic v(p,m)=v(p)m. 23

24 7 Application: Neoclassical Model of Labour Supply Suppose consumption is financed out of Labour income wh, 0 H L. A fixed non-labour income y. The consumption bundle consists of Leisure l L H at price w. Goods other than leisure x =(x 1,x 2,..., x k ), at prices p. The optimisation problem is, max l,x u(l, x) s.t. wl + p.x = y + wl and 0 l L The Marshallian demands are l (w, p,y+ wl) and x (w, p,y+ wl). A dual EMP at utility level U leads to Hicksian solutions h l (w, p,u) and h (w, p,u). 24

25 Consider the Slutsky equations for l, W.r.t. a rise in p i,itisasbefore, l = h l l m x i W.r.t. a rise in w, as this also increases the worker s income, l w = l + l m w m constant m w µ h = l w l m l + l m L = h l w + l m H i.e. assuming that leisure is a normal good, the sign of l w is now ambiguous. Intuition: More expensive leisure vs. higher income. Can then lead to a backwardbending labour supply curve. 25

26 8 Example (1) UMP max x max x,λ The Lagrangian: FOCs: u(x) = x 1 x 2 s.t. p.x m L(x,λ)=x x λ(p 1 x 1 + p 2 x 2 m) L 1 = 1 2 x x λp 1 =0 L 2 = 1 2 x x λp 2 =0 L λ = p 1 x 1 p 2 x 2 + m =0 The Marshallian demands are µ 1 x m =, 1 m 2p 1 2 p 2 and the indirect utility function is v(p,m)= µ 1 m 2 p 1 1 µ 2 1m 2 p m = 2 p1 p 2 26

27 Check Roy s Identity: v p 1 v m = 1 4 mp p p p = 1 m = x 1 2p 1 (2) EMP min h,λ min h The Lagrangian: p.h s.t. p h 1 h 2 U L(h,λ)=p 1 h 1 + p 2 h 2 + λ(u h h ) FOCs: L 1 = p 1 λ 1 2 h h =0 L 2 = p 2 λ 1 2 h h =0 L λ = U h h =0 The Hicksian demands are µ r h p2 = U,U p 1 27 r p1 p 2

28 and the expenditure function is r r p2 p1 e(p,u)=p 1 U + p 2 U =2U p 1 p 2 p 1 p 2 Check Shephard s Lemma: e p 1 = U (3) Duality identities r p2 p 1 = h 1 m e(p,v(p,m)) = 2v(p,m) p 1 p 2 v(p,m)= 1 m 2 p1 p 2 U v(p,e(p,u)) = 1 e(p,u) 2 p1 p 2 e(p,u)=2u p 1 p 2 x 1 h 1(p,v(p,m)) = 1 r m p2 = 1 m 2 p1 p 2 p 1 2 p 1 h 1 x 1(p,e(p,U)) = 1 2U p 1 p 2 2 p 1 = U r p2 p 1 28

29 (4) Slutsky s equation. For own-price effects, x 1 = 1 m p 1 2p 2 1 h 1 = 1 p 1 2 p p m = 1 2 p1 p 2 4 x 1 m x 1 = m = 1 m 2p 1 2p 1 4 p 2 1 m p 2 1 h 1 x 1 p 1 m x 1 = 1 4 m m p 2 1 4p 2 = 1 m 1 2 p 2 1 = x 1 p 1 For cross-price effects, x 1 x 1 p 2 =0 h 1 p 2 m x 2 = 1 2 = 1 2 p p m = 1 2 p1 p m = 1 m p 1 2p 2 4p 1 p 2 h 1 x 1 p 2 m x 2 = 1 4 m p 1 p 2 m 1 m =0= x 1 p 1 p 2 4p 1 p 2 p 2 29

30 9 Appendix 9.1 Quasi-Concavity and Quasi- Convexity Demonstrate using a Cobb-Douglas utility function u(x, y) =x 1/2 y 1/2,whichis a concave function on the non-negative quadrant. The monotonic transformation g(u) =u 4 yields a new utility fn u(x, y) =x 2 y 2. This still represents the original preference orderings, but is no longer concave. Thus concavity and convexity of a function are cardinal and not ordinal properties. A function f defined on a convex subset X < n is quasi-concave if a < the upper level set C a + {x X : f(x) a} is a convex set. 30

31 Similarly, f is quasi-convex if a < the lower level set C a {x X : f(x) a} is a convex set. Here the upper level sets are the upper contour set of the indifference curves. These upper level sets are convex in both cases, i.e. the quasi-concavity property is preserved by a positive monotonic transformation. Thus quasi-concavity and quasi-convexity are ordinal properties. If f(x) is concave (convex) then it is also quasi-concave (quasi-convex). 31

