Lesson 6: Extensions and applications of consumer theory. 6.1 The approach of revealed preference

Transcription

1 Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Lesson 6: Etensions and applications of consumer theor 6.1 The approach of revealed preference The basic result of consumer theor (discussed in previous lesson) is that the own substitution effect is non-positive. This proposition can also be derived using another approach, revealed preference, which proceeds without an reference to indifference curves or utilit functions. Originator: Paul Samuelson, A B

2 Microeconomics I. Antonio Zabalza. Universit of Valencia 2 Suppose that at prices (, ) p p and income m, bundle A is chosen. Suppose we have another bundle B that can also be afforded b the consumer (that is, lies on the bl or inside the bl). Can we sa anthing about the preferences of this consumer respect these two bundles? Answer: es. If at these prices and income, despite that both bundles were affordable, A was chosen, it must be because the consumer prefers A to B. In mathematical terms, is the bundle chosen (A) when prices and ( ) income are (,, ) p p m. (, ) is the bundle B, which is affordable at these prices and income, but is not chosen. Bundle B is affordable: p + p m Bundle A is chosen : p + p = m Substituting the second equation into the first, p+ p p + p If this inequalit holds, then we sa that bundle A is revealed preferred to bundle B. In words: Bundle A is revealed preferred to bundle B if bundle A is chosen when B could have been chosen.

3 Microeconomics I. Antonio Zabalza. Universit of Valencia 3 The concept of revealed preference is not empt. It ecludes the possibilit of some tpes of behaviour. Supose bundles A and B in the following graph. (p, m) A B (p, m) If at (, ) pm A is chosen, at ( p, m) B could not be chosen. Wh? Because if at the second set of prices B was chosen, we would be saing that both A is revealed preferred to B, and B is revealed preferred to A, which is a contradiction. The concept of revealed preference allows us to state the following result:

4 Microeconomics I. Antonio Zabalza. Universit of Valencia 4 Weak Aiom of Revealed Preference (WARP) p+ p p + p ; p + p < p+ p. If Then That is, if at (, ) B, at ( p, p) p p A is revealed preferred (rp) to, if B is chosen, it must be because at these latter prices A is not affordable. A new look to the substitution effect Suppose that at (,, ) p p m the bundle (, ) is chosen. Now suppose that prices change and we compensate the consumer so that he can still afford his old bundle; that is the budget data are now ( p, p, m ). What will the new equilibrium be? R (,) S (, ) T

5 Microeconomics I. Antonio Zabalza. Universit of Valencia 5 Can it be to the left of the initial equilibrium (segment RS)? No. Bundle (,) was revealed preferred to all bundles on that segment. Now, being still bundle (,) affordable, no other bundle in that segment can be chosen. Can it be to the right of the initial equilibrium (segment ST)? Yes. Bundle (,) was not revealed preferred to the bundles on this segment, so the in principle could now be revealed preferred to the initial bundle and be chosen. If this is so, it must mean that as a result of a change in prices and income as the one depicted, a compensated change in prices, the new bundle will be such that necessaril more of the good that has become relativel cheaper will be bought, and less of the good that has become relativel more epensive will be bought. This is the negative substitution effect. In maths Call the new bundle (, ). If the consumer has been compensated so that he can afford the old bundle, it must be that p + p = p + p (1) If the new bundle is chosen, it must be because at the new prices is revealed preferred to the old

6 Microeconomics I. Antonio Zabalza. Universit of Valencia 6 bundle, and that at the old prices if it was not chosen it was because it was not affordable. So, p + p > p + p (2) Subtract (1) from (2) p p + p p p > p+ p p p ( p p )( ) + ( p p )( ) > 0 Suppose that the price of good remains constant, so that the second term in the sum is zero. Then ( p p )( ) > 0 In the figure the first parenthesis is positive (the price of goes down). For the whole epression to hold, the second parenthesis also has to be positive; That is, the quantit of must go up. This is the substitution effect, without the need of indifference curves or utilit functions. If p and the consumer is compensated so that he is able to just afford his old bundle, then. 6.2 Inde numbers One of the most immediate use of revealed preference analsis is for the derivation of indices of welfare change. If prices and income change, normall the consumption bundle also changes. Can

7 Microeconomics I. Antonio Zabalza. Universit of Valencia 7 we sa whether welfare has increased or decreased as a result of this change? This is the sort of question that the theor of inde numbers tries to answer. Definitions: Base period: ( p b, b ) ( b, b p ) Final period: ( p t, t ) ( t, t p ) Quantit indices Their objective is to measure how the consumption of a basket of goods has evolved over time. The measure the change in real consumption, holding prices constant. Goods are heterogeneous and therefore cannot be compared directl. We must convert bundles of goods into the value of these bundles. That is, to weight each good b some price. In general the structure that these indices (we will call them I q) take is the following one: I w + w t t q = b b w + w The important thing to notice is that the weights remain constant in both periods; all that varies are the quantities consumed. In general we will interpret these indices in the following manner:

