Question

The Taxpayer Assistance Center (TAC) provides taxpayer assistance to help in the preparation of individual tax returns. The amount of assistance (A) that can be provided is a function of the number of professional accountants (P) and trained tax preparers (T) that are employed. The function is given by: A = 500P +200T +50PT

(a) TAC has budgeted $700,000 for a new office to pay for the accountants and the tax preparers. It pays an accountant $100,000 and a tax preparer $50,000. Using the Lagrangean multiplier approach calculate the optimal staff combination that can maximize the amount of tax assistance. Determine the optimal amount of assistance provided by TAC. Explain your calculations.

(b) Using Excel-Solver verify your answer to (a). (Note: Provide the screenshots of the spreadsheet along with Solver windows as shown in the Solver supplement so that all Solver commands can be seen.)

Answer 1

(a)

The amount of assitence is given as

$A=500P+200T+50T$

The budget constraint of the firm is given as

$100P+50T=700$

$2P+T=14$

The firms objective is to

maximize $A=500P+200T+50T$

subject to $2P+T=14$

The required Lagrange function is given as

$z=\left (500P+200T+50PT \right )+\lambda (14-2P-T)$

The FOCs are

$\frac{\partial z}{\partial P}=500+50T-2\lambda P=0$ ...................(1)

$\frac{\partial z}{\partial T}=200+50P-\lambda T=0$...............(2)

$\frac{\partial z}{\partial \lambda }=14-2P-T$......................................(3)

From (1) and (2)

$\frac{10+T}{4+P}=2$

$\therefore 2P-T=2$.................(4)

Solving (3) and (4):

$P=4 ;\ T=6$

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(b)

At equilibrium

$T=2P-2$

$\therefore A=500P+200(2P-2)+50P\left ( 2P-2 \right )$

$\therefore A=500P+400P-400+100P^2-100P=100P^2+800P-400$

$\therefore \frac{A}{100}=P^2+8P-4=P^2+8P+16-20=(P+4)^2-20$

$\therefore \frac{A}{100}-20=(P+4)^2$

$\therefore \frac{\sqrt{A-2000}}{10}=P+4$

$\therefore P=\frac{\sqrt{A-2000}}{10}-4$

$\therefore T=\frac{\sqrt{A-2000}}{5}-10$

$\therefore C=100000P+50000T=100000\left (\frac{\sqrt{A-2000}}{10}-4 \right )+50000\left (\frac{\sqrt{A-2000}}{5}-10 \right )$

$\therefore MC=50000\left (\frac{1}{10\sqrt{A-2000}} \right )+25000\left (\frac{1}{5\sqrt{A-2000}} \right )$

$\therefore MC=5000\left (\frac{1}{\sqrt{A-2000}} \right )+5000\left (\frac{1}{\sqrt{A-2000}} \right )=\frac{10000}{\sqrt{A-2000}}$

This implies the firm has increasing marginal cost.

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(c)

A1 f Microsoft Excel 14.0 Answer Report 1 Microsoft Excel 14.0 Answer Report 2 Worksheet: [Book1]Sheet1 3 Report Created: 5/16/2016 1:45:39 PM 4 Result: Solver found a solution. All Constraints and optimality conditions are satisfied. 5 Solver Engine 9 Solver Options + 13 14 Objective Cell (Max) 15 16 SB$3 output 17 18 Cell Name Original Value Final Value 0 4400 19 Variable Cells 20 Cell Name Original Value Final Value Integer 21 SB$12 P 0 4 Contin 6 Contin 22 $B$13 T 23 24 25 Constraints 26 27 $B$11 Budget 28 $B$12 P 29 $B$13 T 30 31 32 0 Cell Name Cell Value Formula Status Slack 700000 $B$11<-$C$11 Binding 0 4 $B$12>=$C$12 6SB$13--$C$13 Not Binding Not Binding 6

B3 f100000 B12+50000*B13 Solver Parameters 3 outputk...i- 4400 Set Objective: 5 Decision O Max O Min Value Of By Changing Variable Cells: $8$12:$8$13 9 Constraints 10 11 Budget 12 P 13 Т 14 15 16 17 Subject to the Constraints: B$11 $C$11 $B$12 $C$12 $8$13 > $C$13 700000 700000 Add 4 Delete Reset All 19 Load/Save Make Unconstrained Variables Non-Negative 21 Select a Solving Method GRG Nonlinear Options 23 24 25 26 27 Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. 29 30 31