& / . . . / . / . . . / .1 . . / . . . . . & ( . & ) / ( . ) / /& / . . ( . ) ) . . . ( . ) . ) . ) / . . . . ( . . ) 1 Theory of Microeconomics Main texts ; 1-Theory of microeconomics ,Walters and Layard 2- Theory of microeconomics , Henderson and quandt Suggested texts; 1-Microeconomic theory, Mas-Colell, Winston , Green. 2-Microeconomic theory, Kreps
2 Cylabsis; Main texts ; 1-Theory of microeconomics ,Walters and Layard consumer theory, ch 5 &6 producer theory and perfect competition ch 7 imperfect competition ch 8 2- Theory of microeconomics , Henderson and quandt consumer theory ch 1,2,3 producer theory ch 4,5, perfect and imperfect competition ch 6, 7,8 Suggested texts;
1-Microeconomic theory, Colell, Whinston , Green. Consumer theory ch 1,2,3,4 Producer theory ch 5, 6 perfect and imperfect competition ch 10,11, 12 2-Microeconomic theory, Kreps consumer theory ch 2,3,4 producer theory ch 7 , 19 , 20 perfect and imperfect competition , ch 8 , 9 , 10 3 W & L CH 5 Utility and consumer demand 4
demand Introduction In this chapter we concerned with the partial equilibrium analysis and not general equilibrium. Partial equilibrium analysis is the main subject of microeconomics. We are looking for the theoretical foundation of the individual demand functions , in other words ; What properties would should we expect from individual demand function. To answer it we should develop a theory of consistent choice which helps us to measure the welfare change and construct cost of living indices. Utility and consumer demand W & L CH 5 5 5-1 preference and choice Basic assumptions; 1-each individual has a preference ordering over the alternative consumption bundles which reflects his taste . He has limited income and he does not spend all of his income at once . So he has limited opportunities to choose . So he will choose the highest rank in his preference ordering that is available. This is utility maximization which leads to the Laws of demand. The key idea is; consistancy . CONSISITANCY ; the same available ordering should be chosen
at any time. Taste should not change during the period of analysis. Each individual has different taste from the other one. Different taste means different individuals. W & L CH 5 Utility and consumer demand 6 5-1 preference and choice Ordering; It is the result of examination . Every thing is compared with the other one . But, when variables(commodities) are infinitely divisible examination of all the elements is not possible. So we should have rules in order to be able to compare every pair of goods. Those who obey these rules have rational behavior. W & L CH 5 Utility and consumer demand 7 5-1 preference and choice Two rationality assumptions;
1-Comparability assumption. Confronting with two alternatives A and B, the individual should be able to choose one of the followings ; Prefer A to B, or B to A , or be indifferent between them . Otherwise, he is not rational. 2-Transitivity assumption; If at a certain point of time an individual prefers A to B and B to C , he should prefer A to C. if not , he is not rational. W & L CH 5 Utility and consumer demand 8 5-1 preference and choice The utility function. If an individual could rank all the alternative bundles from x1xn and being consistent in his decision , then we could say that he owns a utility function. It is possible to assign different sets of numbers to x 1..xn as long as those xs which have higher ranks be assigned higher numbers and still have the same utility function. As long as ranking does not change and x1 has a higher rank than x2, then any number that we assign to x 1 should be greater than x2s number.That is,
If x1 is preferred to x2 , then u(x1)>u(x2) and If x1 and x2 are in the same rank, u(x1)=u(x2) , so they belong to an indifference curve or indifference locus. W & L CH 5 Utility and consumer demand 9 5-1 preference and choice Thus, if we start with any system of numbering u(x), we can equally well use any other system w(x) provided that , if u is higher for one bundle than another , w is also higher. That is if w=f(u), f should be a strictly increasing function ,(f >0). So , u=x, u=2x, u=x2 ,or any other function which has u >0 represent the same taste and consequently the same utility function and consumer behavior. The same demand function will be resulted from each of them. Definition ; any utility function which could be replaced by any strictly increasing transformation function , is called ordinal utility function. W & L CH 5 Utility and consumer demand
10 5-1 preference and choice If we are concerned with the magnitude of the utility, and want to keep certain distance between the different utility levels , the only kind of transformation which does not change the utility function , is linear increasing transformation function . if u1-u0=u2-u1 , then we should have w=a +bu in order to have w1-w0=w2-w1. So any utility function which could be relaced only by its linear increasing transformation function is called cardinal utility function. In cardinal utility function we have to deal with the specific mathematical form of utility function ., while in ordinal utility function we do not need it. Except in welfare economics we do not need to deal with cardinal utility function. W & L CH 5 Utility and consumer demand 11 5-1 preference and choice Shape of the indifference curve. Once we have found that it is rational to assume that every one could have a utility function and an indifference locus, we have to
find out the shape of this locus. Our aim is to show that under certain rational assumptions ,the shape of the indifference curve is strictly x quasi-convex; Indifference curve y W & L CH 5 Utility and consumer demand 12 5-1 preference and choice 1-indifference curves are downward sloping This can easily proved if we assume that the individual always prefer to have more of a commodity. In this way , we have ruled out the satiation . This is called assumption of dominance. x a E E b
y W & L CH 5 Utility and consumer demand 13 5-1 preference and choice 2-Indifference curves are convex towards origin In order to show this , we have to prove that the substitution effect is negative (the compensated demand curve is downward sloping). As it is shown in the figure ; as long as the compensated demand curve is downward sloping , the indifference curve should be convex. W & L CH 5 Utility and consumer demand 14 U1x Convex to the origin Y
Px/Py Px1/Py E1 E Ux X3 X2 X X1 Compensated demand Px Px1 Total effect Px Substitution effect W & L CH 5 Ordinary demand X3 X2
X1 X Utility and consumer demand 15 5-1 preference and choice apple Lexicographic ordering is ruled out, other wise we loose the continuity assumption ; I prefer that bundle which has more oranges regardless of the number of apples. Only if the number of oranges are the same, I prefer that bundle which has more apples. E We can not find any other point with the same utility as point E , u(E) oranges W & L CH 5 Utility and consumer demand
16 5-1 preference and choice Suppose that Px will increase and Py remains unchanged. Income will also increase to compensate for the income effect of increase in Px . Budget constraint change from A B to A B. The initial equilibrium point is E . In order to show the negative substitution effect we should be able to prove that the new equilibrium point lies on E B. W & L CH 5 Utility and consumer demand 17 5-1 preference and choice When price of x increase and income is compensated q x will decrease , which means negative substitution effect. qy B Consistent with E
Starting e B point E Px/Py inconsistent with E Px/Py< P1x/Py P1x/Py e X2 W & L CH 5 X1 qx A Utility and consumer demand A 18 5-1 preference and choice
We actually observe the consumer equilibrium points like E, e , and e (or the expenditure pattern of the consumer ) and then conclude about the shape of the indifference curve. This is called revealed preference approach we will discuss this in detail latter in this chapter .So in the normal markets at any point which consumption occurs (like E or e ) indifference curve must be convex to origin (corner solutions are ruled out). W & L CH 5 Utility and consumer demand 19 5-1 preference and choice 3-indifference curves are strictly convex to the origin. What we have proved so far, does not rule out the cases like the following. In other words, strict convexity has not been approved yet. Consumption does not x2 occur in this range Px2=1 P1x1 P2x1
P3x1 W & L CH 5 Px1 x1 Compensated demand curve is downward sloping but it is broken x1 Utility and consumer demand 20 5-1 preference and choice The condition for strict convexity of indifference curves . For all x0(x10, x20) and x1(x11,x21) for which u(x0)=u(x1) we should have the following relation if indifference curve is strictly convex. u[xxx0+(1-) xx) x1]>u(x1)=u(x0) , 0<<1x<<11 That is ; the utility of a combined basket of any two baskets ; x0(x10,x20) and x1(x11,x21) is greater than the utility of each of them, when u(x0)=u(x1)
W & L CH 5 Utility and consumer demand 21 5-1 preference and choice U(x) U(x2) U[xx1+(1)x2] u(x1)+(1-)u(x2) U(x1) x1+(1- )x2 0< <1 x xx11 W & L CH 5 xx2 2 Utility and consumer demand 22 U(x)
U[xx0+(1)x1] u(x0)+(1)u(x1)=u(x0)=u(x1) U(x0) x1 U(x1) Indifference Surface x01 x0 X 2= )x1 Indifference curve x12 x11 W & L CH 5 x0+(1- x1 x02 x22
x12 Utility and consumer demand 23 x2 4-Indifference curves have well defined first and second derivatives. What we have proved so far has not ruled out the cases in which the indifference curves have kinked point . In kinked points the derivatives are not defined , so we should have well defined first and second derivatives for all the points on the indifference curves. W & L CH 5 Utility and consumer demand 24 y p1x/py p2x/py px3/py px4/ py
Indifference curve E x p 1x p2x p 3x p 4x W & L CH 5 x Utility and consumer demand 25 5-1 preference and choice The assumptions for having well defined downward sloping strictly convex indifference curve . Assumption of rationality; completeness and transitivity. Assumption of dominance; more is preferred and no saturation point. Negative substitution effect; consumers do not buy expensive goods even if they are being compensated
(consistancy). Strictly convex indifference curve; Combination of two equal utility valued basket of commodities is always preferred to each one of them. Differentiable indifference curves. No kinks in indifference curves. We should not have the same quantity for only a specific range of prices. W & L CH 5 Utility and consumer demand 26 Y X W & L CH 5 Utility and consumer demand 27 5-1 preference and choice Q5-1 Can indifference curve cross? If not which assumption rule this out. No they can not. The transitivity assumption rule this out.
