Problem Set1. A firm's production function is º=50L-0.01Ľ' , where Ldenotes the size of the workforce. Find the value of MPin the case when:(a) L=1, (b) L=10, (c) L=100, (d) L=1000Does the law of diminishing marginal productivity applyto this particular function?2. Show that the price elasticity of demand is constant fordemand functions of the formAP =Q"where A and n are positive constants.3. The demand and total cost functions of a good arerespectively 4P+Q-16=0 andТС %3D 4 + 20 —1020a) Find expressions for TR, (profit) 1 , MR, and MC interms of Q.b) Solve the equationdn= 0ÕPand hence determine the value of Q which maximizesprofit.c) Verify that, at the point of maximum profit, MR=MC.4. The cost of building an office complex, x floors high, in aprime location in Accra is made up of three components:(a) GH¢10 million for the land(b) GH¢'/, million per floor(e) Specialized costs of GH¢10000× per floor.How many floors should the office complex contain ifthe average cost per floor is to be minimized?5. The supply and demand equations of a good arerespectively given by 3P-Q. = 3 and 2P+Q, =14The government decides to impose a tax, t, per unit. Findthe value of t (in Ghana cedis) which maximizes thegovernments total tax revenue on the assumption thatequilibrium conditions prevail in the market.6. A firm's demand function for a certain good is given byP = 100e-0.10. Its total cost function is TC = 100e-010 + 50 . Whatoutput level maximizes the firm's profit? 7. An individual's utility function is given byU = 1000x, + 450x, + 5x,x, – 2x – xwhere is the amount of leisure measured in hours perweek and is income earned measured in cedis per week.Determine the value of the marginal utilities, when *=138 and = 500. Hence estimate the change in utility if!!the individual works for an extra hour, which increasesearned income by GH¢15 per week. Does the law ofdiminishing utility hold for this function?8. A firm's total cost function is given byTC = 3Q° +2Q,Q, +7QWhere 9 and l: denote the number of items of goods 1and 2, respectively that are produced. Using thesubstitution method, find the values of and 2: whichminimize costs if the firm is committed to producing 40goods of either type in total.9. A monopolistic producer of two goods, 1 and 2, has ajoint total cost functionTC = 10Q + QQ, +10Q,where 9 and 2: denote the quantity of items of goods 1and 2, respectively that are produced. If P, and P, denotethe corresponding prices then the demand equations areP = 50 -Q + Q,P, = 30 + 20, - Q,Using the Lagrange multiplier approach, find themaximum profit if the firm is contracted to produce atotal of 15 goods of either type. Estimate the new optimalprofit if the production quota rises by 1 unit.

Question

Problem Set1. A firm's production function is º=50L-0.01Ľ' , where Ldenotes the size of the workforce. Find the value of MPin the case when:(a) L=1, (b) L=10, (c) L=100, (d) L=1000Does the law of diminishing marginal productivity applyto this particular function?2. Show that the price elasticity of demand is constant fordemand functions of the formAP =Q&#34;where A and n are positive constants.3. The demand and total cost functions of a good arerespectively 4P+Q-16=0 andТС %3D 4 + 20 —1020a) Find expressions for TR, (profit) 1 , MR, and MC interms of Q.b) Solve the equationdn= 0ÕPand hence determine the value of Q which maximizesprofit.c) Verify that, at the point of maximum profit, MR=MC.4. The cost of building an office complex, x floors high, in aprime location in Accra is made up of three components:(a) GH¢10 million for the land(b) GH¢'/, million per floor(e) Specialized costs of GH¢10000× per floor.How many floors should the office complex contain ifthe average cost per floor is to be minimized?5. The supply and demand equations of a good arerespectively given by 3P-Q. = 3 and 2P+Q, =14The government decides to impose a tax, t, per unit. Findthe value of t (in Ghana cedis) which maximizes thegovernments total tax revenue on the assumption thatequilibrium conditions prevail in the market.6. A firm's demand function for a certain good is given byP = 100e-0.10. Its total cost function is TC = 100e-010 + 50 . Whatoutput level maximizes the firm's profit? 7. An individual's utility function is given byU = 1000x, + 450x, + 5x,x, – 2x – xwhere is the amount of leisure measured in hours perweek and is income earned measured in cedis per week.Determine the value of the marginal utilities, when *=138 and = 500. Hence estimate the change in utility if!!the individual works for an extra hour, which increasesearned income by GH¢15 per week. Does the law ofdiminishing utility hold for this function?8. A firm's total cost function is given byTC = 3Q° +2Q,Q, +7QWhere 9 and l: denote the number of items of goods 1and 2, respectively that are produced. Using thesubstitution method, find the values of and 2: whichminimize costs if the firm is committed to producing 40goods of either type in total.9. A monopolistic producer of two goods, 1 and 2, has ajoint total cost functionTC = 10Q + QQ, +10Q,where 9 and 2: denote the quantity of items of goods 1and 2, respectively that are produced. If P, and P, denotethe corresponding prices then the demand equations areP = 50 -Q + Q,P, = 30 + 20, - Q,Using the Lagrange multiplier approach, find themaximum profit if the firm is contracted to produce atotal of 15 goods of either type. Estimate the new optimalprofit if the production quota rises by 1 unit.

Accepted Answer

Note: Since more than one individual questions are asked, we are going to answer the first question…