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

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Published on 24 Feb 2015
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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.
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