W & L CH 5 Utility and consumer demand 28 5-1 preference and choice Q5-2 (I) a householder can buy x1 and x2 only and p2=1.in situations A and B he behaves as follows . Is his behavior consistent? Situation income p1 x1purchased A 40 1 20 B 60 2 25 W & L CH 5 Utility and consumer demand 29 When he choose A ,he can choose B also
when he choose B he can choose A also Q5-2 solution (I) x2 60 so, A and B are not consistent M=x1p1+x2p2 40=(1)(20)+(1)(20) 40 60=(2)(25)+(1)(10) A 20 B 10 W & L CH 5 20 25 30 40 Utility and consumer demand
x1 30 5-1 preference and choice Q5-2 cont. (ii) Another householder behaves as follows. Again p2=1 . Situation Income p1 x1purchased A 40 1 20 B 61 2 15 Is he consistent? Can you say in which situation he is better of. W & L CH 5 Utility and consumer demand 31 A and B are consistent. Q5-2 solution
(ii) Whenever he can choose A & B he choose B When he choose A he can not choose B x2 61 He is better off in B M=x1p1+x2p2 40=(1)(20)+(1)(20) 40 31 61=(2)(15)+(1)(31) B A 20 W & L CH 5 40 15 20 30. Utility and consumer demand 5
x1 32 5-1 preference and choice Q5-3 By drawing the indifference curves, show how the rational consumer might consume at a boundary point (say x1=0) where indifference curves were non convex. W & L CH 5 Utility and consumer demand 33 5-1 preference and choice Q5-3, solution Y non convex Indifference curves Max utility point
X Budget constraint W & L CH 5 Utility and consumer demand 34 5-1 preference and choice Q5-4 (I) by drawing the indifference curves , show how he might consume at any interior point where indifference curves were non convex if he were faced with a rising marginal cost of x1 in terms of x2. (ii) What first and second order condition apply at this point? (iii) What market situation is this person likely to be in . W & L CH 5 Utility and consumer demand 35 5-1 preference and choice Q5-4
X2 Utility max (ii) F.O.C. (p1/p2) MRSx1x2= MRTx1x2= S.O.C. dMRSx1 x2 /dx1< dMRTx1 x2/ is in a monopsony situation . (iii)dxHe 1 He should pay higher price for x1(p1/p2) If he wants to buy more of it. X1 indifference curves W & L CH 5 Budget constraint Utility and consumer demand 36
5-1 preference and choice Q5-5 Which of the following types of remark corresponds to non convex indifference curve : A : I would rather spend all my time in the country or all in the town, rather than divide myself between the two. B: I prefer a mixture of town and country life to being restricted to one or the other . (ii) which type of preferences will produce a more stable economy. W & L CH 5 Utility and consumer demand 37 5-1 preference and choice Q5-5 , solution (I) Type A will produce a non convex indifference curve, since it results to corner solution. (ii) Type (ii) will produce a more stable economy , since it contains less fluctuations . W & L CH 5
Utility and consumer demand 38 a- Revealed preference and consumer welfare lecture notes , MIT . b- chapter seven revealed preferences. W & L CH 5 Utility and consumer demand 39 5-2 The laws of demand Max u(x1,x2) S.T. P1x1+p2x2=m L=u(x1,x2)+(m p1x1 p2x2) U1= P1 (u1/u2)=p1/p2 U2= P2 F.O.C. m=p1x1+p2x2 = (u1)/(p1)= (u2)/(p2)=marginal utility of income. For utility to maximized, marginal utility of income from expenditure on each commodity should be
the same. du=u1dx1+u2dx2=0 P1dx1+p2dx2=0 (u1/u2)=p1/p2 W & L CH 5 Utility and consumer demand 40 5-2 The laws of demand du=u1dx1+u2dx2=0) d2u=u11dx1dx1+2u12dx1dx2+u22dx2dx2<0 (dx1/dx2)= S.O.C. (p2/p1) u11(dx1/dx2)2+2u12(dx1/dx2)+u22<0 u11(- p2/p1)2+2u12(- p2/p1) +u12<0 u11p22 - 2u12p1p2 + u22p12<0 U11 U12 - P1 U21 U22 -P2 >0 - P1 - P2 0 Having FOC & SOC we will get demand X1=f 1 (p1,p2,m), X2=f 2 (p1,p2,m), W & L CH 5 Utility and consumer demand
41 5-2 The laws of demand First law : Homogeneity of degree zero With respect to all prices and income Xi=f(p1,p2,..,m) homogenous of degree homogenous of degree zero in prices and income Applying Eulers theorm; j(dxi/dpj)pj+(dxi/dm)m=0, divide by x jeij+im =0 y E M/py= nM/(npy) W & L CH 5 Utility and consumer demand unchanged M/px=nM/(npx) x 42 5-2 The laws of demand 2- Income effects must add up. ipixi=m , ipi(dxi/dm)=1 i(PiXi/m)(dxi/dm)(m/xi)=1
ivii =1 Generalized Engels law since vi<1, some of i>1,and some are less than one. Engel proved it by looking at household budget expenditure data. If i>1, share of the commodity(i) increases with increase in income , so it is luxury. If i<1 , share of the commodity(i) decreases with increase in income , so it is necessity If i<0 , consumption of (i) decreases with increase in income , so it is inferior. If i>0 , consumption of (i) increases with increase in income , so it is normal. W & L CH 5 Utility and consumer demand 43 5-2 The laws of demand 3-Negative substitution effect When price of x increases and income increases simultaneously to keep the consumer at the same utility level, consumption of x will decrease. (dx1/dp1)= (dx1/dp1)u +[x x1(dx1/dm)] (dx1/dp1)u = substitution effect x1(dx1/dm)=income effect=compensation needed to /dm)= keep him at the same utility level.
(dxi/dpj)= (dxi/dpj)u xj(dxi/dm) (dxi/dpj) (pj/xi) = (dxi/dpj)u (pj/xi) (pj xj /m) (m/xi ) (dxi /dm) eij = e*ij vj im W & L CH 5 Utility and consumer demand 44 5-2 The laws of demand In order to prove that the substitution effect is negative, we take total differential from the first order condition and find substitution effect U11 u12 -p1 dx1 dp1 U21 u22 -p2 dx2 = dp2 -p1 -p2 0 d - dm+x1dp1+x2dp2 dx1= dp1(D11/D)+ dp2(D21/D)+(-dm+x1dp1+x2dp2)D31/D Dij = co-factor of the element in ith row and jth column. (dp1=dp2=0) , W & L CH 5
dx1/dm= - D31/D Utility and consumer demand 45 5-2 The laws of demand X1dp1=dm change in income required for the compensation of price change. (dx1/dp1)u=D11/D [x(dp2=0), x1dp1=dm] Income effect = [x x1(dx1/dm)] = -[x(dm/dP1)(dx1/dm)] (dx1/dp1)= (dx1/dp1)u +[x x1(dx1/dm)] (dx1/dp1)u=D11/D = substitution effect dx1/dp1=D11/D+x1D31/D [x(dp2=0), dm=0] Total effect=substitution effect+income effect D>0, D11=-p22<0 , =ui/pi>0, D11/D<0 U11 u12 -p1 U21 u22 -p2 = D -p1 -p2 0 u22 -p2 p2 0 Utility and consumer demand 46
5-2 The laws of demand X1 Derivation of slutsky equation. Xi =xi(p1 ,p2 ,m) i=1,2 Xi= ordinary demand, i=1,2 U=U(X1,X2 )=U[xx1(p1,p2,m),x2(p1 ,p2 ,m)] u=u(p1,p2,,m) , m=m(p1,p2,u) X1=X1(p1 ,p2 ,m)=X1C (p1,p2,m(p1,p2,u)) U(X1* ,X2* ) = U* X1* m(P1 ,P2 , U*)=m* X2 X2* = X2(P1 ,P2 , m* )= Xc2 (P1 P2 , U* ) dx1c/dp1=dx1/dp1+(dx1/dm)(dm/dp1)u dx1c/dp1=dx1/dp1+(dx1/dm)x1 Sub effect =total effect income effect [x x1(dx1/dm)=income effect] W & L CH 5 Utility and consumer demand
47 5-2 The laws of demand D11/D<0 substitution effect<0 dx1/dm= - D31/D -X1(dx1/dm)=x1D31/D =income effect which is less than zero for inferior goods. (dx1/dp1)= (dx1/dp1)u x1(dx1/dm) If absolute value of income effect of an inferior good (dx1/dm<0) is greater than substitution effect , demand curve is upward sloping and the good is giffen good. Which is very rare. Since for Giffen good we should have eii = e*ii viim >0 , so | e*ii| <| viim|, since Vi <1, income effect of an inferior good should be very large which is not happening usually. W & L CH 5 Utility and consumer demand 48 Increase in price leads to x2 u 1
increase in quantity demanded E X12,>X11 p12>p11 substitutio n Income effect E Total effect W & L CH 5 u2 X11 X12 Utility and consumer demand x1 49 4- symmetrical cross Effect. dxi/dpj=Dji/D xjdxi/dm=(dxi/dpj)u xjdxi/dm
uij=uji , so the uij matrix is symmetrical and so, Dij=Dji , so, (dxi/ dpj)u= (dxj/dpi)u A matrix with (dxi/dpj)u as its elements is called Slutsky matrix. Gross and Hicks &Allen substitute; dxi/dpj>0 Gross substitute dxi/dpj< 0 Gross complement If good i has big income elasticity(large share in the household expenditure like housing) and good j has small share(like entertainment) i could be gross substitute for j (dxi/dpj)>0 , but j could be gross complement for i (dxj/dpi)<0. Pj , Xi (sub), Xi (inc.) but (inc ) <<1 (sus) Xi ,( dxi /dpj )>0 Pi , Xj (sub), Xj (inc.) but (inc ) > (sus) Xj ,( dxJ /dpI )<0 W & L CH 5 Utility and consumer demand 50 5-2 The laws of demand If If If If (dxi/dpj)u>0 Hicks-Allen substitute (dxi/dpj)u<0,Hicks-Allen complement
