Chapter 6 In our starting model: There are 2 goods The prices and income are given This is a world without time (the consumer can t save)

Transcription

1 Chater 6 In our starting odel: There are goods The rices and incoe are given This is a world without tie (the consuer can t save) (Consuer) Deand is deterined through UTILITY MAXIMISATION given: TASTES (references, utility function, indifference curves, arginal utilities) OPPORTUNITIES (udget set) This sile odel which we egin with is an eale of: Partial analysis all other factors are ket constant. Individualistic one erson s consution and utility doesn t ake any difference to anyone else s utility. No interersonal coarisons of utility can e ade. Static- the odel refers to one single oint in tie. With far reaching rationality assutions individuals have full and accurate inforation aout arkets, goods and rices, and not least they know their own inds erfectly. Many of these restrictions can e relaed in ore sohisticated odels ut then the results ay also e less clear-cut. Note that the only kind of deand that is under discussion is effective deand deand that is coined with urchasing ower. Needs that cannot ake theselves felt on the arket fall outside the scoe of the theory.

2 Make sure you reeer fro chaters -5: Indifference curves when goods are: Perfect coleents Perfect sustitutes and when consuers have: Quasilinear references or Co-Douglas utility functions The consuer s est choice"(when there are goods) will occur either: At a tangency oint (fig. 5.) where MRS -MU/ MU - / As a corner solution ( kinky oint, fig. 5., where the utility function isn t differentiale) In a oundary oint (fig 5.3) tastes & udget constraint deand The things that are given in the odel are: Goods, Prices, Incoe Utility function U(, ) Find i i (,, ) i, The consuer s deand for good i as a function of incoe and (oth) rices. (all other relevant factors are assued to e constant)

3 . Incoe/deand Noral good: i i > 0 Inferior good i i <0 (Own) Price/deand Ordinary good i i i < 0 i Giffen good i i i > 0 i

4 Incoe consution curve -the consuer s est choice of (undles of) goods at different levels of incoe Engel curve - deand for good at different levels of incoe Incoe consution curve Engel curve (noral good)

5 Indifference curves when goods are erfect sustitutes Engel curve for the cheaer good when the two goods are erfect sustitutes / Sloe Engel curve, when goods are erfect coleents / ( ) Sloe If deand for a good increases in asolute ters with a rise in incoe, the roortion of incoe sent on it ay increase or decrease.

6 Eenditure on food and housing ake u a larger roortion of the sending of low incoe households than of high incoe households, even though the richer households on average send ore oney on oth food and housing than the oorer. Noral good: i Inferior good i hoothetic Luury good / (share of incoe sent on ) Necessary good / Share is constant references are Hoothetic references When references are hoothetic the choice etween undles of goods deends only on the roortions of goods in the undles. Irresective of the level of incoe, the share of the udget that is sent on the good is the sae. In other words, eenditure on the good is roortional to the consuer s incoe. The Engel curve is a straight line through the origin

7 Eale: Calculating deand with a Co-Douglas utility function: Maiise u(, y) a y a suject to qy () u(,y) takes its aiu at the sae oint (, y) as the utility function V(,y) ln u(,y). Therefore finding the and y that solve () is equivalent to finding those that solve: aiise V(,y) ln u(, y) a ln (-a) y We will use the condition for arginal utilities. (Alternatively, we could use the Lagrangeethod the calculations are retty uch the sae.) V V y a a y V MRS V y a a y At the otial oint MRS - /q a y q a y a ( q y / q a a q a ) ( a ) ay / Insert into the udget condition ( a) y ay a a ( ) a a a a a a a We can see that a Co-Douglas utility function ilies hoothetic references - akes u a constant share of -

8 Bengt s utility function is U(, y) ln - stas - eer Bengts udget rice of stas rice of eer Bengt s udget Quasilinear references Eale 6.3 a.what is Bengt s deand for eer and stas?.is it true that Bengt would send every krona in additional incoe on stas? c.what haens to deand when Bengt s incoe changes? d.what haens to deand when and increase? Solution: a) Given rices, and, find the quantities and which aiise Bengt s utility! Necessary condition: MU MRS MU MU MU / MRS Therefore the otiu occurs when Money left to uy for: if > ) Yes, if > he won t uy any ore eer when increases. c)-d)

