Micro I. Lesson 5 : Consumer Equilibrium

Transcription

1 Microecono mics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 5 : Consumer Equilibrium 5.1 Otimal Choice If references are well behaved (smooth, conve, continuous and negativel sloed), then at the otimal choice of the consumer, Sloe of i.c. = Sloe of b.c. MRS = Price ratio MU = MU Otimal Choice Wh? Consider a situation in which the above equalit does not hold. Also, remember that moving along the bc is equivalent to using the market. Moving along the ic shows the minimum amount of I need to comensate for loss of. If the market gives me an amount of that is greater than the minimum I require, I will follow the market.

2 Microecono mics I. Antonio Zabalza. Universit of Valencia 2 C U(C) B U(B) B A U(A) B

3 Microecono mics I. Antonio Zabalza. Universit of Valencia 3 At A, for instance, if I give u 1 unit of (distance AB ), the market gives me 1 unit of (distance B B). But I would be satisfied with less; sa, 0.25 units of (distance B B ). Then, it is otimal for me to trade in the market, and go to oint B where m utilit U(B) is higher than that at oint a, U(A). If I kee aling this reasoning I end u at oint C. (Check that ou understand this). Point C reresents the best I can do, given m oortunities. Point C therefore reresents the otimal choice, the equilibrium, of the consumer. Going beond oint C, would lower again m utilit. Mathematics: Maimization of utilit subject to a given budget constraint. Method of Lagrange. Ma U, = U (, ) st.. + = m L= U(, ) + λ m

4 Microecono mics I. Antonio Zabalza. Universit of Valencia 4 First order (necessar) conditions δl δu = λ δ δ = 0 (1) δl δu = λ δ δ = 0 (2) δ L = m = 0 (3) δ This is a sstem of 3 equations in 3 unknowns:, and λ. From (1) and (2) we have that λ λ δu δ = = δu δ = = MU MU MU MU MU = = MU Then, to solve for and we consider equation (3) to get this more simlified form of the above sstem. MU MU = m = + (4) (5)

5 Microecono mics I. Antonio Zabalza. Universit of Valencia 5 This is a sstem of 2 equations with two unknowns (,). (Notice that in general the marginal utilities will deend on and ). Equation (4) is the equalit of sloes of bc and ic discussed above. What the result above sas is that this condition is not enough; we need also that the budget constraint is fulfilled (equation 5). Solving this sstem will, in general, give us the two demand functions for and that we are after. = (,, m) = (,, m) The urose of this lesson is to find out how the three variables (,, m ) influence the demand for and. Before, we give an eamle of this derivation for a articular utilit function: Cobb- Douglas.

6 Microecono mics I. Antonio Zabalza. Universit of Valencia 6 Eamle: Cobb-Douglas (CD) utilit function: a b U(, ) = a b Ma U =, st.. + = m Necessar conditions: a b L= + λ m δ δ δ δ L a a 1 b λ = = 0 (1) L b a b 1 λ = = 0 (2) δ L = m = 0 (3) δ Eliminate λ from (1) and (2), and together with (3) ou obtain a b = a 1 b a b 1 m = + (4) (5) Sstem of two equations with two unknowns.

7 Microecono mics I. Antonio Zabalza. Universit of Valencia 7 Equation (4) can be eressed in the form MU =, MU which for this articular case is a 1 b a a b 1 b = or, a =. b So the sstem, in this simlified form is a = (4) b m = + (5) Solving for and for, we find the two demand equations: a m = a + b b m = a + b With a CD utilit function, the demand for each good deends on income (ositivel) and its own rice (negativel). It does not deend on the rice of the other good.

8 Microecono mics I. Antonio Zabalza. Universit of Valencia 8 Another characteristic of this utilit function is that the arameters of the function give information about the eenditure shares on each good. Share of eenditure on : Share of eenditure on : a = m a + b b = m a + b Sufficient condition Equations (1), (2) and (3) are the necessar conditions. The are not sufficient. For instance, consider the following situation: Point of tangenc A Point of maimum utilit At A, the first order conditions are met and et utilit is not maimized. You need also another set of conditions which are sufficient. These conditions boil down to the requirement that references have to be conve. See that in the figure the are concave.