(dxi/dpj)u >0 , then(dxj/dpi )u >0 commodity i is hicks-Allen substitute for j, then j is also a Hicks Allen substitute for i too. In defining the Hicks Allen substitute ,the consumer remains on the same indifference curve (satisfaction level ). Every good must have at least one Hicks-Allen substitute, but may have no complements. eij=e*ij vjim, Sum over change in all prices j eij= j e*ij j vjim j vj =1 j e*ij = j eij +im =0 the same concept as the first law. e*ii <0 so at least one of e*ij should be positive . W & L CH 5 Utility and consumer demand 51 5-2 The laws of demand e*ij can not serve as a measure for measuring the degree of substitubility , because it lacks the symmetrical property; e*ij =(dxi/dpj)u(pj/xi)#(dxj/dpi)u(pi/xj)=e*ji Define Allen elasticity of substitution; ij=(e*ij/vj) ij=(e*ij/vj)=(dxi/dpj)u(pj/xi)(m/pjxj)= (dxi/dpj)u(m/xixj) =(dxj/dpi)u(pi/xj)(m/pixi)=
=(e*ji / vi ) = ji W & L CH 5 Utility and consumer demand 52 5-2 The laws of demand Q5-6 Derive the demand curve for x1 , x2 ,assuming ; (I) u=x1x2 , (II) u=x1a1x2a2 What are the price and income elasticity; Max x1x2+(m p1x1 p2x2 ) x1= p2 x1=m/2p1 x2= p1 x2=m/2p2 p1x1+p2x2=m (II) x1=a1/(a1+a2)(m/p1) x2=a2 /(a1+a2)(m/p2) (I)e11=(dx1/dp1)(p1/x1)= (-m/2p2)[xp1/(m/2p1)] = -1,e22 =-1 e12= e21=0 1m=(dx1/dm)(m/x1)=(1/2p1)[xm/(m/2p1)]=1 2m=1 the same number will be found for part (II). W & L CH 5
Utility and consumer demand 53 5-2 The laws of demand Q5-7 (I) - Prove that the utility maximization quantities x1 and x2 are the same whether we maximize u(x1,x2) or w=f[xu(x1,x2)]. (II) - The marginal utility of an inferior good is negative , which is why demand falls with income. True or False? Explain. W & L CH 5 Utility and consumer demand 54 5-2 The laws of demand (I) Max u(x1,x2) S.t. m=x1p1+x2p2 u1/u2=p1/p2 Max f[xu(x1,x2)]+(m p1x1 p2x2) fu1 p1 = 0 fu2 p2 = 0 u1/u2=p1/p2
(II) False. marginal utility is negative when consumption is saturated .For inferior goods income effect is negative since consumer substitute better quality good for it. W & L CH 5 Utility and consumer demand 55 5-2 The laws of demand Prove that the SOC condition for utility maximization is equal to d(dx2/dx1)/dx1>0 Subject to (dx2/dx1)=-( u1/u2). U11 U12 - P1 U21 U22 -P2 >0 - P1 - P2 0 p22u11 - 2p1p2u12 + p12 <0 W & L CH 5 Utility and consumer demand 56 5-2 The laws of demand d(dx2/dx1)/dx1= d[x-u1(x1,x2)/u2(x1,x2)]/dx1>0
x2 MRS=u1/u2= - dx2/dx1=-(-5)=5 MRS=u1/u2= - dx2/dx1=-(-3)=3 x1 W & L CH 5 Utility and consumer demand 57 5-2 The laws of demand d(dx2/dx1)/dx1= d[x-u1(x1,x2)/u2(x1,x2)]/dx1 = {[x-u2(u11dx1+u12dx2)+u1(u21dx1+u22dx2)]/dx1} /u22 = [x-u2(u11+u12dx2/dx1)+u1(u21+u22dx2/dx1)]/u22 substituting [xdx2/dx1]= -(u1/u2) we will get (- 1/u23)(u22u11 2p1p2u12 + u12 ) which should be greater than zero . Substituting FOC as U1 = p1 ,U2 = p2 we will get ; 2 2 )<0 >0,
+ p1demand W&L ( CHp 52 u11 2pUtility and 1p2u 12consumer 58 5-2 The laws of demand Q5-9 (I) If x1=p1a1p2a2mb1 what are the values of e11, e12, and 1m? What restriction must govern the magnitudes of a1 , a2, and b to satisfy law 1. (II) Is this function (and a similar one for x2) necessarily consistent with law 2. (I), a1 + a2 + b1 =0 (II) No, it is not necessarily consistent with law 2 . Since we should have v1b1+v2b2 =1 W & L CH 5 Utility and consumer demand 59 5-2 The laws of demand Q5-10
we have estimates for British working class families of the income elasticity of demand for the following goods; rice, condensed milk, butter, margarine, tea, coffee, drink, carpets, theater visits. (I) which goods would you expect to be luxuries, which inferior, which non-inferior non-luxuries. Luxuries: carpet, theater visit , alcoholic drink. Inferior ; margarine, condensed milk Non-inferior, non-luxuries ; rice butter, tea coffee, soft drink. W & L CH 5 Utility and consumer demand 60 5-2 The laws of demand Q5-10 (II) If you would like to distribute a given amount of money in subsidies to these goods, so as to benefit the poor as much as possible, which goods would you subsidies and why? Necessities should be subsidize, because necessities have a larger portion in the basket of commodities of poor people. W & L CH 5
Utility and consumer demand 61 5-2 The laws of demand Q5-10 (III) prove that a subsidy to a necessity, financed by a proportional tax, is progressive. The progressive tax is the one in which the net-benefit decrease with increase in income .The benefit which a person receives from subsidy (assuming everyone has the same price elasticity of demand) is proportional to his consumption of good. X=pa1ma2 Benefit from subsidy=B=bX=bpa1ma2 Cost of subsidy=dm=proportional tax (B C)/m = bpa1ma2-1 - d If a2<1, (necessity), then when m increases, (b c)/m will decrease ,which is a indication of a progressive policy. W & L CH 5 Utility and consumer demand 62 5-2 The laws of demand Q5-11 (I)
Suppose that the gross price elasticity for a good is 1 , its income elasticity 2 , and its income share 1/4 .what is the compensated price elasticity. e*ii= eii + vi im=-1+0.25(2)=-0.5 W & L CH 5 Utility and consumer demand 63 5-2 The laws of demand Q5-11(II) Suppose that in some country the effect of one percent rise in the price of oil is to reduce the demand for scooters by 0.1 percent. The following data are available. oil=j scooters=i V 5% 1% m 2 3 Are oil and scooters Hicks-Allen substitute or complements ? Can you think of any explanation for this result ? How would a rise in scooter prices
affect the demand for oil. W & L CH 5 Utility and consumer demand 64 5-2 The laws of demand Q5-11 (II) e*ij=eij+vj im=-0.1+0.05(3)=0.05>0. i (scooters) is Hicks-Allen substitute for j (oil) .when price of oil (j) increases, taking into account the income effect, the scooter consumption will decrease along with gasoline consumption ( eij =-0.01<0) .But if only the compensated effect is taken into account, the consumer wants to remain at the same utility level and is willing to replace scooters for car in order not to walk more than before. So e*ij> 0. eji=e*ji - vi jm =viij-vi jm=vie*ij/vj - vi jm=-0.01 W & L CH 5 Utility and consumer demand 65 5-2 The laws of demand
Q5-12 (I) The following additional elasticity formula also holds iviij= - vj (Cournot aggregation condition). Prove this by differentiating the budget equation. ipixi=m ipi(xi /pj)+xj=0 i[x(pixi)/m](xi /pj)(pj/xi)=(-pjxj)/m W & L CH 5 Utility and consumer demand 66 5-2 The laws of demand Q5-12 (II) Prove iviij*=0 by differentiating the budget equation and adjusting m to hold u constant. ipi(xi /pj)+xj=m/pj dm=xjdpj, m/pj=xj i(pixi/m)(xi /pj)u(pj/xi)=0. W & L CH 5 Utility and consumer demand
67 5-3 Duality and Welfare change If we want to measure the changes in the cost of living and real income we could use the dual approach to utility maximization, that is cost minimization. Max u(x1 ,.xn) These are the same function .. ) S.t. m=pixi xi=xi(p1,pn, m) ordinary demand One to one relation u=u(x1 ,,.. Xn)= u[xx1 (p1,..pn, m) ,,.. Xn (p1,..pn, m) ] between m and u u(p1,..pn, m) indirect utility function indicates the maximum Utility obtained at each level of income and prices. Min m=pixi s.t. u(x1,x2,.. xn)=u0 xic=xic(p1,pn, u) compensated demand m= pixi=pi xic(p1,pn, u)=m(p1,pn, u)=C(p1,pn, u)=consumer cost function indicates the minimum money cost at which a given utility can be obtained at given prices W & L CH 5 Utility and consumer demand
68 Duality and welfare change x2 u0=u0(x1,x2 ) u0=u0(x10(p10,p20,m), x20(p10,p20,m))= u0(p10,p20,m0) , m0(p10,p20,u0) x02 p10/p20 0 x1 =x1(p10,p20,m0(p10,p20,u0))=x1c(p10,p20,u0) 0 W & L CH 5 m0=x1xp110+x2p2 Utility and consumer demand 69
5-3 Duality and Welfare change Characteristics of the indirect utility function u=u(p1,p2 m) 1- decreasing in p and increasing in m 2- homogeneous of degree zero in prices and m 3- quasi-concave in price changes. The lower is p1, the more p2 has to rise to compensate you for a further fall in p1. P2 U0 = U(P1 , P2 , m0 ) W & L CH 5 Utility and consumer demand P1 70 5-3 Duality and Welfare change Characteristics of the consumer cost function C =C(p1,p2,u) 1-Increasing in prices and increasing in utility. 2-Homogeneous of degree one in prices. If all prices double ,the income (expenditure) needed for a given utility doubles.