9 0 0 ) ( ) ( ) ( > d d d d d d d d ε ε ε ε

10 Price consution curve Shows the undle of goods that is the consuer s otial choice at any given rice (other rices assued to e constant) Deand curve Shows the quantity of one good the consuer will e uying at different rices. Ordinary good - negative sloe Giffen good ositive sloe Reservation rice The rice at which the consuer is indifferent etween uying one unit ore and not uying one unit ore. Assue that the consuer s utility is a function of two variales, her consution of one articular good and her consution of everything else easured uy the oney she sends on it. u ( 0, ) u (, r ) () u (, r ) u (, r ) () The left side of () is consuer utility when she does not uy the good at all and can send her entire incoe,, on other goods. The right side of () is consuer utility when she uys one unit of the good at rice r and has -r left to uy other things. Price change and deand The rice of one good can affect deand for another even if they are not erfect sustitutes or coleents: If goods & y are sustitutes /y > 0 (y ) If goods & y are coleents /y < 0 (y )

11 K is revealed to refer (, ) efore (y, y ) eans that K can choose either ut chooses (, ) (it is assued that K aiises utility and has a unique otial choice) (y, y ) (, ) (Matheatically the assution that there is a unique est choice can e eressed y saying that the consuer has strictly conve references the utility function is such that indifference curves are strictly conve.) y y y y (the entire incoe of the consuer is sent) (the cost of the undle y is within what the udget allows) The rincile of revealed reference: Let (, ) e the chosen undle when rices are (, ), and let (y, y )e soe other undle such that y y. Then, if the consuer is choosing the ost referred undle she can afford, we ust have (, ) >(y, y ) (age ) (, ) >(y, y ) and (y, y ) >(z, z ) (, ) >(z, z ) Or, in other words: If (, ) is referred to (y, y ) ( chosen when (y, y ) is also feasile and (y, y ) is referred to (z, z ) ( chosen when (z, z ) is also feasile) we infer that (z, z ) cannot e referred to (, ) ( will not e chosen when (, ) is also feasile). We say that (, ) is indirectly revealed referred to (z, z ) (, ) (y, y ) (z, z )

12 WARP The Weak Aio of Revealed Preferences (.4) If (, ) is directly revealed referred to (y, y ), and the two undles are not the sae, then it cannot haen that (y, y ) is directly revealed referred to (, ). SARP The Strong Aio of Revealed Preferences (.8) If (, ) is revealed referred to (y, y ) (directly or indirectly), and the two undles are not the sae, then it cannot haen that (y, y ) is directly or indirectly revealed referred to (, ). Indeation: How do you coare living standards in two different countries? It is not so easy to coare actual consution since ore ay e consued of one good in country A and ore of another in country B. To aggregate one wants to easure the value of the average consution undle of eole in each country. But if relative rices are different, the result you coe u with will deend on whether you use the rices in country A or the rices in country B. This is an eale of an indeation role. It does not have one right answer. Instead one has to choose inde according to the question one wants to answer. Quantity indeces Laseyres inde Paasche-inde L q t y y t y y Pq t t y t t y t t y y Lq< etter at tie Lq>? P q > higher utility at tie t P q <? Laseyre-inde L t t y y y y If L <M higher utility at tie t* Price indeces Paasche -inde P t t t t y y t y y If P >M higher utility at tie ** t M-inde t t y y t t M y y

13 * Coare L and M. Their denoinators are identical. Therefore, L <M ust ily that the nuerator of L is saller than the nuerator of M. t t y y < t t t y y t t This eans that it would have een ossile for the consuer to consue (, y ) instead of ( t, y t ), given the rices that eisted at tie t and her udget at tie t. The consuer did in fact choose ( t, y t ) and, thus, revealed her reference for ( t, y t ) relative to (, y ). If M < L the consuer could not afford to uy (, y ) at tie t. The cost would have eceeded her udget. There is no choice etween (, y ) and ( t, y t ) to e oserved and no conclusions can e drawn as to whether the consuer refers (, y ) or ( t, y t ). ** Coare P and M. The nuerators are identical. Therefore, P > M ilies that the denoinator of P is saller than the denoinator of M. t y y t < y y In other words, the udget at tie was large enough to allow the consuer to uy what she in fact ought at tie t, ut not at tie. At tie the consuer had a choice etween (, y ) and ( t, y t ). By actually choosing (, y ) she revealed that she referred this undle to ( t, y t ). If it had een the case that M > P the consuer would not have een ale to urchase ( t, y t ) at tie and, there is no choice etween the for us to oserve and we can t tell which undle the consuer refers. E 7.4 Year Clu Creones Everton Salary.5 illion 5.4 illion Laseyre CPI: L a) Were 5.4 in 994 enough to urchase what he ought in 986 for.5? ) What was the iniu incoe required in 994 in order to do so? c) If instead we had een told that the Paasche inde P.8?

