1 Deriving demand function


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1 Econoics II: Micro Fall 2009 Eercise session 2 VŠE Deriving deand function Assue that consuer s utility function is of CobbDouglass for: U (; y) = y () To solve the consuer s otiisation roble it is necessary to aiise () subject to her budget constraint: + y y (2) To solve the roble Lagrange Theore will be used to rewrite the constrained otiisation roble into a nonconstrained for: a L (; y; ) = y + ( y y) (3) The rst order (necessary) conditions will result in: Cobining (4) and (5) will result in: which, cobined with (6) will give: y = (4) y = y (5) = + y y (6) y y = (7) ( ) = (8) and nally, after soe rearrangeents becoes: = (9) + This is the deand function for the good. When the rice of the good ;, is ed then (9) is the Engel curve for the good : It is easy to see that this was an eale of hoothetic references: It is enough to check the incoe elasticity to be equal to unity: = ( + ) = =
2 Rewriting (9) as: gives the Inverse Deand function! = + (0). Quasilinear references Reark Quasilinear utilities have the for u ( ; 2 ) = + v ( 2 )! Suose the agent is aiising the following utility function: U (; y) = + y () subject to standard budget constraint (2). Assuing that a rational agent will send all her oney on urchasing the goods (ore rigorous alternative is to set u Lagrangian function), the otiisation roble will beocoe: a y y y + y The rst order (necessary) condition after rearranageents reads: 2 y = (3) 2 y This is the deand function for the good y. It is indeendent on the incoe level, i.e. the agent is going to consue eactly the sae aount of the good y as long as the rices reain constant. On the other hand the agent is sending all her leftover oney on urchasing good : Fro (2) and (3) the deand function is: = which is of the usual for: = ( ; y ; ) : Q: Are and y substitutes or colients? (2) 4 y (4) 2 Eercises 2. True/False Clai If the Engel curve for a good is uward sloing, the deand curve for that good ust be downward sloing. 2
3 TRUE: Uward sloing Engel curve Noral good (negative incoe e ect Slutsky ) downward sloing deand curve Clai 2 If the deand function is q = 3 ( is the incoe, is the rice), then the absolute value of the rice elasticity of deand decreases as rice FALSE: The elasticity is: = 3 q = 2 q 3 = q q = : Thus has constant elasticity equal to unity. Note: Any utility function of the for q = A " has constant elasticity equal to ": Clai 3 An increase in the rice of Gi en good akes the consuers better o. FALSE: Increase in rice of any good akes the consuer oorer and thus worse o. (A grahical reresentation ay be helful!) Clai 4 The deand function q = If the rice goes fro 0 to 20, the absolute value of the elasticity of deand increases. TRUE: The elasticity of deand is: " = 0 q : " =0 = = 9 ; " =20 = = 4 : 4 > 9 Clai 5 In case of erfect coleents, decrease in rice will result in negative total e ect equal to the substitution e ect. FALSE: In case of erfect colients there is no substitution e ect, and the total e ect is equal to the incoe e ect. Clai 6 When all other deterinants are held ed, the deand for a Gi en good always falls when incoe is increased. TRUE: To rove the clai we need to show that Gi en good is always an inferior good. We are going to use the version of Slutsky equation that we had in class and ilustrated in Figure 3 (Note: the gure is illustrative and does not elain Gi en good). Thus: = s + Slutsky = X old X new total e ect s = X old X int substitution e. = X int X new incoe e. 3
4 Figure 3 X old X new X int BC( o, o) BC( n, i) BC( n, n) Total effect Substitution effect Incoe effect As we can see, the gure illustrates a case when the rice of the good went down, viz. = o n > 0 and we can rewrite the Slutsky equation as = s + and check for the signs. We know that substition e ect is always negative. We also know that for the Gi en good the total e ect is ositive. Thus the incoe e ect should be ositive: sgn = (5) In order to rove the clai we need to show that sgn = (6) or (sae as) < 0 that is the deand falls when incoe increases. Thus we need to see that sgn [] = sgn [] 4
5 Fro the budget constraint we know that when the rice goes down, the agent gets richer, i.e. = o n > 0 =) n o > 0 as the shift of the budget constraint is aralel to right. Thus we have Again fro (5) and (7) directly follows (6). Q.E.D. = o n < 0 (7) Clai 7 If the goods are substitutes, then an increase in the rice of one of the will reduce the deand for the other. False: According to the de nition! 2.2 Probles Proble Deand functions for beer is given: q b = 30 b + 20 c where is the incoe; b and c are the rices of beer and cake, resectively; q b is the deanded quantity.. is beer a substitute or colient for cake? c = 20 > 0 =) substitute) 2. assue incoe is 00, and cake costs, what is the deand function? (A: q b = b ) 3. write the inverse deand function. (A: b = 4 30 q b ) 4. at what rice would 30 beers be bought? (A: b = = 3) 5. Draw the inverse deand. (Hint: It s a linear function) 6. Draw the inverse deand when c = 2: (Hint: It s arallel to the above, but higher.) Proble 2 Suose the deand function is q = ( + a) ; a > 0; < :. Find the rice elasticity of deand. A = q ( + a) = ( + a) (+a) = +a 2. Find the rice level for which the elasticity is equal to ? A : +a = : = a + 5
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