3- C(p1,p20,u0)/p1=x1(p1,p20,u0) = (compensated demand curve) If p1 increase by one very small unit ,the cost to the consumer of maintaining his original utility is almost equal to x1 (the compensation which needed) W & L CH 5 Utility and consumer demand 71 5-3 Duality and Welfare change 4- 2C(p1,p2 ,u0)/p1p2= 2C(p1,p2 ,u0)/p2p1 (x1/ p2)u= (x2/ p1)u Symmetry of the cross substitution effect. 5- Roys identity; getting ordinary demand from indirect utility function. P1 and m will change and p2 remains constant If dm=x1dp1 then u0 =u(p1,p2,m) (u/p1)dp1+( u/m)dm=0 dm=x1dp1 - [x(u/p1)/( u/m)]= dm/dp1=x1 x1(p1,p2,m)=-[xu(p1,p2,m)/ p1]/[xu(p1,p2,m)/ m] W & L CH 5 Utility and consumer demand
72 5-3 Duality and Welfare change Welfare change and price indices It is often more practical to work from indirect utility function (consumer cost function) .we use consumer cost function to measure welfare change Suppose p1 rises from p10 to p11. If a is the compensation needed to make the consumer as well as before. a=C(p11, p20,u0) - C(p10, p20,u0) if a>0 then welfare has decreased (price increase) if a<0 then welfare has increased (price decrease) welfare change = dw = - a W & L CH 5 Utility and consumer demand 73 5-3 Duality and Welfare change Welfare change=C(p10, p20,u0) -C(p11,p20,u0) = p11 [xC(p1, p20,u0)/p1] dp1
p10 CV= p11 [xx1(p1, p20,u0)] dp1 p10 CV = compensating variation , it is the change in the area between the compensated demand curve and the price line. When price increases the welfare change is negative, and it is equal to S(a) in the figure below. W & L CH 5 Utility and consumer demand 74 Duality and welfare change p1 Compensating variation
P1 P 1 1 0 )X10 - X11-( CV= - [xarea(a+b)-area(b)] = p 1 1 p10 c a W & L CH 5 a x11 b b
- [xpp1x10 +(1/2) pp1(px1)u] = -pp1[xx10 +(1/2)(x1/p1)upp1] X1(p1,p20,u0) (x1/p1)u= (x1/p1) +x1(x1/m) x1 0 x 1 consumer demand Utility and 75 Duality and welfare change If two prices change simultaneously , the CV should be calculated for each of them according to the relevant demand curves. Suppose that commodity 1 and 2 are substitute for each other, and p1 decreases from p10 to p11 due to a government project. The CV could be calculated as follows; W & L CH 5
Utility and consumer demand 76 Duality and welfare change p1 s1 1 X1(p1,p20,u0) s1 3 4 p10 P11 p12 X2(p10,p2,u0) p2 2 s2
2 p20 cv1 cv3 5 p2 1 p2 2 cv2 cv4 X1(p1,p21,u0) X1(p1,p22,u0) x1 X2(p11,p2,u0) X2(p12,p2,u0) x2 Government project ; shift of the s1 to s1
W & L CH 5 Utility and consumer demand 77 Duality and welfare change CV1=C(p10,p20,u0) - C(p11,p20,u0) CV2=C(p11,p20,u0) - C(p11,p21,u0) CV3=C(p11,p21,u0) - C(p12,p21,u0) CV4=C(p12,p21,u0) - C(p12,p22,u0) Total = CV=CV1+CV2+CV3+CV4 CV= C(p10,p20,u0) - C(p12,p22,u0) CV is independent of the path of prices chosen when integrating under compensated demand curve. ( we can either compute CV1 first or any other CVi , and then compute the other CVs . The results are the same. ) Cost of living index in period i= I I = C(p1i,p2i,u0) / C(p10,p20,u0) W & L CH 5 Utility and consumer demand 78 Duality and welfare change
Q5-14 Suppose u=xy and x is wheat.find the consumer cost function, if py=1. Suppose income is $100 and the price of wheat falls from $1 to $0.25 due to the success of Indus Valley project. Evaluate consumers gain in terms of CV . What is the price index(for u=u0) after the project(setting the pre project price index at unity)? W & L CH 5 Utility and consumer demand 79 Duality and welfare change Q5-14, solution; Max u=xy , s.t. m=xpx+ypy , py=1, x=m/(2px) , y=m/(2py) p=px/py = px u=xy=m2/(4p) , C=m=(4pu) u=u0=1002/[x4(1)], C=(4pu0)=100p CV=C(p0,u0) - C(p1,u0) CV=100 100(0.25) = 50 pice index for u=u0 ; [xC(p1,u0)/ C(p0,u0)]=50/100=0.5 W & L CH 5
Utility and consumer demand 80 Duality and welfare change Q5-15 If u=xy, py =1 ,m=100, derive the compensated demand curve for x and compare its slope with the ordinary demand curve .suppose p(=px/py) falls from 1 to 0.25 . Use the method of integration to obtain the CV. Confirm that the result is the same as you obtained in question q5-14. W & L CH 5 Utility and consumer demand 81 Duality and welfare change Q5-15 solution C=(4pu0)=100p [xQ5-14] x(p,u0)= (c/p)=50p-1/2 eii*= (x/p)(p/x)=-1/2 x(p,m0)=m/(2p)=50p-1
eii= -1 , ordinary demand is more elastic than compensated demand . CV= [x50p-1/2] dp=[x100p ] = 100-50=50 W & L CH 5 Utility and consumer demand 82 Duality and welfare change Q5-16 Suppose that at the same time as the change in question 5-14 occurred, the price of y had risen to 2.25. Is this sufficient to cancel out the advantages of the fall in price of x ? What is the CV for the two price changes and what is the new price index? W & L CH 5 Utility and consumer demand 83 Duality and welfare change Q5-16, solution Max u=xy
s.t. , m=xpx+ypy x=m/(2px), y=m/(2py), u=m2/(4pxpy) C=m=(4pxpyu0)1/2=100px1/2py1/2 CV=C(p0,u0) - C(p1,u0) CV=100-100(0.25)1/2(2.25)1/2=100-75=25 W & L CH 5 Utility and consumer demand 84 Duality and welfare change Measurement of welfare change Consumer surplus ; Consumer surplus is the difference between what the consumer is willing to pay ,and the amount which he actually pays for a commodity. The willingness of the consumer can be measured by the area under the compensated demand curve p Compensated X(p,u 0) demand Q
q0 W & L CH 5 Utility and consumer demand 85 Duality and welfare change The amount which the consumer actually pays for the commodity is pqp Consumer surplus x p qx q Consumer surplus can not be negative. It could be zero when demand is horizontal .it can be illustrated more by the following example. W & L CH 5 Utility and consumer demand 86
change Compensated demand (p=willingness to pay) P; 10 9 8 7 6 5 4 3 2 1 0 Q; 0 1 2 3 4 5 6 7 8 9 10 (p=10,Q=0) first buyer is willing to pay 9 (p=9 ,Q=1) CS=0 but he (p=8,Q=2) CS=1 pays 8 , so consumer surplus is (p=7,Q=3) CS=2+1=3 equal to 1 ) (p=6,Q=4) CS=3+2+1=6 (p=5,Q=5) CS=4+3+2+1=10 W & L CH 5 Utility and consumer demand 87 Duality and welfare
change px if price=5 cs for the first one=4(95) 10 9 cs for the second one =3 8 7 6 5 cs for the third one =2 4 3 2 1 0 1 2 3 4 5 6 7 8 9 W & L CH 5
cs for the fourth buyer=1 qx 10 cs for the fifth buyer=o Utility and consumer demand 88 Duality and welfare change Y, other commodities my2=maximum amount he is willing to pay for x1 amount of x . He is not willing to pay more than this ,since he will be worse than before when he was not buying anything (u=u0). m=total income y1 E y2 Buying x1 amount of x from the
(py=1) market increase the utility from u0 to u1. He is gaining additional utility in form of the consumer surplus. my1= the amount that he actually pays for buying x1 amount of x in the market U 0 my2 my1 = y1y2 = consumer surplus P=(px/py)=px W & L CH 5 x1 Utility and consumer demand U1 x 89 Duality and welfare change
CS= x1 0 [xMRSxy(x,u0) p]dx MRSxy(x,u0) = maximum is willing to pay for each unit of x P=the amount actually paid for each unit of x If p=5, q=5 , my1=(5)(5)=25 my2=5+6+7+8+9=35 = X=5 [xMRSxy(x,u0) y1y2=cs=35-25=10 W & L CH 5 0 Utility and consumer demand 90 Duality and welfare change Suppose that the consumer is asked what is the maximum that he is willing to pay if he could have x1(=5)units of x or nothing . He is willing to pay at most an amount equal to my2. Any amount greater than that would
make him worse than before (u=u0). So the price for each unit of x is p*= my2/x1. P* is all or nothing price for buying x1 units of x . If we repeat this for different size of x , we will get all or nothing demand curve. W & L CH 5 Utility and consumer demand 91 Duality and welfare change Px(y per x) x * P*P(x)=(1/x) p(x)dx x 0 P* x px All or nothing demand curve All or nothing demand ordinary demand
P* P =35/5=7 = 35/5=7 * xx x Px= 25/5=5 If W & L CH 5 x=5 x1 Utility and consumer demand 92 Duality and welfare change to Variations Besides consumer surplus , two other measures could be used to measure the welfare changes . Compensating variation (CV) and equivalent variation (EV) . 1-Compensating Variation (CV) The amount of money we can take away from an individual after an economic change , while leaving him as well off as he was before
it. Suppose that the change is the availability to buy x from the market. t=0 when he does not have access to market t=1 when he has access to market if the consumer moves from t0 to t1 , he gains welfare from purchasing in the market Otherwise he would not do that . W & L CH 5 Utility and consumer demand 93 Duality and welfare change y (Py=1) t=0 ( before buying from market) e located at point m when x=x0 t=1 ( after buying from market) Located at point E or any other point on
the budget line Total income CV B m Equilibrium E point u1 c A X 0 W & L CH 5 q s x1