8 Microeconomics I. Antonio Zabalza. Universit of Valencia 8 If I q > 1, real consumption has increased. If I q < 1, real consumption has decreased. Choice of weights If weights are base prices: Lasperes quantit inde, L q. If weights are final prices: Paasche quantit inde, P. q L P p p b t b t q = b b b b p + p p + + p t t t t q = t b t b p + p Inferences about changes in welfare Sa, P > 1 p + p > p + p t t t t t b t b q Clearl, from the principle of revealed preference (prp) we can conclude that t, t is rp to b, b.the consumer (or the ( ) ( ) countr) to which P q refers, is better off at t than at b.

9 Microeconomics I. Antonio Zabalza. Universit of Valencia 9 P < 1 p + p < p + p t t t t t b t b q Here the prp cannot be applied because the bundle b b, is not affordable at the final ear prices. ( ) L < 1 p + p < p + p b t b t b b b b q or p + p > p + p b b b b b t b t From the prp we conclude that b, b is rp to t, t. Therefore welfare has ( ) ( ) decreased between b and t. The consumer (countr) is worse off. L > 1 p + p < p + p b b b b b t b t q Here the prp cannot be applied since the final bundle is not affordable at the base prices. Price indices The same can be applied to the measurement of the evolution of prices. This ma be more familiar to ou (recall IPC). Now what varies are the prices and what stas put (the weights) are the quantities. As

10 Microeconomics I. Antonio Zabalza. Universit of Valencia 10 before there are two tpes of indices depending on whether se use base or final ear prices as weights. L P p p t b t b p = b b b b p + p p + + p t t t t p = b t b t p + p In the case of price indices, prices in the numerator differ from tose in the denominator. Consequentl, the prp cannot be applied (see that for the prp to be applicable, the set of prices at which ou compare two bundles has to be the same). However, if ou have additional information on how ependiture has evolved over time (that is, on how income has evolved), then something can be said. Define a new inde M: p + p M = p p t t t t b b b b + See that M is just the ratio between ependiture (income) in the final ear and ependiture (income) in the base ear. With this information, there are two cases in which we can sa something definite:

11 Microeconomics I. Antonio Zabalza. Universit of Valencia 11 If P p > M Welfare has gone down If L p < M Welfare has gone up Proof that if Pp > M then welfare has gone down. p + p p + p t t t t t t t t > b t b t b b b b + + p p p p t t t t b b b b b t b t t t t t ( p + p )( p + p ) > ( p + p )( p + p ) p + p > p + p b b b b b t b t b b t t (, ) is rp to (, ) Eercise: Proof that if Lp < M then welfare has gone up. Eplain wh in the other two cases, Pp < M and Lp > M, revealed preference cannot help ou to sa whether welfare improves or worsens.

12 Microeconomics I. Antonio Zabalza. Universit of Valencia Budget constraints with fied endowments Suppose that instead of having income, the consumer has a given endowment (, ) of the two goods. To fi ideas, think of a farmer or of a countr that produces oil. The budget constraint then is: p + p = p + p Solving for we have the formula for the new budget line: p = + p The slope is, as before, the relative price. The vertical intercept however is a bit more complicated: it depends on the endowments and on the price ratio. Another important propert of this budget line is that alwas goes through point E: the endowment bundle (, ). (You should check this propert b finding the value of when = ). This propert means that a change in the price ratio will rotate the budget line around this point E. If becomes relativel cheaper, the bl will rotate anticlockwise. If it becomes relativel more epensive, the rotation will be clockwise. p p

13 Microeconomics I. Antonio Zabalza. Universit of Valencia 13 + p p p p E p p Of course, if the endowments change, with prices constant, the bl will shift in a parallel fashion, as with a conventional income effect. (Check that ou understand that). Equilibrium The equilibrium of the consumer need not be at point E. If, for instance, the equilibrium is at the right of point E, the consumer will be buing good (we will sa he is a net buer of ) and selling good (we will sa that he is a net seller of ). And vice versa if the equilibrium is at the left of point E. How can the consumer bu? With the resources he obtains selling. B selling he gets resources b p. And with this resources, the amount of ( ) p + p

