Delhi School of Economics Course 001: Microeconomic Theory Solutions to Problem Set 2.

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1 Delh School of Economcs Corse 00: Mcroeconomc Theor Soltons to Problem Set.. Propertes of % extend to and. (a) Assme x x x. Ths mples: x % x % x ) x % x. Now sppose x % x. Combned wth x % x and transtvt of %, we get x % x, whch contradcts the fact that x x. Hence t s not tre that x % x. Combnng, we get x x. (b) x x x ) x % x % x ) x % x. x x x ) x % x % x ) x % x. Combnng, we get x x.. x x x ) x % x % x ) x % x. (a) Sppose x % x. Combned wth x % x and transtvt of %, we get x % x, whch contradcts the fact that x x. Hence t s not tre that x % x. Combnng, we get x x.. The contnt axom s volated. Consder the followng seqence of bndles: (x n ; x n ) (a; 0) and ( n ; n ) (a n ; b), where a; b > 0. For the preferences descrbed, (xn ; x n ) ( n ; n ) for an n. However, (x n ; x n ) (a; 0) and ( n ; n ) (a; b). Then (a; b) (a; 0),.e., preference s reversed n the lmt, whch s a volaton of contnt. These preferences cannot be represented b a contnos tlt fncton. 4. Ths s the Stone-gear tlt fncton. (a) After power and log transformatons: + (b) Solton wll be nteror. From the condton MRS prce rato, we get x p x The other eqaton s that of the bdget lne: p x + x Solvng these two eqatons gves s the Marshallan demand fnctons: 5. I wll skp some of the easer detals. x + p p x + p (a) For Cobb-Doglas preferences, note that MU as x 0, hence the solton wll alwas be nteror for an strctl postve prce vector. Also, snce the tlt fncton s strctl qasconcave, SOC wll alwas be sats ed. The FOC, sng the standard tangenc condton (MRS prce rato) gves x j p j x p j

2 Ths can be rewrtten as p x k for all () where k s some constant. The vale of k can be solved b sng the above n the bdget constrant: nx nx k ) k snce Usng ths n (), we get the Marshallan demand fnctons: x (p; ) p Plggng back these demands n the tlt fncton and smplfng, we get the ndrect tlt fncton: v(p; ) p For the dal problem (expendtre mnmzaton), () mst stll be sats ed bt the vale of k wll be d erent. To nd ths vale, nsert () nto the constrant of the problem to get p k Upon replacng back n (), we get the Hcksan demand fnctons: p x h (p; ) Plggng back these expressons nto the expresson for spendng,.e., n P n p x h, we get the expendtre fncton e(p; ) p (b) The tlt fncton s of a pecewse lnear form: p (x ; x ) ax + x for x x x + ax for x > x The slopes above and below the 45-degree lne are a and a respectvel. If a >, the nd erence crves are concave to the orgn. In ths case, optmal soltons wll alwas be at a corner. Leavng the detals to o, I wll focs on the case of convex preferences (a < ). Here, the optmal choce s ether at a corner or at the knk, dependng on the slope of the bdget lne. Spec call, Marshallan demands are gven b (x ; x ) 0; p > a ; f a < p < p + p + a ; 0 < a p In the case where the prce rato s ether a or a, the bdget lne concdes wth one of the arms of the nd erence crve and an choce on that arm s optmal. The ndrect tlt fncton s v(p; ) a > a (a + ) f a < p < p + a a < a p

3 The Hcksan demand fnctons are The expendtre fnctons s (x h ; x h ) e(p; ) a 0; > a a a + ; f a < p < a + a a ; 0 > a > a (p + ) a + p a > a f a < p < a (c) The tlt fnctons are concave to the orgn, hence the pont of tangenc represents a mnmm rather than a mm. Obvosl there wll be a corner solton. Marshallan Demand fnctons are: (x ; x ) ; 0 < p 0; p > When p, ether corner s optmal. Indrect tlt fncton: v(p; ) > Hcksan demand fnctons: Expendtre fncton: x h ; x h p ; 0 < 0; p > e(p; ) p p f p p f p > (d) Ths s the case of perfect sbstttes whch elds corner soltons for almost all prce vectors (draw the pctre). Marshallan demands are: (x ; x ) 0; p > a b The ndrect tlt fncton s ; 0 p < a b v(p; ) b a b a p < a b

