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

a) [5 pts] What is the equation that shows the relationship between the amount of labour

used and labours Average Physical Product (APP)?

b) [10 pts] Using the above information, how much labour (L) needs to be used to maximize

TPP? What is the level of production at this input level? (Hint: What is the characteristic

of the TPP function when the TPP is being evaluated at an output level that maximizes

the TPP value?)

Also, in mathematics, a quadratic equation has the general form: ??² + ?? + ? = 0

where the constants (numbers) a, b, and c are respectively known as the quadratic term,

linear term and constant. There are two values that result in the above equality (i.e. Z ). If

you want to find these two values, the following equation is used:

? = (−? ± √(?² − 4??)) / 2?

where the values of a, b, and c are obtained from the quadratic equation shown above.

Pick the value of Z that makes the most sense as your answer.

c) [10 pts] At what labour use is the APP of labour maximized? What is the level of

production obtained using this amount of labour? (Hint: What is the characteristic of the

APP when the APP is maximized? )

d) [10 pts] At what values of labour use do the stages of production begin and end? (Hint:

What characterizes the boundaries of each stage of production? )

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

TPP = 0.4? + 0.09?² − 0.0035?³

(a)

APP = TPP / L = 0.4 + 0.09L - 0.0035L²

(b)

TPP is maximized when dTPP/dL = MPP = 0.

MPP = 0.4 + 0.18L - 0.0105L² = 0

0.0105L² - 0.18L + 0.4 = 0

Solving this quadratic equation,

L = [0.18 {(-0.18)² - (4 x 0.01015 x 0.4)}] / (2 x 0.4)

L = [0.18 {0.0324 - 0.0168}] / 0.8

L = [0.18 {0.0156}] / 0.8

L = [0.18 {0.1249}] / 0.8

L = [(0.18 + 0.1249) / 0.8] OR L = [(0.18 - 0.1249) / 0.8]

L = (0.3049 / 0.8) OR L = (0.0551 / 0.8)

L = 0.38 OR L = 0.07

The higher the value of L, the more the output, therefore L = 0.38.

(c)

APP is maximized when dAPP/dL = 0.

0.09 - 0.007L = 0

0.007L = 0.09

L = 12.86

TPP = (0.4 x 12.86) + (0.09 x 12.86 x 12.86) - [0.0035 x (12.86)³] = 5.14 + 14.88 - 7.44 = 12.58

(d)

(i) Stage I ends when APP = MPP.

0.4 + 0.09L - 0.0035L² = 0.4 + 0.18L - 0.0105L²

0.007L² = 0.09L

L = 12.86 (assuming L 0).

(ii) Stage II ends when MPP = 0.

0.4 + 0.18L - 0.0105L² = 0

L = 0.38 [Using part (b)]

NOTE: As per Answering Policy, 1st 4 parts are answered.