14 Microeconomics I. Antonio Zabalza. Universit of Valencia 14 he spends p ( ) on. Will the former resources be enough to pa for the latter ependiture? Or, in other words, is this buing and selling consistent with the budget constraint? Yes. See that the revenue he obtains b selling has to be equal to the ependiture on. ( ) = ( ) p p But this is precisel the budget constraint, as ou can check ourself with a bit of manipulation of this equation. Response to price changes Suppose the consumer is a net buer of, ( ) 0 >, and that p (good is normal). Will he consume more or less? E q q q

15 Microeconomics I. Antonio Zabalza. Universit of Valencia 15 See that in this case both the substitution effect (from q to q ) and the income effect (from q to q ) work in the same direction and, therefore, the consumer consumes more. Suppose now that for the same price change as above, the consumer instead is a net seller of, <. Will he consume also more? ( ) 0 q q E q Here we see that the substitution and the income effect work in opposite directions. Depending on the size of the income effect, the final equilibrium q could be to the right of the initial equilibrium (more

16 Microeconomics I. Antonio Zabalza. Universit of Valencia 16 consumed) or to the left (less consumed). So in this case it is not possible to sa anthing definite. Wh do we find these different results depending on whether the consumer is a net buer or a net seller of good? It must have to do with the income effect, because the substitution effect works in the same direction in both cases. Remember that the price of goes down. If ou are a net buer of, this price fall means a positive income effect (ou are richer because of this price fall). However, if ou are a net seller of, this price falls makes ou poorer, and therefore the income effect in this case is negative. This could never happen in the previous lesson in which income is not in terms of phsical endowments but in terms of mone.

17 Microeconomics I. Antonio Zabalza. Universit of Valencia 17 Switch from net buer to net seller and vice versa Even without knowing anthing on preferences we can sa quite a few things about the behaviour of the consumer concerning his desired position as seller or buer b using the principle of revealed preference (prp). Net buer of ( ) > 0; p A C E q O B D B the prp the new equilibrium must be on segment ED. The bundle q is revealed preferred (rp) to all the set of opportunities OAB; so, since CE and q are still affordable at the new prices, CE will not be chosen. If the price of a good goes down, a net buer of this good will remain a net buer.

18 Microeconomics I. Antonio Zabalza. Universit of Valencia 18 Net buer of ( ) > 0; p C A E q O D B The prp cannot sa anthing on the new equilibrium. The bundle q is rp to all the set of opportunities OAB. But at the new prices q is no longer affordable, so we cannot sa what bundles in new bl will be or not preferable to q (remember that to be able to make a comparison both bundles have to be affordable). Eercise Equipped with the above reasoning show (with the corresponding graphs) that for a net seller: a) If the price of goes down the prp cannot help ou to tell whether or not the consumer will remain a net seller or not.

19 Microeconomics I. Antonio Zabalza. Universit of Valencia 19 b) If the price of goes up, a net seller of will remain a net seller of. 6.4 Labour suppl A particular application of a budget constraint with endowments is the analsis of labour suppl. We measure labour suppl b H (hours of work per period of time; sa, per da). Clearl there is a maimum of hours per da that we could work, namel 24. This is our endowment of time. The other good is C, consumption; the more we work, the more we can consume. Considering H and C would be like considering a bad and a good. To make the analsis more similar to the previous eposition of consumer theor and work with two goods, instead of considering hours of work, we will consider hours of leisure and call it L. The endowment of leisure is the maimum amount of leisure that we could conceivabl have (24). Call this endowment L. Then the hours that an individual works is this maimum amount minus the actual leisure that he consumes, H = L L

20 Microeconomics I. Antonio Zabalza. Universit of Valencia 20 If out of the 24 hours possible, I decide to consume 16 hours of leisure (8 to sleep, 3 to eat and 5 to enjo mself), it means that I work 8 hours (=24-16). The utilit function is then defined as U (, ) = U C L C wl P w slope = P wh C = P L L L H To build the budget constraint consider the following: If I work H hours (equivalent to having L hours of leisure; L= L H) and the hourl wage is w, then m income will be wh. With income wh, if the price of C, consumption, is P, I will be able to spend on consumption

21 Microeconomics I. Antonio Zabalza. Universit of Valencia 21 wh=cp (1) which means that the real units of consumption will be wh C = P In the etreme, suppose I work all da H L=0 and = L, then C wl = P This gives the vertical intercept of the budget constraint. So, the slope of the budget constraint is the vertical intercept divided b the horizontal intercept. That is, Slope wl P = = L w P The slope of the budget constraint is the real wage. B now it should be eas to derive the formula for the budget line. From (1) we have

