f(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) =

Transcription

1 Problem Set 3: Unconstraned mzaton n R N. () Fnd all crtcal ponts of f(x,y) (x 4) +y and show whch are ma and whch are mnma. () Fnd all crtcal ponts of f(x,y) (y x ) x and show whch are ma and whch are mnma. () The gradent at any crtcal pont s f(x,y) (4(x 4)x,y) 0 Thus y must be zero. x can be zero or ±. Ths gves us three crtcal ponts. The Hessan s x H(x,y) Thecrtcal pont (0,0) s nether a mum nor a mnmum, snce the Hessan has egenvalues 0 and. At (±,0), the Hessan has egenvalues and 3, so these crtcal ponts are mnma. () The gradent at any crtcal pont s f(x,y) ( 4(y x )x x,(y x )) 0 The second equaton gves y x, and substtutng ths nto the frst equaton gves 4(x x ) x 0 so x 0, y 0 s a crtcal pont. The Hessan s 4(y x H(x,y) )+x 4x 4x Evaluatng at (0, 0) yelds H(x,y) 0 0 whch has egenvalues and, so (0,0) s nether a mum nor a mnmum. 3. Suppose a frm produces two goods, q and q, whose prces are p and p, respectvely. The costs of producton are C(q,q ). () Provde necessary and suffcent condtons for a producton plan (q,q ) to be proft-mzng and show how q vares wth p. If C(q,q ) c (q ) + c (q ) + bq q, explan when a crtcal pont s a local mum of the proft functon. () If C(q,q ) c (q )+c (q )+bq q, how does q vary wth b and p? How do profts vary wth b and p? () The frm s proft-mzaton problem s p q +p q C(q,q ) q,q The FONCs are p C (q,q ) 0 p C (q,q ) 0

2 The SOSCs are that C (q,q ) C (q,q ) C (q,q ) C (q,q ) s negatve defnte. Wth the functonal form C(q,q ) c (q ) + c (q ) + bq q, we get a Hessan c (q ) b b c (q ) whch must satsfy c (q ) < 0, c (q ) < 0, and c (q )c (q ) b > 0 to be negatve defnte. So b cannot be too large relatve to the second dervatves of c (q ) and c (q ), or the economes of scope make the crtcal pont characterzed above a global mnmum. () Profts are easy. The proft functon s The envelope theorem mples V(b,p ) p q +p q c (q ) c (q ) bq q (q,q )(q,q ) V b q q < 0 V p q > 0 To compute the comparatve statcs, we totally dfferentate the FONCs, p c (q ) bq 0 p c (q ) bq 0 and re-wrte the result as a matrx equaton, c (q ) b b c (q ) q / b q / b q q Cramer s rule then mples q q det b b q c (q ) deth q c (q ) bq deth The denomnator deth s postve because we are at a mum, and the numerator s postve f q c (q ) bq > 0. Ths s ambguous, so we should expect changes n the spllover effect to have uncertan changes on frm behavor. Smlarly, c (q ) b q / p b c (q ) q / p 0 And usng Cramer s rule, q p b 0 c (q ) deth c (q ) deth > 0 Ths s unambguously postve (actually, ths s just the law of supply, rght? We ll see ths always holds for prce-takng frms).