4 The Hcksan demands are: x h ; x h 0; b a ; 0 > a b < a b The expendtre fncton s e(p; ) b p a a b < a b (e) Frst, focs on cases where there s an nteror.solton. Applng the prncple M RS prce rato, we get + x p + x Combnng wth the eqaton of the bdget lne, we get the Marshallan demands p + (x ; x ) ; p + p p Stckng wth the nteror solton case, the other fnctons are eas to derve (o can explot the smmetr to smplf calclatons): x h ; x h v(p; ) + (p + ) p e(p; ) p p ( + ) s s ( + ) p ( + ) ; p Some caveat has to be added to the solton, however. Sometmes, there wll be corner soltons. Note from the Marshallan demand expressons above, whenever < p, we have x < 0. Ths s nadmssble snce negatve qanttes are not allowed. In ths case, the optmal choce wll nvolve x 0 and all the ncome spent on good,.e., x. The opposte corner solton arses whenever < p. A compact and accrate descrpton of the Marshallan demands wll then be p + x (p; ) 0; mn ; p p p + p x (p; ) 0; mn ; The other fnctons have to be smlarl adjsted to take accont of corner soltons. The detals are ommtted. 6. Frst, we arge that wth addtvel separable tlt, all goods mst be normal goods. Sppose x falls when ncome goes p. For an par (; j), the FOC s 0 (x ) 0 j (x j) p p j If x decreases, b concavt, x j mst also decrease to keep the MRS constant (remember prces haven t changed, onl ncome). Hence, f an one good s nferor, all goods are nferor. Bt ths volates the monotonct axom, so we have a contradcton. Hence none of the goods can be nferor. The argment s completed b notng onl nferor goods can be G en goods. 4

5 7. The agent solves (takng log transformaton of the tlt fncton to smplf calclatons): sbject to the bdget constrant L;F m + w(t ln L + ln F L) pf Sbstttng ths nto the objectve fncton, the problem redces to L and the FOC for nteror solton s ln L + ln [m + w(t L)] ln p L w m + w(t L ) The solton for optmal choce of lesre s m + wt L mn w ; T snce total lesre cannot exceed the tme endowment T. Therefor labor sppl (T wt m ; 0 w L ) s gven b For strctl postve labor sppl, we need w > m T The agent needs a hgh enogh wage and ns cent non-wage ncome to be wllng to work. Food demand s obtaned b replacng (T L ) n the bdget constrant: F p (wt + m); m 8. The agent s problem s (agan takng log transformaton): sbject to the bdget constrant x ;x ln x + ( ) ln x p x + x 5p + Usal tangenc condton (MRS prce rato) gves x ( )x p (a) Usng ths n the eaton for the bdget lne, we get the demands: x (5p + ) p x ( ) (5p + ) (b) The agent s a net ber of good f x > 5,.e., p < 5( ) 5

6 9. Ths exercse focses on the cash-vs-knd debate. The man pont s that the best means of delverng ad depends on whether the donor jdges the recpent s welfare b the recpent s preferences (welfarsm) or the donor s (paternalsm). (a) The tlt fncton can be transformed nto: whch gves rse to demand fnctons (x; ) x I x(p x ; p ; I) p x (p x ; p ; I) I p Indrect tlt s obtaned b sbstttng demands nto the tlt fncton: (b) From the demand fnctons: x,. I v(p x ; p ; I) (p x ) (p ) (c) Let the sbsd be s per school. Banana Repblc s demand fnctons become: I x(p x ; p ; I) (p x s) (p x ; p ; I) I p Here s 0. Usng ths, we can calclate: x 4,. The vale of the sbsd s xs 40. The tlt attaned s 6. (d) Calclate the expendtre fncton for these preferences: The FOC are Solvng, we get the Hcksan demand fnctons e(p x ; p ; ) mn (p x x + p ) sb to x : x p x p and x x(p x ; p ; ) (p x ; p ; ) p p x px p Usng these expressons gves s the expendtre fncton e(p x ; p ; ) + (p x ) (p ) : Insert the prces and target tlt ( 6 ) to calclate the mnmm expendtre. The d erence between ths amont and the ncome I 0 s the cash grant that wll be necessar. Show that ths s less than the 40 Bleedng Heart was spendng n sbsdes. (e) For an cash award c, the demand fncton for schools wold become x(p x ; p ; I) I + c p x To ncrease the demand for schools from to 4, we mst have c 0, whch s mch more than the 40 spent b the agenc nder the sbsd scheme. 6

7 0. Ths qeston tres to nderstand the e ect of tax rates and laws on chartable donatons. (a) It s eas to show that at the optmm: x ( t) x ( )( t) A tax cts ncreases chartable donatons becase t has a pre ncome e ect. (b) The problem can be wrtten as ln x + ( ) ln x sbject to x + x ( t) + tx The bdget constrant can be rewrtten as x t + x where t can be nterpreted as the prce of own consmpton n the sense of opportnt cost (the amont of chartable donatons forgone for ever rpee spent on own needs). Ths gves rse to optmal choces: x ( t) x ( ) (c) A tax ct makes own consmpton cheaper and hence the sbsttton e ect tends to ncrease sel sh spendng and redce donatons. However, the tax ct also leaves more dsposable ncome n the agent s pocket and so the ncome e ect tends to ncrease donatons. For Cobb-Doglas tlt, the two e ects exactl cancel each other ot. (d) Yes. It s the propert that nder Cobb-Doglas preferences, total spendng on each good s a constant. In general, ncome and sbsttton e ects ma not cancel ot exactl and the e ect of tax cts (when donatons are tax dedctble) wold be theoretcall ambgos. 7