22 Microeconomics I. Antonio Zabalza. Universit of Valencia 22 ( ) wl L = CP CP+ wl= wl This is the budget line: the goods are C and L and the endowment of leisure is L. Solving for C we have C wl = P w L P A straight line with vertical intercept wl P and slope wp. Response to an increase in the wage An increase in w will rotate clockwise the bl around the horizontal intercept (check ou understand this). The substitution effect is as alwas: the higher the wage, the less leisure is consumed (the more hours of work are supplied). This is the movement from A to B. But see that there is also a positive income effect: we are all net sellers of time (of leisure); if w, the price of time goes up, we are richer. Where is the final equilibrium? Here we have a problem, because in this case the two effects work in opposite directions. If we suppose that leisure is a normal good (as we should), then the final equilibrium, D, can be to the right of A or to the left of A. Normall it will be to the left of A (we will work more, when

23 Microeconomics I. Antonio Zabalza. Universit of Valencia 23 w goes up). But this is not an unambiguous conclusion. We might feel so rich as a result of the increase in w that we ma decide to work less (in that case the final equilibrium would be to the right of A). Empiricall, this tends to happen for people who alread work a lot and therefore have, even before the increase of the wage, quite a lot of income. C 1 wl P 0 wl P B D A L L In the graph we have depicted the more frequent case: an increase in w results in an increase in hours of work. But see that there is no reason wh we could not draw an indifference map such that point D was at the right of point A.

24 Microeconomics I. Antonio Zabalza. Universit of Valencia 24 Eercise to do in the practical class Suppose that in addition to an endowment of leisure of L ou also have a given endowment of nonlabour income M. That is, ou also have some rent (mone) per period of time that does not depend on the hours ou work. What is the budget constraint in the (C,L) space? 6.5 Consumption over time So far we have analsed the consumption of goods in the present moment of time. Now, in the last application of consumer theor, we analse the consumption of one good in different periods of time. To be more precise. Suppose we have a composite good, consumption, which we denote as above b C. We want to see what will be the amount of consumption now, C 1, and the amount of consumption in some future period of time (tomorrow, net week, net ear), C 2. That is, the consumer s utilit function is U = U( C, C ) 1 2

25 Microeconomics I. Antonio Zabalza. Universit of Valencia 25 Suppose, to simplif things, that prices are 1 in both periods. At the end of this section we will rela this assumption. Suppose also that the consumer has a source of income in each period. Income in period 1 is m 1 and income in period 2 is m 2. m1 and m 2 are the endowments the consumer has. The last assumption we make is that the consumer can lend and borrow mone at a rate of interest of r. Budget constraint Maimum amount of consumption in period 2 (vertical intercept). ( ) C ma = m + m (1 + r) m 2 is the mone the consumer has in period 2. If in period 1 has saved some mone and has lent it, in period 2 he will have m (1 + r). 1 Maimum amount of consumption in period 1 (horizontal intercept). C (ma) = m m2 (1 + r) Here the difficult thing to understand is the second term. This term is the amount of mone I could

26 Microeconomics I. Antonio Zabalza. Universit of Valencia 26 borrow now on the epectation that in period 2 m income will be m 2. Wh? If toda I borrow, net period I will have to return (1+r). To pa this, net period I will have m 2. So, the I can borrow now, so that net period I will be able to repa with m income then, is: (1 + r) = m m2 = (1 + r) 2 C 2 m + m (1 + r) 2 1 slope = (1 + r) C 2 A E m 2 B C 1 m 1 m m (1 ) r C 1

27 Microeconomics I. Antonio Zabalza. Universit of Valencia 27 The slope, as usual is (minus) the vertical intercept divided b the horizontal intercept. Check that this gives (1+r). The rate of interest, r, is the relative price in this case: is the price of present consumption relative to future consumption. If r goes up, present consumption becomes relativel more epensive and the bl rotates clockwise around E. If r goes down, present consumption becomes relativel cheaper, so the bl rotates anticlockwise around E. Remember that the endowment is alwas available: whatever the change in prices, E is alwas affordable (in this and in all other applications). Formula for the budget line To derive the formula for the bl, suppose the equilibrium is at point A. The consumer saves ( m1 C1). This means that the future consumption he will be able to have is Or, C = m + ( m C )(1 + r) [ ] C = m + m (1 + r) (1 + rc )

28 Microeconomics I. Antonio Zabalza. Universit of Valencia 28 A straight line with vertical intercept [ m + m + r ] and slope (1+r). 2 1 (1 ) Eercise to do at the practical class Suppose the consumer s equilibrium is at point B (that is, he is a borrower). Derive the formula for the budget line. It should, of course, be the same as the one derived for a saver. Another (important) wa of epressing the budget line If in the previous formula of the bl ou multipl both sides of the equation b (1+r) and rearrange the terms ou have C C m + = m + (1 + r) (1 + r) This form for the budget line sas that the present value of consumption must be equal to the present value of income. Wh is the epression on the left of the equalit sign the present value of consumption, and wh the epression on the right is the present value of income?