u 0 u0 x Utility and consumer demand maximum amount that could be taken in order to make him not worse than before(u0)= maximum he would be willing to pay for the change from t=0 to t=1. Increasing his utility from u0 to u1 Px/Py=Px 94 Duality and welfare change If the change was denying access to the market,(t1 t0), CV is equal to em (CV<0). He should be compensated for an amount equal to (em) in order to make him as well off as before. For a welfare gain (t0 t1) , (CV>0) is the amount that he would be willing to pay for
the change. For a welfare loss (t1 t0) , (CV<0) is the amount he would need to receive as compensation for the change. W & L CH 5 Utility and consumer demand 95 Duality and welfare change 2-Equivalent Variation (EV) Equivalent riation. The equivalent variation (EV) is the amount of money we would need to give to an individual , if an economic change did not happen , to make him as well off as if it did. Suppose that the change was a welfare gain defined as the availability to buy from the market ( t0 t1). while he is not able to buy from the market . The amount of compensation (EV>0) is equal to ( em ) which make him as well off as he has access to the market and buy x1 units of x . If the change was a welfare loss defined as preventing him to buy from the market while he was buying from the market (t1 t0), the amount of money which is needed to compensate him if the change happens is mc (EV<0). In other words he
is willing to pay mc at most to avert the change. Or we can take from him at most an amount equal to mc . W & L CH 5 Utility and consumer demand 96 Duality and welfare change (Py=1) e y t=0( before buying from market) t=1( after buying from market) located at point m when x=0 Located at point E or any other point on theThe budget line amount
EV CV m B Equilibrium E point u1 c A W & L CH 5 q u0 x1 x s
Utility and consumer demand which is needed to upgrade his utility to u1 while he is not buying anything from the market . Px/Py=Px 97 Duality and welfare change For a welfare gain,both CV and EV are positive and for a welfare loss both are negative. The difference between CV and EV can be seen easily from the consumer cost function. Suppose that the change in consumer situation in change in prices from p0 to p1 . CV= C(p0,u0) - C(p1,u0) C(p0,u0) =A= his expenditure for maintaining u 0 with p0 at t=t0 (what he had before change in price which is equal to his income). C(p1 ,u0 ) = B the expenditure which he needs to maintain u0 with p=p1 at t=t1 (what he needs after change in price ) (A B )=CV is the amount which could be taken away from consumer after change in price and let him be as well off as
before W & L CH 5 Utility and consumer demand 98 Duality and welfare change EV=C(p0,u1) - C(p1,u1) C(p0,u1)=A=the expenditure which he needs to maintain u 1 with p=p0 at t=t0,before change in prices. C(p1,u1) =B (his original income = C(p0,u0) )= the expenditure which he should spent after price changes from p 0 to p1 , in order to reach the utility level u 1 when income does not .change (A-B) =EV should be given to consumer (before the change in price at t=t0 ) in order make him as well off as after the change (u1) As it is seen CV measures the welfare change based on the previous (original) utility level before the change (t=t 0, u=u0) while EV measures the change based on the future (final) utility level after the change ( t=t 1, u=u1 ). W & L CH 5 Utility and consumer demand 99
Duality and welfare change So far the move is considered from t0 to t1. Now suppose that we consider the move from t1 to t0 . It could be shown that CV of move from t1 to t0 equals minus EV of move from t0 to t1. That is ; CV(1 to 0 )=C(p1,u1)C(p0,u1) = - EV(0 to 1) EV(1 to 0 )=C(p1,u0)C(p0,u0) = - CV(0 to 1) Move from t0 to t1 is a welfare gain, gain while move from t 1 to t 0 is a welfare loss. loss As it can be seen from the figure , EV for a welfare loss must be finite ,so must the CV for a welfare gain. W & L CH 5 Utility and consumer demand 100 Duality and welfare change (Py=1) y e EV
CV m What is the relative size of EV and CV? for normal goods , we have MRSxx/y>0 (as income increase, compensated price or MRS will increase too), so EV>CV as it is MRS B>MRS EV=me>AB>mc= shown; A CVis also true for welfare loss This MRSE>MRS B For Welfare gain(t0 to t1); q EV0= em> CV0=mc E
c A q s x1 W & L CH 5 u u1 For Welfare loss(t1 to to); 0 CV1= - EV0=- em<- mc= CV0=EV1 x Utility and consumer demand Px/Py=Px 101 Duality and welfare change What is the relative size of EV and CV?
y for inferior goods , we have e EV m E CV MRSxy/y<0 (as income increase, compensated price or MRS will ), so EV
s A u0 x W & L CH 5 Utility and consumer demand 102 Duality and welfare change What is the relative size of EV and CV? y for goods with zero income effect e , we have MRSxy/y=0 (as income increase, compensated price or MRS will not change ), so EV=CV as it is shown; EV=me=AB=mc=CV MRSB =MRSA
EV m consumption doesincreases not change When income CV B u1 c A u0 x W & L CH 5 Utility and consumer demand 103 5-3 Duality and Welfare change EV and CV could deal with changes in external
circumstances other than price changes. In these cases ; u=u(x,y,d) and C=C(p,u,d), where d=dummy variable for external change . Which measure(EV or CV) is best? There is not a positive answer. It depends to the distributional value judgments. But if Kaldore criterion were to be used ,this would be equivalent to the criterion that CV>0 provided that the act of compensation did not alter the structure of relative prices. W & L CH 5 Utility and consumer demand 104 5-3 Duality and Welfare change Q5-18 (I) If your utility function was u(x,y,d,), what expression indicates how much you would need to be paid to compensate you for your child being run over? Is this the CV or EV or what ? What is its maximum value? Solution; t=1 child is alive , accident t=0 child is dead move is from t=1 to t=0, so CV1 should be
calculated because after change we want to measure the change in welfare. CV1 is negative because child is dead and welfare has been reduced. Compensation is needed to bring his utility to when his child is alive ( EV0 will be positive when we move from t=0 to t=1 . When child is dead how much we should pay him to for the the child) . W & Lcompensate CH 5 Utility anddeath consumerof demand 105 5-3 Duality and Welfare change C=C(P0 ,U0 , 1) , child is alive ,( before change or accident ) C=C(P0 ,U0 , 0) , child is dead ,( after change ) U0 = U(X , Y) when X=X(Px0 Py0 ,M ) , CV1 = - [xC(p0,1,u0) - C(p0,0,u0)] =- EV0 CV1 = - EV0 could be taken from him after his child is dead ( a negative amount ) Or , EV 0 could be given to him to compensate him for the loss of his child. ( a positive amount ) it may
be infinite. Utility and consumer demand 106 5-3 Duality and Welfare change Q5-18(ii) The government is considering building an office block outside a householders back window.Two economists propose different approaches to evaluate the householders loss of daylight. A says: ask him how much he would pay us not to build. B says : ask him how much we should have to give him before he would agree to our building(assuming he had the right to stop us). Which question would produce the bigger answer and why? Which economist do you think is right? W & L CH 5 Utility and consumer demand 107 5-3 Duality and Welfare change There is not an office
block , t=1 There is an office block , t=0 dayligh t y No daylight U 1> u 0 y1 EV0 CV 0 m x1 y0 x0 No daylight t=0,
t=1 x=0 x=1 W & L CH 5 A:maximum amount he would like to pay for the light is equal to the value of light for him which is equal to CV0 as shown in the figure.(in CV ,value of the light has been evaluated after the change ) B:maximum amount which he is ready to accept and give the permission to build the building is equal to EV0 as shown in the figure. ) in EV ,value of the light has been evaluated before the change ) As it is kwon EV is greater x than CV when x (light) is daylightconsidered as a normal good. Both A and B are right 108
. There is no positive answer to it . Utility and consumer demand 5-3 Duality and Welfare change Q5-19 Return to question 5-14. What is the EV?Is it larger than CV? Solution; x=m/(2p) , y=m/2 demand functions u=xy=m2/(4p), u1=1002/4(0.25) C(p,u1)=(4pu1)1/2=(100/0.5)p1/2=200p1/2 EV=C(P0,U1) C(p1,u1)=200 200(0.25)1/2=100 EV=100>CV=50 W & L CH 5 Utility and consumer demand 109 5-3 Duality and Welfare change Q5-20 Suppose that a two world two person world inhabited by persons A and B.A new government policy would shift production from (x0,y0) to (x1,y1), but it so happens that (x1,y1) could be distributed