32 u(x,y) = (x,y)^ u(x,y) x u(x, y) =x 1/2 y 1/ y u(x,y) = (xy)^2 u(x,y) y x u(x, y) =x 2 y 2 32

33 9.2 Envelope Theorem (1) For Unconstrained Optimisation max x,y u = f(x, y;φ) Given the solution (x,y ) at a parameter value φ, v(φ) =f(x (φ),y (φ); φ) is the maximum-value function. Then dv(φ) dφ = f φ i.e. (x, y) can be treated as constants. Proof FOCs are, f x = f y =0 Totally differentiate f(x, y; φ) w.r.t. parameter φ at the optimal point (x (φ),y (φ)), dv(φ) dφ = f x (φ) x φ + f y (φ) y φ + f φ First two terms disappear from the FOCs. 33

34 (2) For Constrained Optimisation max x,y u = f(x, y;φ) s.t. g(x, y; φ) =0 Given the solution (x,y ) at φ, v(φ) = f(x (φ),y (φ); φ) is the MVF. Then dv(φ) dφ = L φ where L(x, y, λ; φ) is the Lagrangian function. Proof The Lagrangian optimisation is, max x,y,λ L = f(x, y;φ) λg(x, y; φ) FOCs L x = f x λ g x =0 L y = f y λ g y =0 L λ = g(x,y ; φ) =0 34

35 Totally differentiate f(x, y; φ) w.r.t. φ at (x (φ),y (φ)), dv(φ) dφ = f x (φ) x φ + f y (φ) y φ + f φ Using first two FOCs, dv(φ) dφ = λ g x (φ) x φ + λ g y (φ) y φ + f φ Differentiating the third, g x (φ) x φ Substituting this back, + g y (φ) y φ + g φ =0 dv(φ) dφ = f φ λ g φ = L φ 35

36 9.3 Quasilinear Utility Function Linear in one or more of the goods: U(x 0,x 1,..., x k )=x 0 + u(x 1,...,x k ) For a two-good case with p 0 =1, max U(x 0,x 1 )=x 0 +u(x 1 ) s.t. x 0 +p 1 x 1 m x 0,x 1 By substitution U = u(x 1 )+m p 1 x 1 FOC is u 0 (x 1 )=p 1. This is independent of m; i.e. given relative prices the same amount of x 1 is consumed no matter the income, i.e. parallel ICs in the direction of x 0. In the FOC the demand of good 1 is only a function of the price of good 1 can write the demand function as x 1 (p 1 ). The demand for good 0 is then x 0 = m p 1 x 1 (p 1 ). Substituting back into yields the indirect utility function V (p 1,m)=v(p 1 )+m where v(p 1 )=u(x 1 (p 1 )) p 1 x 1 (p 1 ). 36

37 9.4 Homogeneous Functions Definition A function u(x) is said to be homogeneous of degree k if u(tx) =t k u(x) t >0 Proporties If u(x 1,x 2 ) is a C 1 homogeneous function of degree k, a. The partial derivatives are h.d.k 1. b. The slope of the tangent line to the level sets is constant along each ray from the origin. c. (Euler s Theorem)Forall(x 1,x 2 ), u u x 1 (x)+x 2 (x) =ku(x) x 1 x 2 Property b implies that if u(x) is a utility function, then the MRS is constant along rays from the origin, and that the income elasticity of demand is identically 1. Problem: homogeneity is not an ordinal property, e.g. u = x 1 x 2 is homogeneous, but v = x 1 x 2 +1isn t. 37

38 9.5 Homothetic Functions Definition A function v(x) is said to be homothetic if it is a monotone transformation of a homogeneous function, i.e. if there is a monotonic transformation g(z) and a homogeneous function u(x) such that v(x) =g(u(x)) x in the domain. Then a monotonic transformation of a homothetic function is homothetic, and hence it is now ordinal. As MRS is an ordinal concept, homothetic functions also have property that the slopes of the ICs are constant along each ray from the origin. In fact the converse is also true, i.e. if the slopes of the ICs are constant along each ray from the origin, then the function is homothetic (but not necessarily homogeneous). 38

39 Definition A utility function v(x) is homothetic if the MRS is h.d.0. If the MRS is constant over income then a consumer s demand for each good is linear in his income. Thus if we define homotheticity by restricting the homogeneity of the underlying function u(x) to degree 1 (as Varian does), then doubling consumption of each good doubles the level of utility. Thus utility will also be linear to his income. Hence the indirect utility function is of the form v(p,m)=v(p)m where v(p) =v(p, 1). 39