2. MANAGERIAL ECONOMICS 2
1. a. Special characteristics of Q = K⅔L⅓ and how it is different from a Linear
Production function and a Leontief production function.
Production function in economics defines the output produced based on factors of production or inputs
used. As one of the key concept of neoclassical theory, it is used in relating marginal product and
determining efficient allocation of production of production resources which is the main focus of
economics. The aim of production function is ensure efficiency allocation and use of production inputs
such as capital and labour including the allocation of income to the factors of production. Technological
problems associated with production efficiency from the perspective engineer are also addressed by the
Given a production function Q = K*L with diminishing returns labor and capital to increasing returns of
a single production factor but with constant returns to scale implies that increasing capital and labor by
100% leads to the increase in output by 100%. Linear production function in the basic and simplest
production functions with labor as its sole factor of production. The Leontief production function is
used in situations with fixed proportions of inputs used; where by increasing one output without
increasing the others in the same proportion will not change the output. This is given as Q = min (aX1,
b. Assuming that the firm hires 10 units of capital and 20 units of labour, use the production
function given in part a to compute the average and marginal products of labour. (8 marks)
Given that Q = K⅔L⅓ and K = 10 units and L = 20 units,
Q = 44.46 units = TP
Marginal product of labor denoted by APL = Q/L = 44.49/20 = 2.22
MPL = 2L-2/3
When L = 20
MPL = 2x20-2/3
c. How the firm would use the marginal product of labour to determine the units to hire.
3. MANAGERIAL ECONOMICS 3
From the calculations in b, the slope of average product after making substitutions would be =
Based on the slope, three conclusions can be made:
i. The point at which average product is less than labor’s marginal product, the average productivity of
labour will be on the increase as denoted in the equation > 0.
ii. When average product of labour is less than its marginal product, there will be a decrease in labour’s
average product depicted in the following equation < 0.
iii. The point at which average product and marginal product of labour meet is the minimal value for
average product denoted by = 0.
2. A firm producing Q units of output with fixed quantity of capital (K) and labour (L):
a. The average and marginal products of labour as per example.
Q TPL APL MPL
16 8 2 0.010
36 16 2.25 0.002
65 24 2.7 0.001
97 32 3.0 0.0006
4. MANAGERIAL ECONOMICS 4
137 40 3.4 0.0004
177 48 3.6 0.0002
209 56 3.7 0.0002
233 64 3.6 0.00014
249 72 3.45 0.00012
257 80 3.2 0.00010
a. Graphs of the total product, average product and marginal product curves and their
The total product curve illustrates relationship between variable input like labour and total product as
depicted in the graph.
Production output is positive until the ninth laborer is hired meaning the maximum number of workers the
firm can hire is eight.
The average product curve illustrates in a graphical way the relationship between quantities of input that is
variable to average product assuming other inputs fixed as put in the diagram below. It illustrates what a
unit input produce. It is best applied in analyzing production in a short-term.
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The hump-shape of the graph reflects an increase and subsequent decrease in marginal returns
demonstrating law of diminishing marginal returns.
Marginal product curve as illustrated in the diagram below represents others factors including production
inputs constant, quantity in units of variable inputs a firm uses to produce extra unit of outputs. It is best
used in analyzing a firm’s short-run production.
Initially, the marginal productivity increases until two workers are hire before falling meaning the first two
workers leads to increase in marginal returns as opposed to the extra ones whose contribution is decreasing
marginal returns also known as law of diminishing marginal returns.
c. The total variable cost, average variable cost and the marginal cost of the firm if the cost of
capital is $1,000 and labour costs $10.00 per hour.
TFC TVC AVC TC MC
1000 80 5 1080 -
1000 160 4,44 1160 4
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1000 240 3,69 1240 2,76
1000 320 3,30 1320 2,5
1000 400 2,92 1400 2
1000 480 2,71 1480 2
1000 560 2,68 1560 2,5
1000 640 2,75 1640 3,33
1000 720 2,89 1720 5
1000 800 3,11 1800 10
a. Graphs of the TVC, AVC and MC curves in relation to product curves in part b.
It can be concluded from the discussion that AVC reaches its low point when the slope of the graph is
tangent to TVC and given that that slope measures both AVC and MC, AVC’s minimum point is where
it intersects MC. MC is normally above AVC when AVC is rising while MC is normally below AVC
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when AVC is decreasing and the higher the output, the narrow the gap between AVC and ATC because
AFC is falling. At the same time, MC intersects AVC and ATC at their minimum points.