14 n n Answer: a) L, n n i.e. consution 986 at 994 rices divided y consution in 986 at 986 rices. Assuing that Liar did not save (sent his whole incoe) his consution in 986 cost.5 illion n n L(... n n < that is to say, his 994 incoe was sufficient. ) P (994 consution at 994 rices) divided y (994 consution at 986 rices) If P.8 his 994 consution would have cost hi 5.4 illion/.8 3 illion in 986. This is ore than he could afford in 986. But that is not an answer to the question. To answer we need to know what his 986 consution cost in 994, and that we do not know. )

15 Eercise 7.3 Solution: Ec Other goods ursr idrag kuong Food The lue line is the faily s udget line without enefits 750 can e sent on either food or other goods. The red line is the udget line if the faily receives 50 in cash enefits. The green line is the udget line if the faily gets the otion of uying u to 50 worth of couons which reduce the rice of food to a half. Given that food is a noral good, which art of the green and red lines could e chosen?

16 Ea Aril Question 3 Peder and Pekka live in different countries with different relative rices of goods. Both of the send the whole of their incoe on two goods, and y. Peder has to ay 4 daler for each unit of and 5 daler for each unit of y. Pekka ays 9 arks/unit for and 7 arks er unit of y. Peder uys 8 units of and units of y. Pekka uys 0 units of and 9 units of y. a) Write the equations for the udget constraints of Peder and Pekka and show the in a diagra. ) Is the inforation that you are given in this question sufficient to deterine whether Peder would refer Pekka s consution to his own? Would Pekka refer Peder s consution undle? Elain your answer! c) What econoic rincile do you need to answer question )? Answer: 3. a) Peder s consution costs 4*85* 67 daler His udget constraint is 45y67 Pekka s udget constraint is 97y53 ) At the rices that Peder ays, Pekka s consution undle would cost 4*09*585 daler. This eans that Peder can t uy this consution undle and we haven t got enough inforation to deterine which of the two undles that Peder would refer if hed id have the choice. Pekka would have to ay 8*9*749 in order to uy the coination of goods that Peder consues and this falls within his udget constraint. We can therefore oserve Pekka choosing etween the two undles and conclude that he does refer his own to that of Peder. c) The Weak Aio of Revealed Preferences (Varian.4) If (, ) is directly revealed referred to (y, y ), and the two undles are not the sae, then it cannot haen that (y, y ) is directly revealed referred to (, ).

17 Chater 8 Price and deand Eale: Assue that there are two goods: consution and leisure oth of which increase utility. The rice of leisure (oortunity cost) is the wage rate, w. Choice of leisure when w? Leisure costs ore sustitution effect Leisure is ore affordale incoe effect Sustitution effect to do with relative rices Incoe effect to do with what you can afford Incoe and sustitution effects of a rice decrease Old udget New udget S I Pivot line

18 - - [ - ] is called coensated incoe. It is the sallest incoe that allows the consuer to uy the original consution undle at the new rices. Sustitution effect s (, ) (, ) Sustitution effect difference etween coensated deand and deand efore the change Incoe effect: n (, ) (, ) Incoe effect new deand inus what deand would e with coensated incoe Eale Goods:, y Price of : Price of y: q Incoe: 3 Utility function: 5 5 U (, y) y a) If 5, q0, 000, calculate the deand for and y ) If is reduced to 4, calculate c) At what incoe would the consuer e just ale to consue the sae undle (, y) as in a) if 4 d) Calculate the sustitution effect for e) Calculate the incoe effect for f) Draw the two udget lines, indicate the choices and the sustitution and incoe effects

19 Solution: a) With a Co-Douglas utility function U (, y) α y β and incoe, deand is α, β y ( α β ) ( α β ) y 0.4*000/5 80 y 0.6*000/0 30 ) 0.4*000/4 00 c) 4*80 0*30 90 d) With 4 and 90 would e 9. The sustitution effect is 9 80 e) Alternative : Incoe effect Alternative. Incoe effect total effect sust. effect 0 8 sustitution effect > 0 sustitution effect < 0 Noral good: incoe effect. > 0 incoe effect < 0 Inferior good: Reverse! Total effect: n, ) (, ) () ( s, ) (, ) () ( ( Eercise: Add & check!, ) (, ) (3) THE SLUTSKY EQUATION (THE SLUTSKY IDENTITY) Noral good: The two effects ull in sae direction. < 0 > 0 Inferior good: Effects work in different directions. >0 eller <0??