in such a way that ua0 and ub0 were just maintained(I.e., the Caldore criterion is satisfied).In satisfied fact however x1 and y1 will be distributed so that ua1>ua0 and ub1
y a After change to toexpress expresswelfare welfare change changein in terms termsof of CV. CV CV. CV.. A CV CVA=(a =(a c) c) BB CV CV = = -- (a (a b) b)
OB b c ub1 E1 E0 y1 ua1 Ua1 a0 u Ua0 Ub0 Y0 ub0 oA x1 W & L CH 5 X0
11. . OB AA BB CV +CV CV +CV = = bb cc >0 >0 X0 , Y 0 = production before change X1 , Y 1 = x after production change Utility and consumer demand 111 5-3 Duality and Welfare change y a
After change b E1 c o Have to express welfare B change in terms of 1 1 YCV. , X are production of Y and X after change . ub1 a0 u Ua0
y1 CVA=(a c) ub0 oA x1 W & L CH 5 Uu a1 a1 So after change we x CVB = - (a b) CVA +CVB = b c >0 Utility and consumer demand 112 5-3 Duality and Welfare change Laspeyres and Paasche measures of
welfare change welfare change It It is is not not possible possible to to compute compute EV EV & & CV CV without without any any information information about about demand demand or or utility utility function. function. For For this this reason reason we we use use Laspeyres Laspeyres or or Paasche
Paasche measures measures of of welfare welfare change change to to measure measure welfare welfare change change approximately approximately.. Suppose Suppose that that price price of of xx falls falls from from p p00 to to p p11 and and 0 1 consumption consumption of of xx increases increases from from xx0(original (original bundle)
bundle) to to xx1(final (final bundle). bundle). Laspeyres Laspeyres measures measures of of welfare welfare change change is is minus minus the the change change in in the the cost cost of of buying buying the the original original bundle; bundle; LL = = -- (p (p11 p p00)x )x00 =
= -- px px00 or or L=(p L=(p00 p p11)x )x00 = =p p00x x00 p p11x x00 = = what what he he 0 was was spending spending for for buying buying the the original original bundle(when bundle(when p=p p=p 0)) minus minus what what he he needs needs to
to spend spend for for buying buying the the original original bundle(when bundle(when 1 p=p p=p1)) .Paasche .Paasche measures measures of of welfare welfare change change is is minus minus the the change change in in the the cost cost of of buying buying the the final final bundle; bundle;
P= P= (p (p00 p p11)x )x11 = =p p00x x11 p p11x x11 = = what what he he needs needs to to spend spend for for buying buying 0 the the final final bundle(when bundle(when p=p p=p0)) minus minus what what he he will will be be spending
spending for for buying buying the the final final bundle bundle (when (when p=p p=p11)) W & L CH 5 Utility and consumer demand 113 p P CV y e C(p0,u1 ) EV
l c A 0 C B For normal goods, u0U P > EV > CV >L x0 xB a a p Price decrease from p0 to p1 m/p
1 x A p0 Laspeyres measure 0 Welfare gain ; x1 m/p0 Y per x W & L CH 5 C(p1,u0) u1 And welfare gain. Consumer surplus
L=(p0 p1)x0 = p0x0 p1x0 P= (p0 p1)x1 = p0x1 p1x1 c(p0,u0)=p0x0=c(p1,u1)=p1 x1 p1x m L CV=C(p0,u0) - C(p1,u0) EV=C(p0,u1) - C(p1,u1) p0x1 1 x0 b X(p,m) = ordinary demand B C X(p,u0) = compensated =
demand xB x1 x Utility and consumer demand 114 5-3 Duality and Welfare change As is is seen from the figure CV & EV are exact measures of welfare change, while P and L are approximating it. For CV and L , the amount of approximation is equal to (CV L), which can be calculated as follows; CV = the maximum amount which he is willing to pay for the welfare change (price decrease from p 0 to p1). The amount which we can take away from him after change to enable him to remain on the original utility level ( U 0 ) L= The amount which he actually pays for the welfare change ( price decrease ) = {(p0-p1)x0}. The amount which we can take away from him after change to enable him to buy the original bundle ) By defenition ,(CV L) is consumer surplus between x 0 and xB, when price change from p0 to p1. Which is equal to area b b = (CV L) = [xMRSyx(x,u0) p1]dx X So CV could be calculated as follows; X CV=L+(CV L) =(p0 p1)x0 + [xMRSyx(x,u0) p1] dx = a+b
B 0 xB x0 W & L CH 5 Utility and consumer demand 115 5-3 Duality and Welfare change As it seen Compensating Variation for a price change is equal to the change in the area between the compensated demand curve and the price line. Previously this was proved by an assumption that the price change should be small enough. But here is no need for this assumption. CV=x(p,u0)dp Lim(CV/p)p0 = (cv/p)=x(p,u0). As p tends to zero , the area b tends to zero faster than a , and (CV/p) tends to xo . For small enough price changes, L = P = CV = EV , This could be detected from the figure on page 112 (lines are very close together) For welfare gain CV>L , but for welfare loss , L>CV,
(-L>-CV). W & L CH 5 Utility and consumer demand 116 5-3 Duality and Welfare change Q5-22(I) To maintain my present standard of living at existing prices I need $5000 per annum.since I currently buy 100 bottles of Coca Cola a year, an increase in price of Coca Cola equal to $2.50 each year would mean that I need at least $5250 to maintain my standard of living. True or False ? Explain. W & L CH 5 Utility and consumer demand 117 p y L=(p1 p0)x0=($2.50)(100)
e =250=area(a+b) = L > area b = CV L>CV=the real amount of compensation m A C Consumers will substitute for x0xB from other goods , and partly compensate their loss of u0 utility. u1 x1 x0 xB Y per x x
A p1 a b p C X(p,m ) X(p,u0) 0 x1 W & L CH 5 m/p m/p1 0 xB
x0 Utility and consumer demand x 118 5-3 Duality and Welfare change Q5-22 (ii) Suppose the world price of oil is $12 per barrel and Britain becomes able to produce 1 billion barrels a year of North sea oil at cost of $6 per barrel. What is the annual gain to Britain? Is this an approximate or exact measure? Solution; $6 billion. It is not an approximate measure in the sense in which we have been talking. For the consumers price is (presumably) unaffected and there is no substitution towards oil. There is simply a profit of $6 per barrel. W & L CH 5 Utility and consumer demand 119
5-3 Duality and Welfare change Q5-23 (I) what are the relative size of P and EV. Why? p>EV, the substitution effect is not considered in calculating P. (ii) does it matter whether the good is Giffen for purpose of (I)? No we are only concerned with substitution effect. x W & L CH 5 Utility and consumer demand 120 5-3 Duality and Welfare change y Decrease in price, decrease in quantity P>EV , CV>L , but CV>EV Substitution effect is always negative, so P should be greater than EV, and CV should be greater than L P EV
L A CV Income effect B P P W & L CH 5 Utility and consumer demand x 121 p y Q5-23(iii) e Derive a measure of EV compared with just given for CV. m
l c A C 1 u1 u u0 x0 W & L CH 5 x p0 EV = x(p,u1)dp p1 A P0 aa b
P1 P=EV+a m/p 1 m/p0 Y per x EV x1 xB C P = p0x1 p1x1 X(p,m ) X(p,u1) x0 xB
x1 Utility and consumer demand x 122 Q5-24 Suppose u=xy, m=100, py=1. The price of x rises from $0.25 to $1. Calculate L, P, CV, EV.Check that they have the correct relative sizes. Solution; this is question 5-14 and 5-19 in reverse. In Q5-14 and Q5-19 there was price fall from $1 to $0.25, and CV calculated in Q5-14 was 50 ,and EV calculated in q5-19 was100, so EV and CV here is equal to 50 and 100. L=(p0-p1)x0=(-0.75)(100/2(0.25))=-150 P=(p0-p1)x1=(-0.75)(100/2(1))=-37.5 L
Cost-of-Living Indices The most commonly used is Laspeyres price index which weights the prices by quantities in the base year; n n l =(i pikxi0)/(i Pi0xi0), where; there are n commodities(i=1,.n), pik & xik , are price and quantity consumed from commodity xi in period k. The Paasche price index weights the prices by quantities in the final year. P = (ni pikxik)/(ni Pi0xik), W & L CH 5 Utility and consumer demand 124 5-3 Duality and Welfare change Suppose that l =2. It means that the consumer in year k should spend twice as much as the base year(t=0), in order to be able to buy x0 (base year bundle) in year k. Now if p =2, it means that if the consumer wants to buy xk(the bundle in year k) in t=0(base year), he needs to have only half of the expenditure in year k. Laspeyres price index is commonly used in order to adjust the economic variables (like income) for the change in
prices. Nominal income in year k = (i pikxik) should be divided by p in order to make him able to buy XK in year t=0 . Nominal income in year t=0 (i Pi0xi0 )should be multiplied by l in order to make him able to buy x0 in year k. W & L CH 5 Utility and consumer demand 125 5-3 Duality and Welfare change Which one of these indices is right? Laspeyres or Paashe ? The answer of course is neither , for neither corresponds to an accurate measure of welfare change. The laspeyres price index overestimate the rise in the cost of original standard of living(when prices increase) and the Paasche price index underestimates it. The true price index when u0 is the base for comparison is equal to C(p1,u0)/C(p0,u0) The true price index when u1 is the base for comparison is equal to C(p1,u1)/C(p0,u1) W & L CH 5 Utility and consumer demand