3 4. A consumer wth utlty functon u(q,q,m) (q γ )q α + m and budget constrant w p q +p q +m s tryng to mze utlty. () Solve for the optmal bundle (q,q,m ) and check the second-order suffcent condtons. () Show how q vares wth p, and how q vares wth p, both usng the closed-form solutons and the mplct functon theorem. How does the value functon vary wth γ? Brefly provde an economc nterpretaton for the parameter γ. () The mzaton problem s wth FONCs The crtcal pont then s and m w p q p q. The Hessan s H(q,q ) q,q (q γ )q α +w p q p q q α p 0 (q γ )αq α p 0 q p/α q p αp (α )/α +γ 0 αq α αq α (q γ )α(α )q α whch has leadng determnants 0 and ( αq α ) < 0. So ths s, ykes, not negatve defnte. We cannot conclude ths s a mum (actually, we wll be able to prove t later, snce the objectve functon s quas-concave ). () As for comparatve statcs, we are n somewhat awkward terrtory because we are not surethat the proposed crtcal pont s an optmum. We can stll try to use the mplct functon theorem, but we cannot assume that the Hessan has the alternatng sgn pattern assocated wth a negatve defnte matrx. Dfferentatng the closed form soluton yelds q p αp (α )/α To use the IFT, we totally dfferentate the FONCs to get αq α q p 0 q p αq α +(q γ )α(α )q α q p 0 Rewrtng ths as a matrx equaton yelds 0 αq α αq α (q γ )α(α )q α q / p q / p 0 Usng Cramer s rule to solve for q / p yelds 0 αq α q det (q γ )α(α )q α p 3

4 or q α αq p (αq α ) αq α Whch gves the same result as dfferentatng the closed form soluton. The optmzed value of the consumer s utlty s gven by and the Envelope Theorem mples V(γ ) (q γ )q α +w p q p q (q,q )(q,q ) V (γ ) q α < 0 so that an ncrease n γ makes the consumer worse off. The parameter γ s the mnmum amount of q that the agent must consume to get postve utlty from consumng any q or q. Notce that f the consumer can t afford enough γ, he should just put all of hs wealth nto m, whch gves a constant margnal utlty of. Let s thnk about q as the qualty of a prmary good lke a computer, electrc gutar, or camera, and q as secondary or complementary goods lke software, an amplfer, or lenses; anythng that you also requre to make q worth ownng. Then the consumer needs to purchase a prmary good of qualty at least γ, or the payoff s actually negatve, and the consumer regrets makng the purchase, no matter how good the secondary goods are that he also purchases. 5. There s a consumer wth utlty functon u(q,q,m) v(q,q )+m and budget constrant w p q + p q + m. Let v() 0. Frm produces q wth cost functon c q, and frm produces q wth cost functon c q. All frms and the consumer are prce-takers. () Characterze the utlty-mzng demands for the consumer (q d,qd,md ) and the proft-mzng supply for each frm, q s and qs by provdng frst-order necessary condtons and second-order suffcent condtons. When are these condtons satsfed? () Solve for a perfectly compettve equlbrum (the markets for each good clear, so that qk s qd k for k,) q, q, m, p, p, assumng that q,q > 0. Explan how q vares wth c. When are the goods complements, and when are they substtutes? () Suppose there s an excse tax on good, pad by the consumer, so that the effectve prce the consumer pays s p + t. How does a change n the tax affect consumpton of good and good n equlbrum? () The consumer s demand functons are characterzed by the system of FONC s v (q d,qd ) p 0 v (q d,q d ) p 0 and The SOSCs are that the matrx m d w p q d p q d v (q d,qd ) v (q d,qd ) v (q d,qd ) v (q d,qd ) be negatve defnte. Ths requres that v (q d,qd ),v (q d,qd ) < 0, and v (q d,qd )v (q d,qd ) v (q d,qd ) > 0. 4

5 The frms proft functons are lnear n ther quanttes, so that the soluton s π (p c )q 0 p < c q 0, ) p c p > c Snce π (q) s zero, we cannot verfy the SOSC s. () In equlbrum, t must be the case that p c, or otherwse the market wll fal to clear. Ths mples that equlbrum s characterzed by the system of equatons and v (q,q ) c 0 v (q,q ) c 0 m w c q c q If we totally dfferentate the frst two equatons wth respect to c, we get a system v (q,q ) v (q,q ) q / c 0 v (q,q ) v (q,q ) q / c To solve the system, we use Cramer s rule, to get 0 v (q q det,q ) v (q,q ) c v (q,q ) So the sgn s determned by v (q,q ), snce the denomnator s postve. Snce p c n equlbrum, ths also tells us whether the goods are (gross) complements or (gross) substtutes. If v s negatve, the whole term wll be postve, and the goods are (gross) substtutes. If v s postve, the whole term wll be negatve, and the goods are (gross) complements. () If there s a tax on good one, the equlbrum equatons become v (q,q ) c t 0 v (q,q ) c 0 Totally dfferentatng wth respect to t, we get a system v (q,q ) v (q,q ) q / t v (q,q ) v (q,q ) q / t Solvng the system wth Cramer s rule, v (q q det,q ) t 0 v (q,q ) q det t v (q,q ) v (q,q ) 0 0 v (q,q ) < 0 v (q,q ) 5