3. Assume that a firm hires only labour and capital to produce bicycles
a. Explain the cost minimization rule for this firm and why this rule is logical…(8 marks)
Cost is minimized at the levels of capital and labor such that the marginal product of labor divided by the
wage (w) is equal to the marginal product of capital divided by the rental price of capital (r) (MPl/w =
MPk/r). More intuitively, you can think of cost being minimized (and, by extension, production being most
efficient) when the additional output per dollar spent on each of the inputs is the same (or, in less formal
terms, you get the same "bang for your buck" from each input). This formula can even be
extended to apply to production processes that have more than two inputs.
b. Whether the firm is minimizing costs for hiring 100 units of capital at $5.00 a unit and 500 units of
labour whose unit costs is $4.00 when MPl is 20 while MPk is 25.
K = 100 units, L = 500 units, MPl = 20, MPk = 25, w = 5.00.
If the firm is minimizing costs, MPl/w = MPk/r, and in our case:
20/4 = 25/5 = 5, so the firm is minimizing costs.
c. What the firm should do if the price of capital falls to $4.00 in relation to cost minimization rule.
If the price of capital falls to $4.00, the firm should increase the use of capital and decrease the use of labor
to minimize cost.
4. a. Whether the college should take the students’ offer as per the example
If there is no outside offer and the students increased their offer to $55,000, the college should take
the student offer, because the next college attempt to find another partner through advertising will
increase losses, so the college should sell the cafe to students to cover the part of $110,000 losses with
b. why short run ATC is U‐ shaped as well as the long‐ run ATC and the difference between the two
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The long-run ATC curve based on the diagram has many short-run ATC curves. There are three phases in
the long-run ATC curve depicting points of economies of scale, constant returns to scale and diseconomies
to scale. The implications of the Law of diminishing marginal returns on production function leads to u-
shape of Average Total Cost (ATC) curve in the short-run. In the short-run, the firm can only increase or
reduce working hours of its labor and not capital. At the end of the day, more of variable input (combined
with fixed variable (Capital) leads to diminishing marginal returns on labor.
In the long-run also called ‘variable-plant period’, all factors of production are variable meaning there is no
diminishing marginal product. The u-shape of the ATC in the long-run is due to concepts of diseconomies
of scale, constant returns to scale, and economies of scale. Initial stages of the firms growth witness
improvements in efficiency but later on becomes too big for its own benefit leading to increase in costs of
productivity of inputs like.
5. a. Whether the manager shown a preference for the second compensation scheme and if the
company benefited from the same compensation scheme
As we can see from this example, the manager has shown a preference for the second compensation
scheme, because her amount of income now depends on her activity and the profits of the company. So the
company benefited from the second compensation scheme, because its profits will rise, as the manager
spend more time to work harder to receive the higher income.
b. Difference between economies of scope and cost complementarity
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Economies of scope means new efficiencies are produced as more products are produced. For example,
looking at the formula: C (Q1, Q2) = f + aQ1Q2 + Q1^2 + Q2^2, cost of producing quantity (Q1 and Q2)
of product 1 and 2 is C(Q1, Q2). In the same cost function formula; ‘a’ denotes cost complementarity. If a
negative ‘a’ is introduced in the equation, then it would mean aQ1Q2 will indicate cost savings in
production of the two items. The firm can at this point either: produce the two goods together as denoted in
the original formula in cost function: C (Q1, Q2) = f + aQ1Q2 + Q1^2 + Q2^2 or produce the two items
separately depicted in the following cost function formula: C(Q1, 0) + C(0, Q2) = [f + Q1^2] + [f + Q2^2]
= 2f + Q1^2 + Q2^2.
Economies of scope means the differences in cost of producing the two products depicted in the following
cost function formula: C(Q1, 0) + C(0, Q2) - C(Q1, Q2) = 2f + Q1^2 + Q2^2 - [f + aQ1Q2 + Q1^2 + Q2^2]
= f - aQ1Q2. For there to be economies of scale, the above formula must exceed 0 in which f - aQ1Q2 > 0
and hence f > aQ1Q2.
Lack of cost complementarities means cost of producing Q2 increase with extra unit Q1 is added but if the
inefficiency of ‘f’ which is a fixed cost is more than marginal inefficiency, then the situation would still be
regarded as economies of scale since the system has the capacity to produce unit of each product at cheaper