Question

1. [50 pts] Suppose we have the following production function generated from the use of only

one variable input, labour (L):

??? = 0.4? + 0.09?² − 0.0035?³

Where TPP represents the total physical product and L is measured in 1000 hour increments (ie

1.5=1500 hours). The Marginal Physical Product curve is represented by:

??? = 0.4 + 0.18? − 0.0105?²

e) [5 pts] Using Excel, or some other spreadsheet program, graph the production function.

On a separate graph, plot both the APP and MPP functions. In both graphs use the range

of input use between 3000 and 35000 hours of labour using 500 hour increments. (Hint:

Don't forget that labour use is measured in 1000 hour increments. Use a formula and the

copy command to generate the 64 input levels used to generate these graphs within

EXCEL.) Make sure to clearly label your graphs and axis.

f) [10 pts] Assume that the price of your output is $10/unit and the price of labour is $11:00.

What is the profit maximizing amount of labour to use? Would the firm want to produce

given these prices? (Hint: What is the equilibrium condition necessary to maximize

profits from the input perspective?)

BONUS [10 pts] At what level of labour use is the MPP of labour maximized? What is the level

of production at this point? (Hint: What is the characteristic of the MPP when the MPP is at its

maximum? )

Answer 1

e - The following image shows the production function

Production Function 20 15 10 3 5.5 810.5 13 15.5 18 20.5 23 25.5 283 -5 10 -15 20 25 30

The following image shows APP and MPP

APP and MPP 3 5.5 810.5 13 15.5 18 20.5 23 25.5 28 30.5 33 -1 -2 -3 -4 -5 -6 -7 MPPAPP

f - Px = 10, w = 11

At equilibrium, MPP = w/Px which implies,

11 = 0.4 + 0.18L - 0.0105L^2

Which gives L = 6, 11.1

At L = 6, Profit is given by

Profit = 10x - 66

In order to maximise profit, firm would produce positive quantity of x only if x > or equal to 6.6

Hence, Profit = 0 if q < or equal to 6.6, infinity if q > 6.6

At L = 11.1, Profit is given by

Profit = 10x - 122.1

In order to maximise profit, firm would produce positive quantity of x only if x > or equal to 12.21

Hence, Profit = 0 if q < or equal to 12.21, infinity if q > 12.21

Bonus - Maximizing MPP with respect to L.

MPP = 0.4 + 0.18L - 0.0105L^2

Differentiating with respect to L and setting LHS = 0

0 = 0.18 - 0.0210L which gives L = 8.57

Production at L = 8.57 is 7.83