29 Microeconomics I. Antonio Zabalza. Universit of Valencia 29 Equilibrium The equilibrium position is found as usual: point of tangenc between bl and ic. The problem here is that the ic is not between two different goods, but between the same good at two different time periods. So, we must first interpret the meaning of the slope of this ic. C 2 C 2 C 1 45º C 1 If at the intersection with the 45º line the slope of the ic was 1, it would mean that I value equall present and future consumption. This in fact is not like this. Normall we give more value to present than to future consumption. Wh? Think What would I prefer 1,000 now or 1,500 in five ears time? (Difficult choice? Well, think that in five ears time I ma be dead).

30 Microeconomics I. Antonio Zabalza. Universit of Valencia 30 Normall slope should be larger than 1 in absolute terms, meaning that people value more present than future consumption. C 2 C 2 C2 C 1 slope = = 1+ 1 = 1+ C1 C1 C1 The term ( ) C2 C1 C1 is the etra compensation I need now to give up the consumption of present income. Call it θ, the subjective rate of time preference. Then, the slope of the ic is ( 1 θ ) slope = + Recalling that the slope of the bl is (1+r), the equilibrium position, that is the tangenc between the ic and the bl, is a point where the market rate of interest equals the subjective rate of time preference. r = θ

31 Microeconomics I. Antonio Zabalza. Universit of Valencia 31 C 2 Equilibrium point; r = θ E C 1 Changes in equilibrium We suppose both C1 and C 2 are normal goods. Again, it is an application of revealed preference in the contet of changing endowments. A. First suppose the consumer is a lender m C >. ( ) A.1 Increase in the rate of interest. In that case, b the prp, the consumer will continue to be a lender. Do ou agree with that?

32 Microeconomics I. Antonio Zabalza. Universit of Valencia 32 Concerning the effect on consumption in the two periods, the result is given b the following table: Effects C 1 C 2 Substitution - + Income + + Total? + C 2 C q A q q E D B C 1

33 Microeconomics I. Antonio Zabalza. Universit of Valencia 33 A.2 Fall in interest rate In this case, the prp cannot help us to predict the final segment of the budget constraint where the consumer will be. So, in this case, we must conclude that the consumer could switch and become a borrower. Regarding the qualitative change in the consumption levels of present and future consumption, the relevant table is: Effects C 1 C 2 Substitution + - Income - - Total? - C 2

34 Microeconomics I. Antonio Zabalza. Universit of Valencia 34 A q C q q E B D C 1 Eercise: Complete the analsis when the consumer is a borrower and for the cases in which the rate of interest: a) increases; and b) decreases. Show in particular that in the first case the consumer could switch to being a lender, and that in the second case we can assert definitel that the consumer will remain a borrower.

35 Microeconomics I. Antonio Zabalza. Universit of Valencia 35 Inflation So far we have considered that the price of consumption in the two periods is the same: This in inde terms is equivalent to saing that the price in both periods is equal to 1: p1 = 1 and p2 = 1. In terms of the budget constraint, if we want to consider these prices eplicitl, the equation is pc pc pm + = pm + 1+ r 1+ r Suppose now that there is inflation and that p2 > p1. (If, sa, p1 = 1 and p2 = 1.10, it would mean that between period 1 and period 2, prices have increased 10%). Then the budget constraint is: C2 m2 C1+ p2 = m1+ p2 1+ r 1+ r 1+ r 1+ r C = m + m C p2 p2 That is, the slope of the budget line, the relative 1+ r to price of present consumption changes from ( ) ( 1 r) p2 +. Inflation, therefore, it is like if the discounting factor had changed.

36 Microeconomics I. Antonio Zabalza. Universit of Valencia 36 Let us relate this new discounting factor more eplicitl to inflation. Call π the rate of inflation, so that 1+ r 1+ r = p 1 + π 2 In our eample, π = 0.10, or a 10% rate of inflation. Then, let us call (1+ρ) this new discounting factor. 1+ r = π ρ ρ is the real rate of interest. Approimatel is equal to the nominal rate of interest minus the rate of inflation. ρ r π So, if the nominal rate of interest is 15% and the rate of inflation is 10%, the real rate of interest is 5%. So, when there is inflation, the budget constraint is ( 1 ρ) ( 1 ρ) C = m + + m + C The discounting factor is then defined with the real rate of interest.