Problem 1 of MIDTERM Page 268 #15 Group 6 Ritchie Roi Chua Ramon Jose Miguel Naguiat Mary Jane L. Paglumotan Mark Anthony Salazar Mark Anthony Salazar Charlotte Tang Tong Ya(Nicole)

Problem 1 (Page 268 #15)  Given are the pairs of demand and cost functions for different commodities on which the specified amount of the unit tax is to be imposed. At the level of maximum profit to the monopolist, determine: A.The volume of sales B.The unit prices with tax and without tax C.The amount of tax revenue D.The amount of maximum profit obtainable p=(15-X) 2,0 < X < 15, AC=X 2 – 10X + 65, t u = 10

Solution A.The volume of sales that will maximize profit [Without Tax] p o =(15-X o ) 2 & AC o =X o – 10X o + 65 TR o =pX o =(15-X o ) 2 X o =(15-X o ) 2 X o =(225-30X o +X o ) X o =(225-30X o +X o ) X o TR o =225X o -30X o +X o

Solution AC o =TC/X o TC o = AC o (X o ) =(X o -10X o + 65) (X o ) =(X o -10X o + 65) (X o ) TC o = X o -10X o + 65X o 22 3 O is the total revenue function (w/o tax) & TC O is the total cost function (w/o tax). Therefore, the profit function, P O will be: P O =TR O -TC O Where TR O is the total revenue function (w/o tax) & TC O is the total cost function (w/o tax). Therefore, the profit function, P O will be: P O =TR O -TC O =(225X O -30X O +X O )-(X O -10X O +65X O ) =(225X O -30X O +10X O -65X O ) P O =-20X O +160X O

Solution To get the critical points: P o = -20X o + 160X o P o =-40X o P o =0 0=-40X o X o =4 To check whether X o =4 is a max or min point: P o = - 40 Therefore P o is a maximum at X o =4, which means that the sales volume that will maximize profit is 4 units if no tax is imposed. 2, “

Solution p u =(15-X u ) 2 t u =10 TR u =225X u - 30X u + X u TC u =X u – 10X u + 65X u + 10X u TC u =X u – 10X u + 75X u WITH TAX IMPOSED:

Solution P u =TR u -TC u =(225X u -30X u +10X u -75X u ) P u =-20X u +150X u To get the critical points P U =-20X U + 150X U P U =-40X U P U =0 0 = -40X U X U =3.75 To check whether X U =3.75 is a max or min point: P U = - 40 Therefore P U is a maximum at X U =3.75, which means that the sales volume that will maximize profit is 3.75 units if tax is Imposed ,,,

Conclusion (A): Without tax, the volume of sales should be 4 units to obtain a maximum profit. With tax, the volume of sales should be 3.75 units to obtain a maximum profit. Solution

B. UNIT PRICE Without Tax With Tax p o =(15-x) 2 p U =(15-x) 2 =(15-4) 2 =( ) 2 =(15-4) 2 =( ) 2 p o =121 p U = Conclusion: The unit price with tax is Php while the unit price without tax is Php Solution

C. AMOUNT OF TAX REVENUE: Let T U be the amount of tax revenue. T U =t U X U =10(3.75)=37.50 Conclusion: Amount of tax revenue is Php37.50 Php37.50 Solution

D. MAX PROFIT OBTAINABLE With Tax: P U = -20X U +150X U = -20(3.75) (3.75) = WithoutTax: P o = -20X o +150X o = -20(4) (4) = 320 Conclusion: The maximum profit obtainable is Php281.25, if tax was imposed. While max profit will be Php if no tax was imposed Solution 2 2

Graph P=(15-X) 2 TR= 225X-30X 2 +X 3 TC u = X u 3 -10X u 2 +75X u TC o = X o 3 -10X o 2 +65X o AC o = X o 2 -10X o +65X o AC u = X u 2 -10X u +75X u *All terms with subscript U are WITH TAX, while those with subscript O are WITHOU TAX

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