9 Microecono mics I. Antonio Zabalza. Universit of Valencia 9 Etreme eamles Corner solutions Maimum utilit, but at this oint Sloe of ic > sloe bc MRS > rice ratio A articular eamle of a corner solution is when the two goods are erfect substitutes: Suose U=+ and = 2; = 4; and m = 12. ic sloe: 1 bc sloe: 1/2 3 Maimum U A 6

10 Microecono mics I. Antonio Zabalza. Universit of Valencia 10 Here MU 1 < ; < 1 MU 2 Maimum is obtained at oint A, where onl is consumed. So the demand function in this case is m = Kink solutions Suose the two goods are erfect comlements with the following utilit function: U = min {, } and = 2; = 4; and m = A 4 12 The otimal choice must lie on the diagonal and on the budget constraint. Therefore, the otimal choice is found b solving the sstem

11 Microecono mics I. Antonio Zabalza. Universit of Valencia 11 = m = + The solution is m = = + For the articular eamle used here 24 = = = Changes in the equilibrium osition Now we want to investigate how the equilibrium just studied is altered (dislaced) b changes in the eogenous variables of this roblem. In articular, we want to know how the equilibrium changes when, m, and change. Or to ut it in other words. The result of the revious analsis was the derivation of two demand curves = (,, m) = (,, m) We want to sign the artial effects of the three eogenous variables m, and on the demand of and. We will consider three tes of changes: a) Simultaneous change in rices and income b the same roortion. b) Change in income onl.

12 Microecono mics I. Antonio Zabalza. Universit of Valencia 12 c) Change in one rice onl. Equiroortional change in rices and income If m, and all move b the same roortion, the bc does not change and therefore the oint of equilibrium does not change either. Suose initial bc is m = + Multil all rices and income b k (if, for instance, k=1.1, then all variables increase b 10%). The new bc is km = k+ k But k can be cancelled out b dividing both sides of the equation b k. So, the original bc remains unchanged m = + Change in m onl We know that an increase in m moves the bc out. The osition of the final equilibrium deends on whether the goods are normal or inferior. Suose first that both goods are normal. Then, if there is an increase in income from m 0 to m, more of both goods will be bought. This is reresented in the following figure.

13 Microecono mics I. Antonio Zabalza. Universit of Valencia 13 B 0 m A m The demand curve that treats rices as given arameters and income as a variable, is known as the Engle curve. It takes the form: = m (,, ). m (, ) m 0 m A B 0

14 Microecono mics I. Antonio Zabalza. Universit of Valencia 14 We sa that a good is inferior if when income is raised, holding everthing else constant, less of this good is bought. Sa is inferior. Then, B 0 m A m Eercise: Derive the shae of the Engle curve corresonding to good, when is an inferior good. Proert: The sum of the income elasticities of each good, weighted b its corresonding eenditure share, must be equal to one. We start with the budget constraint: m = + Then, differentiating both sides of the equalit b m, we find:

15 Microecono mics I. Antonio Zabalza. Universit of Valencia 15 dm = + dm m m 1 = + m m m m 1 = + m m m m 1 = s ε + s ε m m As we were looking for, the weighted average of income elasticities must add u to 1. Imlications of this result: a) Not all goods can be inferior. Not all Engle curves can have negative sloe. b) Goods that take a large share of eenditures are unlikel to have either ver large or ver low income elasticities, since the average must equal one. Problem for home: Sa we divide goods in two tes: food and non-food. We know food takes 60% of eenditures, and the income elasticit of nonfood is 2. What is the income elasticit of food? Eercise: Find the sloe and grahical shae of the Engle curve for good when the utilit function is Cobb Douglas. What about when the goods are erfect substitutes? And erfect comlements? You

16 Microecono mics I. Antonio Zabalza. Universit of Valencia 16 will see in this eercise that all Engle curves for these secific cases are straight lines. This is like this because the oints of equilibrium in the goods sace when income rises is also a straight line from the origin. In these cases we sa that references are homothetic. Change in one rice (holding constant m and the other rice) Suose decreases with and m constant. A B 0 ' Normall if the rice of one good goes down, the quantit consumed of that good goes u. and viceversa.