126 p y p1x0 C(p1,u0) e m l c A c(p0,u0)=p0x0=c(p1,u1)=p1x1 = m p0x1 C(p0,u1) C u0 u1 x1 x0
m/p1 m/p 0 x l =(P1x0)/(p0x0) > C(p1,u0)/C(p0,u0) p =(p1x1)/(p0x1) < C(p1,u1)/C(p0,u1) W & L CH 5 Utility and consumer demand 127 5-3 Duality and Welfare change True cost of living index should not be dependent on the level of utility. In order to overcome to this deficiency , we could specify the utility function as a homothetic function(homogeneous of degree one for example). u=u(x1,x2)=u(x1,x2) x1= x1 , x2 = x2 , c=p1x1+p2x2 c = p1x1+ p2x2 = p1 x1 + p2 x2 =c c/u =c/u = c/u = f(price ratio) c=uf(p)
So , true cost of living index would be independent of the utility level at which cost were measured. It is only a function of price ratio . C C(p1 ,u0 )/C(p0 ,u0 )= u0 f(p1)/ u0f(p0 )= f(p1)/ f(p0 )= f(P) U W & L CH 5 Utility and consumer demand 128 5-3 Duality and Welfare change How serious is all this? If we are only using indices to compare adjacent years , it may not matter too much, since prices will not change drastically. The true price index tends to the Laspyeres price index; c(p1, u0)/c(p0 , u0)(p1x0)/(p0x0)=l The problem with the Laspeyres price index is that the same weights used over a number of years. Then , if we compare the Laspeyres index in year 9 (L9) with that in year 8 (L8) ; L9/L8=[x(p9x0)/(p0x0)]/[x(p8x0)/(p0x0)]=( p9x0)/ (p8x0) we are comparing price index for year 9 with year 8 without having consumption bundle in year 8(x8) or year 9(x9) . W & L CH 5 Utility and consumer demand
129 5-3 Duality and Welfare change A preferable approach for solving this problem is to use chain-linked index, as follows; D1=(p1x0)/(p0x0) D2=(D1)(p2x1/p1x1)=[x(p1x0)/(p0x0)] [x(p2x1)/(p1x1)] The only problem with a chain-linked index is that it is not necessarily path independent, in the sense that if we started with p0 and returned in period 2 to the original set of prices we would not find D2=1, regardless of prices in period 1. We would however find D2=1, if the utility function was homothethic. (p2=p0) W & L CH 5 D2=D1[xf(p2)/f(p1)]=[xf(p1)/f(p0)][xf(p2)/f(p1)]=1 Utility and consumer demand 130 Q5-25 The following data shows that the poor were better off in 1971 than in 1965. True of False? (assume that the data relate to the poor, who only consume these two goods). Expenditure per head per week price index
Year 1965 1971 1965 1971 Food 25 30 1 1.3 Other 50 80 1 1.1 Solution; True. Income in 1971 was $110 , and the cost of the 1965 bundle at 1971 prices was 87.5 ; 25(1.3) + 50(1.1)=87.5, in other words welfare has improved; 87.5/75 = (p1x0)/(p0x0)=L<(m1/m0 )= (p1x1)/(p0x0) =110/75 W & L CH 5 Utility and consumer demand 131 Q5-26 Suppose that a Laspeyres price index is used to
construct an index of real income by dividing income into an index of money income. What are the price weight in the resulting measure of real income?(Assume you are comparing adjacent years.) Solution: Y1=[x(m1/m0)/L]=[x(p1x1)/(p0x0)]/ [x(p1 x0)/(p0 x0)] Y1=(p1x1)/ (p1x0) W & L CH 5 Utility and consumer demand 132 5-3 Duality and Welfare change Q5- 27 Laspeyres price indices(and any other) are normally computed for a group of people whose expenditure patterns vary. The weight used are the shares of each good in the expenditure of the whole group. (I) Is group index a simple average of the indices for each individual. (ii) If not which goods have their price change reflected more heavily in the index than if the index were such a simple average? (iii) Does this mean that the index necessarily underestimates the inflation experienced by the poor? (iv)If linear expenditure system is correct, what is the income
level of the representative consumer to whom the published price index relates?(after reading the next section) W & L CH 5 Utility and consumer demand 133 5-3 Duality and Welfare change Solution; Each individual ks own price index is; Lk = (i pi1xik)/(i pi0xik), xik = consumption of commodity i for individual k in period t=0 For two person group ( a and b), the group index is; L= (i pi1xi)/ (i pi0xi) , xi = group consumption of i L = [x(i pi1xia)/(i pi0xia)] [x(i pi0xia)/(i pi0(xia+xib)] + [x(i pi1xib)/(i pi0xib)] [x(i pi0xib)/(i pi0(xia+xib)] As it can be seen the individual indices are weighted by the shares of the individuals in total income. Only if x ia =xib, The group index is a weighted average index. (ii)- The price changes of luxuries are more heavily represented than if rich and poor were weighted equally. The main difference between expenditure pattern of poor and rich are related to the expenditures on luxuries. When income increases , expenditures on luxuries change drastically. W & L CH 5
Utility and consumer demand 134 5-3 Duality and Welfare change (iii) No . prices of luxuries might be rising faster than those of necessities. iv- We need to discover what is the income of an individual k (mk ) who spends the same proportion of his income on each good i as the community as a whole.First , we find the share spent on good i by the community as a whole ; For individual k (k=1,.n) , expenditure on xi is equal to pixik pixik = pi i +imk ij ( pj j) Therefore summing over individuals; K pixik = n[xpi i ij (j pj j) ] + ik mk Overall share=(k pixik)/(k mk) = [xpi i - i j (pj j)]/(k mk/n) +i j Individual share =( pixik )/mk = k[x pi i i( pj j) ]/mk + i If individual share must be equal to overal share , then (m k/n)= mk . The representative individuals income is the average income of the group. W & L CH 5 Utility and consumer demand 135
The structure of the utility function Any demand function must satisfy the restrictions which have been mentioned before hand. But there are many mathematical functions which satisfy these restrictions. Further assumptions are needed to reach to an specific form of demand and utility function . Let us begin from the direct utility function . Homothetic utility function An increasing homothetic function is any increasing transformation of a linear homogeneous function. u=(ax1+bx2)n ( n >0) is homothetic function, since it is increasing transformation of u1=(ax1+bx2) it is also homogenous of degree n. For homothetic function the expansion path is a straight line passing through the origin. In order to prove it , we should prove that MRS is a function of x2/x1 . W & L CH 5 Utility and consumer demand 136 The structure of the utility function U=u(x1 , x2) U(mx1 , mx2) = mau(x1 ,x2) , If m=1/x1 , then U( 1 , x2/x1)= (1/x1)au(x1 , x2) U(x1 ,x2)=(x1)au(x2/x1)
U1=du/dx1=a(x1)(a-1)u(x2/x1) (x2/x12)u(x2/x1)x1a U2 =du/dx2 =(1/x1) u(x2/x1)x1a MRS=u1/u2 = a u(x2/x1)/ u(x2/x1) (x2/x1) = f(x2/x1) On the expansion path homogenous of degree MRS = P1/P2 . As while as p1/p2 will not change , MRS will not change either, and x1/x2 remains constant. So all the points on expansion path remains on a straight line through origin. W & L CH 5 Utility and consumer demand 137 x2 Expansion path MRS(x0)=MRS(x1)=f(x2/x1) =f(p1 / p2) So x2/x1 is constant as while as p1 / p2 does not change. U=(x1)a1(x2)a2 x2
x20 (p1x1)/(p1x1+p2x2)=constant x1 1 u1 x0 u0 m0 x 10 W & L CH 5 x 11 (P1x1)/(p2x2) is constant if p1/p2 is constant. (p1x1)/m = constant = a1 (pixi)/m = ai [x(pixi)/m] is only a function of p1/p2 Pixi = ai m, or xi = a i m /pi m1 x1 ai changes when prices
change Utility and consumer demand 138 The structure of the utility function If pixi=aim , two propositions follow directly ; First ; Engel curve is a straight line. Or marginal propensity to spend on xi out of additional income [x(pixi)/m] is independent of income. Second ;the Engel curve goes through origin. Or the income elasticity is equal to one for each commodity in the consumption bundle. The second proposition violates the generalized Engels law. It can not be accepted. The first proposition is a useful one which could be used where we need to assume that aggregate demand for x to be independent of the income distribution. So we need to specify a demand curve having only the first proposition. If we want to specify a demand curve with income elasticity not equal to one(does not pass through origin), it should be like; xi = A + m/pi This kind of demand curve has an Engel curve and utility function that looks like what has been shown in the figure below (Stone gray utility function). Linear expenditure system is such a demand curve, which is derived from the stone gray utility function.
