Econ 401A Draft 5 John & Ksenia. Homework 4 Answers. 1. WE in an economy with constant rturns to scale and identical homothetic preferences.

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1 Econ 0A Drft 5 John & Kseni Homework Answers WE in n economy with constnt rturns to scle nd identicl homothetic preferences () U( q, q ) ln q ln q ln( z ) ( z ) ln( z ) ( z ) / / / / Choose ie z, z ln z ln z ln z ln z to solve Mx{ U( q, q) z z, z z 6} Mx{ ln z ln z ln z ln z z z, z z 6} L ln z ln z ln z ln z ( z z ) (6 z z ) FOC (fter some work) z z z z, z z z z 6 Therefore 6 6 ( z, z ) (, ), ( z, z) (, ) () A necessry condition for profit-mximiztion is cost minimiztion z solves Min{ r z F( z) q} z After some work the FOC implies tht F r z z MTRS ( z ) 6 r F z z (c) Solve for the mximizing outputs ( q, q ) nd show tht their rtio is / / / / / 6 / / 6 q ( z) ( z) ( ) ( ) () 6 / / / 6 / / q ( z) ( z) ( ) ( ) Therefore q q

2 Econ 0A Drft 5 John & Kseni (d) Mximiztion y the representtive consumer FOC U p q q MRS( q) p U q q (*) (e) Solve for the WE input prices For profit mximiztion F z p ( z ) r 0 nd F z p ( z ) r 0 Since p these determine the input prices Suppose tht commodity is produced in the north of the country nd those who live in the north own the inputs used in its production Commodity is produced in the south of the country nd those who live in the south own the inputs used in the south (f) In CRS economy the equilirium profit is zero so p q r z nd p q r z Appekling to (*) in (d) pq pq Therefore r z r z Since It follows tht Bonus r z r z r ( z z ) r r z r z r (g) The representtive consumer spend hve of his welth on ech commodity Therefore in ech region hlf the welth is spent on ech commodity Welth in ech region is r z r z r Then imports in ech region re worth r S N (h) On the supply side, if inputs in ech region re tripled to (,8) nd (,6) then ecuse of CRS, ll the necessry conditions for profit mximiztion continue to hold if ll prices re unchnged nd output in oth regions triples On the demnd side, the demnd of the representtive consumer triples since totl welth triples So mrkets continue to cler (i) If the two regions ecome independent countries the economies re the sme s in prt (h) so the WE of (h) remins WE

3 Econ 0A Drft 5 John & Kseni Pricing gme () The profit of firm is ( p) ( p ) q ( p) ( p )(60 0 p 0 p ) The profit of firm is ( p) ( p 7) q ( p) ( p 7)(50 0 p 0 p ) For ny p p, the est response y firm is p tht solves Mx{ ( p) ( p )(60 0 p 0 p )} The prtil derivtive of the profit function cn e otined y multiplying out ll the terms nd d=then tking the derivtive or y pplying the product rule p (60 0 p 0 p ) 0( p ) 0 0 p 0p For mximum this must e zero Then p ( ) p p (50 0 p 0 p ) 0( p 7) 90 0 p 0 p For mximum this must e zero Then p (9 ) p () If oth prices re mutul est responses then p (9 ) p nd p ( ) p Therefore 9 0 p (9 ( p )) p p Hence 6 5p 0 nd so p p p nd so p

4 Econ 0A Drft 5 John & Kseni The two est response functions re depicted elow (c) The initil price for p is depicted elow long with the est response y firm (d) If the figure is lown up it is cler tht the Cournot djustment process converges to the equilirium point At ech step the responding price rises (e) From the slide, in the quntity setting gme the equilirium is ( q, q) (0,0) Sustituting into the demnd price functions, equilirium prices re ( p, p) (6,9) These prices re oth (slightly) higher Seled high-id uction with more thn uyers () The id function of uyer or uyer is B( ) k so the mximum id is kmx 0k This is id with proility zero tht uyer wins with proility if he mkes this id Thus he hs no incentive to id higher nd pty more () If uyer ids the proility tht she wins is the proility tht oth of the other uyers id lower Buyer ids lower if B( ), ie k ie The proility of k this is F( ) k 0 k i i

5 Econ 0A Drft 5 John & Kseni Similrly uyer ids lower with proility F( ) k 0 k Therefore W ( ) ( ) 0k The joint proility is the win proility Buyer s expected pyoff is therefore u(, ) W ( )( ) ( ) 00k (c) u ( ) 00k If this is positive t 0k uyer id 0k If not, uyer chooses so tht u 00k ( ) 0 Then B ( ) Therefore the est response is B ( ) Min{,0 k} (d) If k / then B ( ) Tht is, if the other two uyers use the idding strtegy B ( ) then uyer s est i i i reponse is to use this sme strtegy The sme cn e sid for uyer nd uyer so the idding strtgies re mutul est respones/ (e) Suppose ll the other uyers id ccording to the strtegy B ( ) k Arguing s ove, If i i i uyer ids the proility tht uyer s id is lower is F( ) The sme cn e k 0 k sid for uyers nd Thus the win proility is W ( ) ( ) 0k Arguing s in (d) the equilirium idding stregy is B ( ) i i i All py uction (formerly clled the sd loser uction) () Suppose tht uyer s idding strtegy is proility is B ( ) k If uyer ids her win 5

6 Econ 0A Drft 5 John & Kseni / / Pr{ B ( ) } Pr{ k } Pr{ } Pr{ ( ) } F(( ) ) k k k Thus the win proility is / W ( ) / 00k Buyer s expected pyoff is therefore / u(, ) W ( ) / 00k / u ) 0 / 00k ie Thus uyer s est reponse is B( ) k00 k00 / For symmetric equilirium B ( ) B ( ) k00 k nd must e the sme Therefore k nd so k nd so k(00) 00 Repeting the rgument for uyer, if uyer ids ccording to respsonse is B ( ) 00 So these strtegies re mutul est reponses A very similr rgument shows tht for some idders ˆk, i i i B ( ) k 00 00, then uyer s est B( ) kˆ is the equilirium strtegy with three 6