17 Microecono mics I. Antonio Zabalza. Universit of Valencia 17 The demand curve The following grah shows how the demand curve is derived out of the consumer equilibrium. In fact, each oint in the demand curve is a oint of equilibrium for the consumer at different rice levels. A B 0 ' 0 A 1 B Demand curve (,, m )

18 Microecono mics I. Antonio Zabalza. Universit of Valencia 18 Eercise: Draw the demand curves (and identif the sloe of the curve) for CD references, for erfect substitutes and for erfect comlements. The demand curve not alwas is downward sloing. This is an anomal, but in rincile it can haen. B A 0 ' Demand curve (,, m ) 0 A 1 B

19 Microecono mics I. Antonio Zabalza. Universit of Valencia 19 Goods which disla this anomalous behaviour are called Giffen goods. [Illustration concerning consumtion of horse meat]. 5.3 Income and substitution effects Now we want to decomose the effect of the change in one rice in two effects: a) a first effect which is equivalent to a change in relative rices holding income constant. This is called the substitution effect. b) A second effect which is equivalent to a change in income holding relative rices constant. This is called the income effect. 0 0 Suose initiall we are at oint A (, ), with rices (, ) and income m. Therefore, the budget constraint at this oint is: = m (1) 1 Suose now the rice of goes down to, and we ask what is the income the consumer would now need to bu the old bundle of goods. This income, 1 m, is = m (2) 1 0 Clearl m < m. More recisel, the change in income can be found substracting equation (1) from equation (2).

20 Microecono mics I. Antonio Zabalza. Universit of Valencia 20 m m = + ( + ) = ( ) m = So, the change in income needed to bu the old bundle is equal to the initial quantit of times the change in the rice of. Suose after the rice change in we take awa m from the consumer, so his new bc asses through A but is flatter than his old bc. Will he remain at A? Clearl not. He can do better than this b going to B s s (, ). As comared with oint A, the consumer has adjusted his consumtion b buing more of the good that has become relativel cheaer () and less of the good that has become relativel more eensive (). The change from oint A to oint B is the substitution effect. We can reresent it formall as a change in the demand for from the initial osition = (,, m ) at oint A, to the osition s = (,, m ) at oint B. If we denote s b, then s 0 ( ) = s (,, m ) (,, m ) s = (,, m ) is called the comensated demand for. Comensated because is the demand for as

21 Microecono mics I. Antonio Zabalza. Universit of Valencia 21 a result of a fall in the rice of when the consumer is comensated for the increase in income generated b the fall in the rice of. Income effect The movement from A to B is a hothetical movement. At his final choice, the consumer is sending all his income; so he will be at a oint such as C in the final budget line, where the demand for n is = (,, m ). Now, observe that the move from B to C is like a ure income effect (income 1 0 increases from m to m, while rices remain 1 0 n s n constant at and ). If we call =, we have that the income effect is, = n (,, m ) (,, m ) Total effect The total effect is the move from A to C. That is from 0 to n. Or using the same terminolog as above, n 0 = = (,, m ) (, m ) ,

22 Microecono mics I. Antonio Zabalza. Universit of Valencia (,, m ) (,, m ) n C 0 A s B (,, m ) 0 s n

23 Microecono mics I. Antonio Zabalza. Universit of Valencia 23 The Slutsk equation Notice that the total effect can be written as the sum of the substitution and income effects. 0 0 ( n ) = ( s ) + ( n s ) = + s n This eression is called the Slutsk equation. Signing the substitution effect If goes down, as in the figure above, then the change in demand for that results from the substitution effect must be non-negative If < (,, m ) (,, m ) Or, 1 0 s If < 0 Wh is this so? Because the indifference curves are well behaved (continuous, smooth, negativel sloed and conve). Convince ourself that with this te of i.c. it cannot be otherwise. Points to the left of A will lie on lower i.c. and therefore will not be chosen. Conclusion: The substitution effect alwas moves oosite to the rice movement. We sa the substitution effect is negative: if the rice goes down, the demand for the good due to the substitution effect increases, and vice versa.

24 Microecono mics I. Antonio Zabalza. Universit of Valencia 24 Signing the total effect Contrar to what haens with the substitution effect, the total effect can be signed most of the times, but not alwas. It deends on whether the good is normal or inferior. To see that, recall the Slutsk equation. Normal goods = + s n = + s n ( ) = ( ) + ( ) Both substitution and income effects work in the same direction. Consequentl, as rice goes down, quantit demanded goes u, and vice versa. The total effect is negative. Check ou understand wh income effect in this case is negative. m. Price and good demanded due to income effect move in oosite directions.

