Exercise 1: Robinson Crusoe who is marooned on an island in the South Pacific. He can grow bananas and coconuts. If he uses

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1 Jon Riley F Maimization wit a single constraint F5 Eercises Eercise : Roinson Crusoe wo is marooned on an isl in te Sout Pacific He can grow ananas coconuts If e uses z acres to produce ananas z acres of l to produce coconuts, te outputs of ananas coconuts are A z / A z / Tere are ˆ acres of usale l Roinson Crusoe s preferences are represented y te utility function U f ( ) (a) Sustitute for in te utility function so otain a stard one constraint maimization prolem for te maimization of Roinson Crusoe s utility () If ˆ 8 A (4,), solve for te optimal inputs outputs Eercise : Equilirium on te isl Continue wit te data for Eercise wit ˆ 8 A (4,) Let te output price vector e p ( p, p) let te input price e r Since scaling up all tree prices y a common factor as no effect on production or consumption coices, we are free to coose one of te tree prices Let te input price e r Roinson is te owner of ot firms He acts as a price taker cooses inputs outputs to maimize te profit of ot firms (a) Solve for te input dems z ( p ) outputs y ( p ) Hence sow tat for te input market to clear 6 p p () Sow tat te profit-maimizing supply of output is y( p) 8 p

2 9 August 07 (c) Sow tat te total value of te output produced y te two firms is W p y ( p) p y ( p) 8p p Crusoe, now wit is consumption at on, must coose a consumption undle to maimize is utility His udget constraint is terefore p p W (d) As consumer e also acts as a price taker Sow tat e will spend alf of is wealt on commodity Hence sow tat is dem for commodity is ( p) (8 p p ) p (e) Sow tat for te market for output to clear, 6p p (f) Solve for te equilirium output prices ence te equilirium allocation (g) Compare it wit your answer to te previous eercise Remark: It may seem silly to tink of Roinson Crusoe as a price taker since e surely knows tat e is selling commodities to imself But if we were to assume tat tere was a Roinson Crusoe on eac of N isls eac produced te two commodities, te idea of trading wit no individual trade aving an effect on prices no longer seems unreasonale if N is large enoug If in fact te tecnology on eac isl is te same, te equilirium is te same as in te one isl case Eercise : Market equilirium wit Roinson Crusoe Friday Roinson s plantation produces ananas Te production function is q (6 z ) Roinson / as 8 endowment of units of te input (l) Friday s plantation produces coconuts Te production function is q ( z ) / Friday also as 8 units of l Te two individuals ave te same utility function

3 9 August 07, {, } F F u R F Friday s is u ( )( ) (a) If te output price vector is p individual as wealt W sow tat is dem for commodity is W p () Hence eplain wy, for tis utility function, market dem is te same as te dem of a R F single agent wit te same preferences wealt W W W Tus te equilirium prices outputs in tis two agent economy are te same as in an economy in wic tere is one representative agent (c) Solve for te market clearing prices in te representative agent economy (d) In te agent two plantation economy ow many anana coconuts are traded? (e) Suppose Friday only as 4 units of l wile Roinson as How muc l will Friday rent from Roinson in order to produce more coconuts? Eercise 4: Division of a firm wit a udget constraint A firm can produce output r Te division manager is given q f () z z z using te input vector witout violating te udget constraint r z r z z Te input price vector is dollars instructed to maimize output, q f () z Solution: Te sadow profit is L ( z, ) z z ( ) Since revenue is zero if z 0, te profit-maimizing inputs must e strictly positive FOC L ( z, ) z r 0 z L ( z, ) zz r 0 z

4 9 August 07 z zz Ten r r so q Terefore Sustituting for z in te udget constraint, Terefore 4 4 Te maimized output for any udget is terefore q z z 4 ( 4r)( 4r) Tis is depicted elow Figure -: Maimum output wit a fied udget Eercise 5: Te cost function of a firm A firm can produce output q f () z z z using te input vector z Te input price vector is r Te division manager is instructed to produce ˆq units of output at te minimum cost Solve for te firm s cost function, C( q, r ) 4

5 9 August 07 Eercise 6: Utility maimization U( ) ( ) ( ) (a) Solve for te dem functions, ( p, I), if (, ) 0 () Solve for te dem functions if 0 Hint: Define y write down te udget constraint in terms of y rater ten ten appeal to (a) (c) Solve for te dem functions if 0 Eplain wy ( p, I) 0 if income is sufficiently ig (d) Wat condition must e satisfied for a strictly positive consumption undle (e) Wat condition implies tat ( p, I ) 0? Eercise 7: CES utility function U( ) ( ) ( ), 0 (a) Solve for te dem functions, ( p, I ), if (, ) 0 () Solve for te dem functions if 0 (c) Solve for te dem functions if 0 income is ig (d) Wat condition must e satisfied for a strictly positive consumption undle (e) Wat condition implies tat ( p, I ) 0? 5

ECON 200 EXERCISES (1,1) (d) Use your answer to show that (b) is not the equilibrium price vector if. that must be satisfied?

ECON 200 EXERCISES (1,1) (d) Use your answer to show that (b) is not the equilibrium price vector if. that must be satisfied? ECON 00 EXERCISES 4 EXCHNGE ECONOMY 4 Equilibrium in an ecange economy Tere are two consumers and wit te same utility function U ( ) ln H {, } Te aggregate endowment is tat prices sum to Tat is ( p, p)