W & L CH 5 Utility and consumer demand 139 The structure of the utility function a1 Stone gray utility function u=(x1 - 1) ( x2 - 2) a2 0
m1 x1 Engel m0 m1 m Minimum x necessary for Pj j = minimum income necessary for surviv survival W &alL CH 5 Utility and consumer demand 140 The structure of the utility function max u=(x1- 1) a1 ( x2 - 2) a 2 s.t. m - Pj j = p1(x1 - 1) + p2 ( x2 - 2) (x1- 1) =X1 , (x2 - 2)=X2 ,( m - Pj j )=M Solving the optimization; Xi =(ai M)/pi xi =i +ai[x(m - Pj j)/pi];Linear expenditure system; Engel curve is straight, it is defined only for xi>i
aggregate demand is independent of income distribution so long as each consumer of society is consuming at least i of each good , or his income is greater than ( Pj j) W & L CH 5 Utility and consumer demand 141 The structure of the utility function Q5-28; A question on the linear expenditure system. (I) can there be inferior goods in it? (ii)What distinguishes necessities from luxuries? (iii) If i is positive(which is not required in principle but found for most goods in practice), is xi price elastic or inelastic. (iv) if all i are positive, are any goods gross complements? All goods? Solution; (I) no, since , ai>0. (ii) for necessities , at hypothetical zero income , consumption would be positive , so that pixi/m falls as m rises (income elasticity is less than one) .for luxuries the reverse is true. W & L CH 5
Utility and consumer demand 142 The structure of the utility function x Engel For necessities , if m=0, then xi = i + ai [x(m- j Pj j)/pi ] >0 m Pj j (iii)- Inelastic; xi= i + ai[x m/pi - i - (ji Pj j)/pi ] (dxi/dpi)= - ai (m ji Pj j)/(pi2) = (dxi/dpi)(pi/xi)=(1/xi)[x- ai(m- jI pj j-pii)/pi -aii+ i - i)] | | = (1/xi) [x -xi + i (1 ai) ]=(1/xi) [x xi - i (1 ai)] <1 (iv) yes all goods. (dxi/dpj)=(-ai j)/pi W & L CH 5 Utility and consumer demand 143 The structure of the utility function
Additive Additive utility utility functions; functions; Stone Stone gray gray utility utility function function belongs belongs to to aa class class of of function function in in which which utility utility is is treated treated as as aa sum sum of of the the utility utility derived derived from from each each separate separate
commodity. commodity. u=f[xu u=f[xu11(x (x11)) + + uu22(x (x22)) + + .] .] a1 (( x a2 u=(x u=(x11-- 11))a1 x22 -- 22))a2 logu=a logu=a11log log (x (x11-- 11)) +a +a22log(x log(x22 22)) u(x u(x11,, xx22)) = = uu11(x (x11)) + + uu22(x
(x22)) The The marginal marginal utility utility of of each each good good is is independent independent of of the the quantity quantity of of any any other other good. good. (independent (independent wants). wants). Two Two implications; implications; First First ;; there there is is no no cross cross substitution substitution effect.
effect. Second Second ;; the the ratio ratio of of ownprice ownprice elasticity elasticity to to income income elasticity elasticity is is the the same same for for every every good. good. W & L CH 5 Utility and consumer demand 144 The structure of the utility function Proof: u/ xi= ui(xi)= pi =marginal utility of income
log (u/ xi)=log ui(xi) = log + log pi [x(logui)/logxi][xlogxi/logpi]= [xlog/log pi] + 1 [x(log ui)/log xi][xlogxi/logpj]= [x log/log pj] [x(log ui)/log xi][xlogxi/log m]= [xlog /log m] [xlog/log pi]0.] (ji) [xlog/log pj]0 [xlogxi/logpj]= xi pj = 0 [xlogxi/logpi]/[xlogxi/log m]=1/[xlog/log m] [x(xi pi)/xi m] = 1/[xlog/log m] 1/(elasticity of marginal utility of income)= constant Knowing the income elasticity of each good , and price elasticity of one good is enough to find the price elasticity of other goods. W & L CH 5 Utility and consumer demand 145 The structure of the utility function Separable utility function; Direct tests of the additive assumption have been suggested that it is not satisfied. Marginal utility of many complementary goods are not independent from each other. It is more proper to have the broader category of goods included in any one variable and define the separability in some way between these broader category.( health care, clothing , food , entertainment)
u = u[xf r(x1 , x2) , f s(x3 ,x4), .] The fundamental implication of separability is that if goods i and j are separable from all other goods, then (ui/uj)/xk = 0 , (all k i , j) , ( i and j belongs to group r and k belongs to other group s) MRS between coffee and tea is independent from the number of shirts one wears . W & L CH 5 Utility and consumer demand 146 The structure of the utility function If good i belongs to group r , the demand for good i can be expressed in either of two ways. xi=f (p ,m) , p=the whole vector price if i belongs to group r, then demand for x can be expressed as; xi = g(pr , mr), which can be estimated much easier and cheaper than the first one. [xxi/ pj]= [xxi/ mr][xmr/ pj], [xi belongs to r and j belongs to s]
A change in pj will affect all goods in group r in a way that is proportional to their response to expenditure in group i. (mr/pj)= rj = factor of proportion which depends on group r and commodity j only. W & L CH 5 Utility and consumer demand 147 The structure of the utility function (xi/pj)u = (xj/pi)u (xi/pj)u= (xi/mr) (mr/pj)u= (xj/ms) (ms/pi)u= (xj/pi)u (mr/pj)u / (xj/ms) = (ms/pi)u / (xi/mr) = rs (mr/pj)u = rs (xj/ms) (xi/pj)u= (xi/mr) (mr/pj)u= (xi/mr) (xj/ms) rs = substitution effect of change in pj on xi . In other words , change in xi because of change in pj will be revealed trough change in xj and xi because of change in the expenditure pattern of the groups in which commodities i and j belongs to . W & L CH 5
Utility and consumer demand 148 The structure of the utility function Q5-29 .people commonly save a part of their income. Yet we commonly analyze demand for a good as a function only of expenditure ( i.e, income minus saving). What are we implicitly assuming about the position of saving in the utility function? Solution; that all good is separable from it. u=u[x ur(x1,xn) , us(s) ] max ur(x1,xn) s.t. mr=pixi xi=xi(p1, p2 ,.mr) W & L CH 5 Utility and consumer demand 149 The structure of the utility function Household composition and equivalence scales. Consumption occurs within the households and is influenced by the size of household.
A household with a given income is likely to spend more on food , the more mouth it has to feed. Household size has to appear as a variable in utility function, Engel curve , and demand function. For welfare comparisons between families of different sizes , we need to know how to measure the real income after allowing for household composition To abolish the poverty we should know at what income, the families with different sizes would be at the same minimum acceptable level of utility. At a given income level , larger families spend a higher portion of their income on food and lower portion on luxuries. W & L CH 5 Utility and consumer demand 150 The structure of the utility function Max u=u(x1/h , x2/h) s.t. P1(x1/h) +p2(x2/h) = m/h (xi/h)=f(p1, p2, m/h) (xi/h)=g(m/h) ajnj = h , (aadult=1, n=age categories) If some functional form is assumed for g , the weights can be estimated by first guessing values for aj (for example aj0) and estimating the parameters of the g
functions, and then taking the predicted values of g for each household and regressing [xxi/g(m/aj0nj)] on ajnj to re-estimate the values for aj since [xxi/g(m/ aj0nj)]=h=ajnj . A more subtle approach is to allow for aj to be different for each good. W & L CH 5 Utility and consumer demand 151 The structure of the utility function Q5-30 Suppose that ordinary demand curves have been run for (i) single adult households and (ii) households of type k , yielding the estimates of ( i ) xi = ci1mbi (ii) xi = cikmbi What is your estimates of hk? What conditions determines whether , given m , xi is higher or lower for households of type k, assuming hk>1. Solution; Since h=1 , for a single adult , the general Engel curve for household of any size h must be Xi /h=ci1(m/h)bi Therefore for households of size hk , we have X i=hkci1(m/hk)bi=(hk1-bi)ci1mbi But we also know that for households of size hk ; W & L CH 5
Utility and consumer demand 152 The structure of the utility function xi = cikmbi , thus cikmbi = hk(1-bi)ci1mbi , so ; hk=(cik/ci1)1/(1-bi) if bi<1 (xi is necessity) , Since hk>1 , we should have cik>ci1. At given m , larger families consume more necessities than small ones, and fewer luxuries. END OF CHAPTER 5 W & L CH 5 Utility and consumer demand 153
Missing Food: The Case of Post-Harvest Grain Losses in Sub ...
The Size of the Problem Losses are lower than previously estimated but still substantial. Average grain losses are estimated to range between 10-20% (depending on the grain and the season ) Based on estimates by APHLIS, the value of losses...