( q) (2q1 5 q2) 2q2

Transcription

1 John Riley Micro 6 October 08 Homework Answers Answer to ( q) 8q 3 q (q 5 q ) q (a) Necessary conditions for a critical oint ( q) 8 4(q 5 q) 0 q ( q) 3 0(q 5 q) q 0 q Solve two linear equations: Solution q (6,4) Check by substitution (b) The maimiing lines are as deicted below Profit-maimiing lines (c) Profit is increasing in the one indicated so q is not a local maimum (d) Consider solutions (i) with q 0 and (ii) with q 0

2 John Riley Micro 6 October 08 (i) (0, q ) 3q 5q q (ii) ( q,0) 8q 4q (e) ( q) 8q 3 q (q 5 q ) bq (i) b If you choose q (6, 4) Solver indicates a maimum This is not correct! Solver only looks locally in the and directions and both artial derivatives are ero If you start anywhere else the rogram will climb uhill to a local maimum The results are shown above so q (6,0) is the maimum (ii) b Increasing the cost of commodity can only make it even less rofitable so the solution is unchanged Remark: From the class discussion in a coule of days, you will be able to show that the rofit function is concave Thus if you find a q=(q,q) satisfying the necessary conditions, that vector is the maimier

3 John Riley Micro 6 October 08 Answer to : Make F () as big as ossible so make as small as ossible Then it is equivalent to F() make G () F() as big as ossible L ( ) L 0 with equality if 0 Note that the right hand side aroaches infinity as aroaches ero Therefore L 0 Hence and so / / / / / / Therefore / y an identical argument / / / Therefore / / / ( ) So / / / Hence / / / / / ( ) y an identical argument, / / / / / ( ) Therefore ( ) ( )( ) / / / / / / Therefore 3

4 John Riley Micro 6 October 08 q F( ) ( ) / / (*) Remark: We can use this result to ask how big a budget we need to roduce q above, q is feasible with a budget of satisfying units From the result q ( ) / / From (*) maimum outut is a strictly increasing function of the budget Thus with any smaller budget it is not ossible to roduce q Thus the minimum budget is ( ) q / / 3 Walrasian Equilibrium (a) The budget constraint is 0 If the rice vector is scaled u to constraint becomes the budget ( ) 0 Thus the constraint on is unaffected (b) Suly Suose that 4 ev likes commodity four times as much If she sells a unit that gives her 4 dollars so she can then urchase more than 4 units of commodity So she sulies all of her endowment (7 units) If 4 the argument is reversed ev will not suly any units of commodity A similar argument holds for Ale If he sulies all of his endowment of commodity (8 units) If he sulies no units Hence the market suly function is as deicted Note that if the rice, Ale is indifferent as to how many units to trade (c) Demand If the rice is above 4 neither consumer urchases any of commodity If the rice is between 4 and Ale sells her 8 units of commodity and uses the money to urchase commodity 4

5 John Riley Micro 6 October 08 If the rice is below both sells their endowment of commodity and use the money to urchases commodity (d) If the rice is between 4 and Ale sells her 8 units of commodity She uses this to urchase A 8 commodity so Then market demand is A 8 If the rice is below all 0 units of commodity are sold and the funds are sent on commodity A A 0 Therefore ( ) 0 and so demand is Market demand is therefore as deicted below Market demand Since one of the consumers is indifferent as to how much is traded when 4 or there is a range of ossible demand sat these rices (e) From the two figure the equilibrium rice is ( ( ) ) ( ( ) ) 0 (f) If the market for commodity clears so that ( ), then ( ( ) ) ( ( ) ) 0 So ( ) Remark: You could also answer this question by deicting the budget line and level sets of the two consumers 5

6 John Riley Micro 6 October 08 4 Equilibrium trades (a) If the rate at which utility rises with commodity is 6 and the rate at which utility rises with commodity is, then the consumer is willing to give u 6/ = 3 units of commodity for one more unit of commodity Generally, If the rate at which utility rises with commodity is MU and the rate at which utility rises with commodity is MU, then the consumer is willing to give u MU/MU units of commodity for one more unit of commodity This is the consumer s marginal willingness to substitute commodity for commodity MRS A ( ), MRS ( ) ( ) (b) These are both on the 45 line If higher MRS The argument is reversed if then Thus ev has a ( ) ( ) 0 (c) Suose as deicted below that MRS ( ) / as deicted below Then such a consumer will wish to urchase additional units of commodity by selling some of his endowment of commodity On the 45 line the two consumers have a MRS = Thus if the rice ratio / both will wish to buy more of commodity and sell some of commodity then neither will wish to trade so markets clear If (d) If the two consumers have the same endowment above the 45 line then ev has a greater MRS than Ale oth therefore gain if the echange rate lies between the two marginal rates of substitution ev sells commodity and Ale sells commodity If the ratios are equal and greater than then ev s MRS is higher Then raising Ale s ratio above that of ev s increases his MRS so it is not ossible to say who trades what 6

7 John Riley Micro 6 October 08 If the ratios are equal and less than then Ale s MRS is higher Raising Ale s ratio increases his MRS further so now Ale sells commodity and ev sells commodity 7