March 07 Mathematical Foundations John Riley Module Marginal analysis and single variable calculus 6 Eercises Eercise : Alternative definitions of a concave function (a) For and that 0, and conve combination t (), elain why the second definition imlies f f t f t t f t 0 0 ( ) ( ( )) ( ( )) ( )) ( ( )) Multilying both sides by t, it follows that 0 0 ( t) f ( ) ( t) f ( ( t)) ( t) f ( ( t)) ( t) ( t)) f ( ( t)) (b) Use a very similar argument to show that tf tf t t f t t t f t ( ) ( ( )) ( ( )) ( )) ( ( )) Add the two inequalities and then use the fact that ( t) ( t) t 0 to show that the second definition imlies the first Eercise : Equivalence of the first two definitions for a differentiable function (a) Define y( t) f ( ( t)) where t () is a conve combination of 0 and Use the first definition of a concave function to show that y( t) y(0) f t 0 ( ) f ( ) This holds for all small t 0 Taking the limit it follows that 0 (0) ( ) ( ) y f f (b) Use the Chain Rule to show that 0 0 y (0) f ( )( ) (c) Hence show that the first definition imlies the second Given this result and the result in the revious eercise, the two definitions are equivalent
Eercise 3: Alternative definitions of differentiable concave functions Prove that the second definition of a concave function imlies the third Eercise 4: Maimizing a concave function (a) Show that the derivative of following function is continuous ( ), f ( ), ( ), (b) Deict the grah of the function (c) What values of are maimizers? Eercise 5: The sum of n strictly concave functions is strictly concave (a) Above we showed that if the two functions f ( ) and f ( ) are strictly concave then their sum g( ) f( ) f( ) is strictly concave (b) Elain why g( ) f3( ) is strictly concave if f ( ) 3 is strictly concave (c) Use this result to elain why the sum of n strictly concave functions is strictly concave Eercise 6: Profit-maimizing firm A monooly estimates that the market demand function is q( ) 6 q The cost of roduction is C( q) F 6q q 3 (a) What is the demand rice function? (b) Show that the revenue function is concave and the cost function is conve (c) Hence elain why the rofit function is concave
(d) Assuming that the firm is going to roduce, obtain an eression for total revenue Rq ( ) and hence the rofit function ( q) (e) Solve for the outut that maimizes ( q) You should elain why it is a maimum (e) What is the rofit-maimizing outut if (i) F 0 (ii) F 60 Eercise 7: Inut choice Using z units of an inut, the maimum outut of the firm is q f z z / ( ) 0 (a) If the rice of the inut is z and the rice of the outut is write down an eression for total rofit as a function of the number of units of the inut urchased (b) If (, ) (60, ) solve for the rofit maimizing inut and hence outut (c) Reeat this for any air of rices and show that maimized outut is q ( ) 5 Eercise 8: Price taking firm A firm sells roduces such a small fraction of the industry outut that its outut decision has a negligible effect on the outut rice The cost of roduction is C( q) q q q q 4 3 4 (a) Write down an eression for total and marginal rofit if 8 (b) What condition must be satisfied for q to be a critical oint? (c) Show that this condition holds if q and q (d) Show that the condition can be written as follows: ( q)( q)(3 q) 0 (e) Which of the critical oints is a local maimum and which is a local minimum?
(f) Which is the rofit maimizing outut? (g) Deict the rofit function in a neat figure (h) In a searate figure deict the cost function and the revenue function (i) Use one of these figures to elain why there is an interval of oututs that will not be rofit maimizers at any outut rice Eercise 9: Discriminating monooly A monooly sells in the domestic market where the demand function is q 30 The cost of roduction is C( q) 0q (a) Solve for the demand rice function (the inverse maings from quantity to rice) (b) Solve for the rofit-maimizing outut and hence show that the rofit maimizing rice is 40 Euroe eliminates imort duties As a result the firm can sell in Euroe where the demand function is q 33 We ignore transortation costs so that the cost of roducing q units for sale in the USA and q units for sale in Euroe is C( q q) 0( q q) (c) Obtain an eression for the Euroean demand rice function (d) Write down an eression for the total rofit ( q, q) Elain why the rofit maimization roblem can be treated as two indeendent roblems, each with one variable (e) Solve for the rofit-maimizing oututs and hence show that the rofit-maimizing rices are 40 and 65 Eercise 0: Common rice This is a continuation of the revious eercise Manufacturers of cometing roducts comlain that the US is selling its eorts at a lower rice (This is called duming ) In resonse the 3
Euroean Trade Commission rules that the firm can only sell in Euroe if the rice same as in the USA is the (a) Solve for the total demand in the USA and Euroe, q( ) q( ) q( ) and hence show that if the rice in both the USA and Euroe is, then the rice function for total demand is 4 q 3 (b) Solve for the quantity that equates marginal revenue and cost and show that the imlied rice is * 3 (c) What is the new rofit on sales in the USA and in Euroe? (d) Should the firm sell in Euroe? Eercise : Robinson Crusoe Robinson Crusoe is stranded on an island If he does nothing he has a daily suly of coconuts c0 0 If we works hours er day, he can increase the suly to c c0 Robinson does not like work If he works hours his utility is u( c, ) c(8 ) (a) Solve for the number of hours that maimizes his utility (i) if c0 4 (ii) c0 8 (b) For what values of c 0 will Robinson choose not to work? (c) Are there any values of c 0 for which Robinson will work all 8 hours Eercise 3: Necessary conditions for a maimum Suose that f( ) is twice differentiable and has a maimum at (a) Elain why the second derivative cannot be strictly ositive (b) Show that the function f ( ) ( ) 4 is concave (c) Show that f ( ) ( ) 4 has unique maimizer * that that the second derivative is not strictly negative at * 4
Eercise A4: Profit maimization A firm roducing outut has a rofit of ( ) 9 (a) Show that this function has critical oints maimum? (b) What is the rofit-maimizing outut and Which of these is a local 5