25 Microecono mics I. Antonio Zabalza. Universit of Valencia 25 Inferior goods = + s n (?) = ( ) + ( + ) Here the sign of the final effect deends on the relative strength of substitution and income effects. We have two ossibilities: a) Substitution effect dominates in absolute terms. Then the total effect is negative. This is what will usuall haen. b) Income effect dominates in absolute terms. Then the total effect is ositive: as rice goes down, quantit demanded goes u and vice versa. This means that the demand curve is uward sloing. We call this te of goods, Giffen goods. The are ver rare. A conclusion regarding inferior and Giffen goods: a Giffen good must be an inferior good, but no all inferior goods are Giffen goods.

26 Microecono mics I. Antonio Zabalza. Universit of Valencia 26 Eressing the Slutsk equation in terms of rates of change with resect the change in rice. = + s n = + Recall revious result, s n (1) m= If the rice goes down, this eression gives us a negative number (the amount of income that, as a result of the rice fall, has to be taken awa from the consumer so that he can just afford the old bundle A). To identif the income effect from B to C we want to work with the negative of this (negative) amount: secificall, with the amount of mone that is given to the consumer so that he can go from the budget AB to the final budget at C. For this urose we define a new change in income, ( n), which is just the negative of the reviousl defined income change, ( m), n= m n= n = (2)

27 Microecono mics I. Antonio Zabalza. Universit of Valencia 27 Substituting (2) into (1) = n s n Which is the Slutsk equation eressed in terms of rates of change. Notice that now the income effect is eressed directl as a change in due to a change in income and, therefore, for a normal good, is ositive. From this equation we state the Law of Demand: If the demand for a good increases when income increases (that is, if the good is normal), then the demand for that good must decrease when its rice increases. = n s n ( ) = ( ) ( + )( + ) Naturall, this can be said because we know the substitution effect is alwas negative.

28 Microecono mics I. Antonio Zabalza. Universit of Valencia 28 Another wa of measuring the substitution effect: the Hicks substitution effect. The revious wa of measuring the substitution effect was roosed b an economist called Slutsk. Another economist (John Hicks) roosed another wa of identifing the substitution effect. To comare them we define both: Slutsk substitution effect: Change in demand when rices change but the consumer s urchasing ower is held constant so that the original bundle remains affordable. Hicks substitution effect: Change in demand when rices change but the consumer s income is changed so that he can reach his original utilit level. That is, change in demand when rices change but consumer s utilit is held constant at its original level. For small (infinitesimal) changes in rices both measures coincide. For large (non infinitesimal) changes in rices the differ. This can be seen grahicall.

29 Microecono mics I. Antonio Zabalza. Universit of Valencia (,, m ) (,, m ) C A B B (,, m ) 0 sh s n 0 A B 1 B C Demand curve (dc) Comensated dc (Slutsk) Comensated dc (Hicks)

30 Microecono mics I. Antonio Zabalza. Universit of Valencia 30 In the grah we identif three demand curves: Usual demand curve (dc): = (,, m) Comensated dc (Slutsk): = (,,urchasing ower) Comensated dc (Hicks): = (,, u) Check ou understand the ceteris aribus clause of each of these three demand curves. Eercise: How would the grah below look like if good instead of being normal was inferior? (What will be the relative configuration of the three demand curves?)

31 Microecono mics I. Antonio Zabalza. Universit of Valencia Imlications of the MRS conditions Observation of demand behaviour can give us information about the underling references of consumers who disla that behaviour. In a cometitive market, rices are the same for everbod. Thus, if consumers are at equilibrium ositions, MRS = MRS = MRS = = MRS = n Everbod will adjust their consumtion of goods until their own internal marginal valuation (MRS) equals the market s eternal valuation ( ). Marginal changes in consumtion, therefore, will be valued the same for everbod. Eamle (Varian s): Suose that in a cometitive market one bottle of milk costs 1 and one ack of butter costs 2. This means that, MRS MU MU = b = b = An roject (olic) that gives eole goods for more than what the value them is rofitable, and vice versa. m m 2

32 Microecono mics I. Antonio Zabalza. Universit of Valencia 32 Project A: 1 ack of butter is roduced with 3 bottles of milk. This is not a good roject. Using market rices, it is equivalent to saing that 2 are roduced with 3. Peole value more the inuts than the outut of this roject. Project B: 3 bottles of milk are roduced with 1 ack of butter. This roject is OK. Using market rices, it is equivalent to roduce 3 with 2. Here eole value inuts less than outut. Conclusion: Prices are not arbitrar things; rather, the reflect how eole value things at the margin.