20 Noral good not Giffen g. Giffen g. inferior Inferior g. can e Giffen or ordinary Law of Deand: If deand for a good increases with incoe, it does not increase with rice (. 47) Eale: Price decrease of for a good which has a erfect sustitute. Assue: Goods are erfect sustitutes and to After Before After > > 0 / > < 0 / / 0 Sust. eff < < / 0 / 0 Inc. eff.

21 Eale Varian age Ta is t kr / litre. Consuer ays (t)/l 3. Susidy taation incoe Incoe Consution of etrol, rice of etrol, other cons. Before ta:,, y After: (t) y Budget constraint (average consuer): y () (t) y t () y ( ) ( ) (, y ) was feasile efore introduction of the ta WARP (, y) referred to (, y ) Slutsky equation, continued s n () s n () (3) Insert (3) into () s n s n or s n s where n Why switch to? Varian only says that it is convenient. Start fro the second ter on the right side of eq. () (4) (5)

22 ) ( ) ( ) ( n n irresective of 0, But the liit of when 0 is the derivative of with resect to. The second factor n ), ( ), ( is not a difference quotient. But ), ( ), ( n is a difference quotient. As 0, and this difference quotient aroaches the derivative of (deand) with resect to (incoe). If is noral we know that this derivative is ositive.

23 To find the sustitution effect one wants to kee incoe constant and isolate the effect due to change in relative rices. But what is to e eant y keeing incoe constant?. Actual incoe. Preserving urchasing ower (Slutsky) 3. Preserving utility (Hicks) Hicks sustitution effect gives the coensated deand curve

24 Chater 9 What s new in the odel? We no longer assue: A given incoe in oney ters. But instead assue: Given initial endowents Endowents can e ought and sold (ω, ω ) endowents of good and (, ) gross deand (consution) ( - ω, - ω ) net deand For good i: Net uyer i > -ω i i - ω i > 0 Net seller i < -ω i i - ω i < 0 Budget constraint: ω ω ( -ω ) ( -ω )0 (, ) (ω, ω ) sloe - / ω ω > σ σ (ω, ω ) referred endowent to (σ, σ ) But it does not ean that the consution undle (ω, ω ) is referred to the consution undle (σ, σ ).

25 (, ) (ω, ω ) sloe - / Eale K. was K ecoes K s utility Net seller Net seller Net seller Net uyer? Net uyer Net uyer Net uyer Net seller Iossile! (fig. 9.4) Sustitution effect Ordinary incoe effect Endowent-incoe effect

26 Slutsky revisited s endowent incoe effect rice change incoe (udget) change deand change ω so that ω ω s ) ( ω

27 Another way of descriing it: two goods in quantities (, y) rices (, q) endowent (w, u), < (rice decrease) w, u, q unchanged y y Budget restriction efore rice decrease: Incoe w qu y After rice decrease: wqu qy In order to uy (, y) after the rice decrease qy is required Consution of good : Consution of each good is an effect of oth rices as well as of incoe (the value of the endowent) ut since q is constant we can see as a function of only its own rice and of incoe. Total effect: sustitution effect: ordinary incoe effect (, wqu) (, w qu) (, qy) (, w qu) (, qy) - (, qy) endowent-incoe effect (, w qu) (, qy) (, w qu) (, w qu) What do the oints reresent? (, w qu) actual deand efore the change, urchases ade at the old rice and the endowent is evaluated at the old rice the old consution undle (, w qu) (, qy) (, qy) actual deand after the change, urchases ade at the new rice and the endowent is evaluated at the new rice the new consution undle. what deand would e with the new rice if the value of the endowent (the incoe) was only just enough to uy the old consution undle what deand would e with the new rice if the value of the endowent hadn t changed at all in oney ters.

28 Eale : Workook Eercise 9. toatoes 30 auergines Endowent Consution Price Auergine 5 5 kr/kg Toatoes 5 y 5 kr/kg a) Money value of endowent 5*55*5750 ) Since the goods are erfect coleents y Budget constraint 55y750 and y and y 5 and y5 Answer: He sells 0 kg auergine and uys 0 kg toatoes c) toatoes changes to 75 New value of endowent: 5*55*75000 Since the goods are coleents we can see one kg auergines and one kg of toatoes as a coosite good with rice Maiu consution with an endowent value of 000 is 0 units of the coosite good, i.e. he consues 0 kgs of each vegetale. Answer: He sells 5 kg of auergine and uys 5 kg toatoes. d) If the value of his endowent had reained 750 he would have consued 7.5 kg of each (y analogous reasoning to c)). e) Total change in consution of toatoes: The sustitution effect is (rice75, the endowent value needed to uy old undle) (5, 750) To uy y5 at the new rice he would need 5*575*5500 (75, 500) (5, 750) When goods are erfect coleents there can e no sustitution effect! Ordinary incoe effect: (75, 750) (75, 500) Endowent incoe effect: (75, 000) (75, 750) Check: total change in consution of toatoes.

