# MGEA02H3 Chapter Notes - Chapter 5-10: Demand Shock, Hula Hoop, Fixed Cost

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Published on 24 Feb 2015

School

Department

Course

Professor

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Week 5 (Sept 29, Oct 1)

Topics: Production and Productivity. How all production processes are characterized by

diminishing marginal returns to additional use of some variable input, when one input

is fixed in amount. This gives us diminishing marginal product (and diminishing

average product), and explains the costs of production in the short run.

Recommended Reading: part of Chapter 11 “Behind the Supply Curve: Inputs and

Costs” (pp. 343-353).

Study Objectives

1. To appreciate that firms are assumed to try to maximize profits.

2. To understand the concept of a production function representing best-practice

technology. To understand the concepts of total product, average product and

marginal product and their relation to each other.

3. To understand the implications of the Law of Diminishing Marginal Product

(Law of Variable Proportions) for the shapes of each curve.

4. To understand the definitions of short run and long run as they apply to

production theory (short run and long run are defined differently in

macroeconomics, for instance).

Study Questions

1. You are given the following short run production function for the firm (L =

number of workers per day; Q = output per day):

0.01Q = L - 0.005L2

The firm is a perfect competitor and the price of its output is $100.00 per unit. If

labour costs $100.00 per day, the number of workers per day hired by the firm

will be 99 workers. What will the marginal cost of additional output be when this

number of workers is hired?

(A) $6 (B) $10 (C) $12 (D) $15 (E) $20 (F) $24

(G) $26 (H) $30 (I) $36 (J) $40 (K) $44 (L) $48

(M) $50 (N) $55 (O) $60 (P) $64 (Q) $66 (R) $70

(S) $75 (T) $80 (U) $84 (V) $90 (W) $96 (X) $100

(Y) none of the above

2. The cost department of a firm has found that the firm employs 90 units of capital

(fixed in the short run) and requires the following amount of labour to produce

levels of output greater than 30: L = 162 - 1.2Q + .02Q2 Q > 30

The price of capital is $7 per unit and the price of labour is $5 per unit

a) Derive an expression for the marginal product of labour. Does this short run

production function exhibit diminishing marginal productivity?

b) Derive an expression for the average product of labour. Determine the level of

output and labour at which the average product of labour is maximized. What is

the relationship between the average product of labour and the marginal product

of labour at that point?

c) Derive an expression for fixed cost, variable cost, and total cost as a function of

the amount of output produced. Derive an expression for average cost, average

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variable cost, and marginal cost. Find the minimum points for AVC and AC.

Demonstrate that marginal cost is equal to both AC and AVC at their minimum

points, respectively.

d) What is the relationship between the maximum average product of labour and

the minimums in part (c)? Can you show that this is no accident?

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ECONOMICS A02H

ANSWERS TO STUDY QUESTIONS

WEEK 5

1. The production function is 0.01Q = L - 0.005L2 or Q = 100L - 0.5L2. In this question,

MPL = dQ/dL = 100 – L. The marginal cost of output is defined as dTC/dq or the

rate of change of costs as output increases. Loosely speaking, it is the cost of one

more unit of output. The marginal product of labour at the optimum is 100 - L,

and since L = 99, the marginal product of labour = 1 (one more unit of output).

The cost of one more worker is $100, so at the optimum the cost of one more unit

of output is $100. The correct answer is (X). More formally, the marginal cost of

output is given as MC = PL /MPL = 100/(100 - L) = 100/1 = $100.

2a) This question involves a bit of a “trick”. The marginal product of labour is given

by dQ/dL. But the way the function is written, it is much easier to find dL/dQ =

- 1.2 + .04Q

We just invert this expression to write MPL = dQ/dL = 1/[-1.2+0.04Q]

For Q > 30, dQ/dL is positive, but it clearly gets smaller as Q (and L) rises (you

may see this by inspection, or, if you are more careful, by taking the second

derivative, which is given by - .04/[-1.2+.04Q]2 ).

b) The average product of labour is given by APL = Q/L = Q/[162 - 1.2Q + .02Q2].

We maximize the average product of labour by setting dAP/dQ = 0, which

occurs at

[(162- 1.2Q+.02Q2) - Q(-1.2+.04Q)]/[162 - 1.2Q + .02Q2]2 = 0

or [162-.02Q2]/[162 - 1.2Q + .02Q2]2 = 0 or 162 - .02Q2 = 0 or Q2 = 8100

or Q = 90, which is also L = 216, so that APL = 90/216 = 10/24 = 5/12 = 0.4167

At that point, MPL = 1/[-1.2+0.04(90)] = 1/[-1.2+3.6] = 1/2.4 = 5/12 = 0.4167

Obviously the two are equal, which we is normal relationship between the

average and the marginal at a maximum (or minimum).

c) FC = PKK = $7x90 = $630; VC = PLL = $5(162 - 1.2Q + .02Q2) = 810 - 6Q + 0.1Q2

Obviously, TC = FC + VC = 1440 - 6Q + 0.1Q2

We can easily write AVC = (810/Q) - 6 + 0.1Q and AC = (1440/Q) - 6 + 0.1Q

Also, MC = dTC/dQ = -6 + 0.2Q

To find minimum AVC, set dAVC/dQ = 0

This gives us -810/Q2 + 0.1 = 0, or Q2 = 8100, or Q = 90, at which point AVC =

12

To find minimum AC, set dAC/dQ = 0

This gives us -1440/Q2 + 0.1 = 0, or Q2 = 14400, or Q = 120, at which point AC =

18

From the equation for MC, MC = 12 when Q = 90, and MC = 18 when Q = 120

d) We found that the maximum APL occurs where AVC is at a minimum (at Q = 90).

This is no accident, because AVC = VC/Q = PL x (L/Q) = PL/APL

In this problem, the price of labour is a constant. Thus, if APL is maximized,

PL/APL is clearly minimized.