6 6. An agent s tryng to decde how to nvest. There are K rsky assets avalable, and a safe asset that gves a return R > 0. The return on each s normally dstrbuted wth mean µ k, varance σk, and prce p k. A portfolo s a vector π (π,π ) gvng the amount of each asset purchased. () Suppose the covarance between the goods s zero, so that the varance-covarance matrx σ Σ 0 0 σ s dagonal. If the agent mzes the mean-less-the-varance of the portfolo, π µ π Σπ +Rm subject to the constrant w m+ k p kπ k, what s the optmal portfolo? (If π k < 0, what does that mean economcally?). How does the agent s optmal portfolo vary n R? How does the agent s payoff vary n R? () Suppose there are non-trval covarances between the assets, so that σ Σ σ σ σ s not dagonal, but t s symmetrc. Are the frst-order necessary condtons suffcent for a crtcal pont to be a mum n ths problem? How does a change n the covarance between the returns σ affect the optmal portfolo? How does a change n the covarance σ affect the agent s payoff? () If K s arbtrary, derve frst-order necessary condtons and second-order suffcent for the optmal portfolo problem. Are the second-order suffcent condtons satsfed for any varancecovarance matrx Σ? () If the VCV matrx s dagonal, we get the objectve functon whch has FONC s and Hessan π π µ σ π +R(w µ σ π Rp 0 µ σ π Rp 0 σ H 0 0 σ whch s negatve defnte for sure. So the optmal portfolo s π Rp µ σ p π ) The agent s payoff satsfes whle the optmal portfolo satsfes V (R) m > 0 π R p /σ > 0 Bascally, a hgher nterest rate on the safe asset ncreases the value of the agent s wealth, and he s happy to take on more rsk. 6

7 () The objectve s π π µ +π µ π σ π π σ +R(w p π p π ) wth FONCs µ p π σ π σ 0 µ p π σ π σ 0 and SOSCs that the matrx σ H σ σ σ Σ be negatve defnte. Snce Σ s a VCV matrx, t s postve defnte, so that Σ s negatve defnte, and the FONCs are automatcally satsfed. The closed form solutons are π (µ p )σ (µ p )σ σ σ σ π (µ p )σ (µ p )σ σ σ σ The change n the agent s payoff wth respect to σ s V (σ ) π π < 0 so he s unambguously worse off. The change n π wth respect to σ s (µ p )det(σ)+σ (µ p )σ σ (µ p ) det(σ) whch appears to be ambguous. The problem s that σ enters the both the numerator and denomnator of π negatvely, so that an ncrease n σ shrnks the denomnator and numerator at the same tme, so t s hard to tell whch effect domnates as long as µ p > 0. If µ p < 0, then we could get unambguous comparatve statcs, but ths seems to be a werd stuaton (shouldn t the expected return on an asset be above the prce, to compensate the agent for holdng rsk?). () For arbtrary K the objectve s π π µ π Σ π +R(w π p) wth FONCs or µ Σ π Rp 0 π Σ (µ Rp) and the SOSC s are that Σ be negatve defnte. However, Σ s a VCV matrx, so t s postve (sem-)defnte, so Σ s negatve (sem-)defnte, so the SOSCs are automatcally satsfed. If any of the assets were exactly correlated, we could get a postve defnte VCV matrx by smply mergng those assets together, and sayng that f the optmal portfolo requres buyng π k unts of the merged asset, any lnear combnaton of those assets that adds up to π k wll do. 7