29 Eale (fro ea Nov. 3 rd 005) The stor Gudrun in the eginning of January this year felled illions of trees in the Southern arts of Sweden. Because all this tier had to e sold, the rice of wood fell. Thus, forest owners saw a decrease in the value of their endowents of wood and tier. Soe of the coanies that uy wood are forest owner cooeratives and they decided to share the loss fro the stor y aying the sae rice as efore. Private coanies aid the lower rice. The farer Skoglund owns an ale orchard and soe forest land, where he can cut wood. He eats his ales (or other food that he can echange for ales in the village arter econoy) and he heats his house with wood. Therefore we can assue his utility to e a function of the quantity of wood he uses a fuel (w, in cuic eters) and the quantity of ales (a, in sacks) that he consues (eats or arters) U ( w, a) w a He also has the otion of uying and selling wood and ales on the arket. The rice of a cuic eter of wood is 50 kronas and the rice of ales is 400 kronas er sack. His endowent is 5 3 of wood and 30 sacks of ales. a) What is his udget constraint? ) How uch wood and ales will he consue and how uch will he sell/uy on the arket? c) Skoglund is not a eer of the cooerative forest coany so after the stor the rice he could get for his wood was 00 kronas er 3. What is the value of Skoglund s endowent now? How uch wood will he use as fuel? d) If Skoglund had een a eer of the cooerative and could have sold his wood at the rice of 50/ 3 the value of his endowent would have een unchanged. In this case, with the sae oney value of the endowent and a urchasing rice of 00/ 3, how uch wood would he consue? e) What is the new arket value of what Skoglund consued efore the stor? f) How uch wood would Skoglund consue if he had a oney incoe equal to the su you calculated in e), and the rice of wood was 00/ 3? g) Use your answers to questions d)-f) to divide the difference in Skoglund s consution of wood in question ) and c) into sustitution effect, ordinary incoe effect and endowent effect. Answer: a) 50w400a 50*5 400* ) Since this is a Co-Douglas utility function with eonents 0.4 and 0.6 we know that the deand functions are w 0.4*/ w () a 0.6*/ a () where is incoe (the value of the endowent. With the given rices and 5000 w40 and a37.5. Skoglund sells 3 of wood and uys 7.5 sacks of ales. c) The value of the endowent is now 400 and w 00 According to eq. () w 44.8 d) With 5000 and w 00, w 50

30 e) The total effect is an increase in his consution of wood y In the hyothetical situation in d) the value of the endowent is as efore the rice change, ut the relative rices and the urchasing ower of the incoe are those after the rice change. To find the endowent incoe effect we coare this with his deand in question c) when the rice is lower and the effect on the value of the endowent is taken into account. Endowent incoe effect: To uy 40 3 of wood and 37.5 sacks of ales after the rice decrease, requires kr. With 3000 and w 00 with the ivot line as udget constraint - w46. Sustitution effect: Ordinary incoe effect: Check: actual total effect Eale 3: Laour suly Non-laour incoe Cons. Total tie Leisure Working. tie Wage rate (hourly) Y C T R L W Budget: C Y W (T R) Y WL C WR Y WT Assue: Conve references Leisure is noral good Y R, L W? sustitution effect R, L ord. incoe effect R, L BUT endowent-incoe effect R, L

31 Eercise 9.: M non-laour incoe w wage er hour U utility R rest, leisure L laour tie C consution T total tie availale Endowent: M500 T50 Ojective function (to e aiised): UC*R Constraints: CMw*L () LRT () Answer: a) Value of endowent: 50050*T 3000 ) c) MU C R MU R C C R w At the otiu MRS relative rice R C wr C w 50 R (3) Insert (3) into () 50R50050(50-R) (4) R30 R30, () and (3) L 0 and C500 d) Use (4) with syols and (3) Cw(T-R)Mw(T-C/w) CM/wT/ (6) e) (3) RC/w. According to (6) M T R (7) w f) (7) and M 0 R0T/ 5 if T50