production costs - university of ghana ... costs social costs of production, refer to costs to...
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PRODUCTION COSTSIn this section we introduce production costs into the analysis of the firm. So far, our emphasis has been on the production process without any consideration of costs.
However, production activities do involve costs – implicit and explicit.
But cost is a rather complicated concept. It is a term that is open to more interpretations. Note for example the difference between consumers’ cost and producers.
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PRODUCTION COSTSSocial Costs of production, refer to costs to society when its resources are employed to make a given commodity. Since economic resources are limited, when resources are used to produce a certain product, less can be produced of some other product that can be made with those resources.
Private costs, are in contrast with social costs. Private costs are defined to be costs to the individual firm or producer. Take case of pollution!
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PRODUCTION COSTS
Explicit costs include the ordinary items that
an accountant would include as the firms
Expenses, e.g., payroll, payments for raw
materials, etc.
Implicit costs (alternative costs or opportunity
cost) include opportunity costs of resources
owned and used by the firm’s owner. This type
is often omitted in calculating the firm’s costs.
These costs generally come under private costs
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PRODUCTION COSTS
The economic cost of any activity is the value
of the best forsaken alternative.
The firm in order to attract the resources or
“factors” necessary to engage in production,
must pay resource owners amounts sufficient to
induce them to sacrifice their best alternatives
(whether employment or leisure).
But it is important to note that in production,
cost and price are different.
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PRODUCTION COSTSAs we shall see later, costs are important in determining a firm’s optimum profit position.
They are also the basis for a firm’s supply curve.
Having noted all these, it is worth stating the goal of the firm. Economists usually assume that the firm maximises profit, which is defined as the difference between revenue and cost.
This section on costs assumes that the firm under analysis is a competitive or “price-taking firm”.
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PRODUCTION COSTS
Distinguishing between short-run and long-run costs
This is related to the same concept in production theory.
In the short-run some costs are fixed, whilst in the long-run they become variable. This is the fundamental difference between the two.
Nevertheless, the distinction is a matter of degree.
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PRODUCTION COSTSDistinguishing between short-run and long-
run costs
The longer the run contemplated, the greater the range of costs regarded as variable rather than fixed.
Hence, if a firm is not committed to any outlays, it is in the long-run. In the long-run all options are open!
The situation changes to the short-run once a commitment to some factor of production has been undertaken.
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PRODUCTION COSTS
Distinguishing between short-run and
long-run costs
Thus, the short-run is characterised by
fixed and variable costs.
In the long-run, all costs are variable
since all costs depend on the volume of
output.
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PRODUCTION COSTS
Total, Average and Marginal Costs
The total cost can be regarded as the sum, taken
over all resources employed, of factor prices times
factor quantities.
In other words, it is the sum of all costs incurred
by the firm to produce a given level of output.
From the total cost, two other measures emerge:
average and marginal costs.
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PRODUCTION COSTS
Total, Average and Marginal Costs
The average cost (AC) is defined as the cost
per unit of output. Formally, this is defined as:
Where TC is total cost and Q is output
Q
TCAC
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PRODUCTION COSTS
Total, Average and Marginal Costs
The marginal cost (AC) is defined as the
change in total cost resulting from a unit change
in output. Formally, this is defined as:
Where TC is total cost and Q is output
Q
TCMC
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PRODUCTION COSTS
Total, Average and Marginal Costs
Some important conclusions are worth making!
We have noted earlier that in the short-run, the
firm faces both fixed and variable costs. But as the
firm alters its output, only the variable costs
change (why?).
The marginal cost that a firm experiences as it
expands output from given fixed resources are
entirely due to its variable costs.12Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION COSTS
Total, Average and Marginal Costs
This therefore leads to a major conclusion!
Decisions about output are based
entirely on marginal costs; fixed
costs are totally irrelevant to any
output decisions.
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PRODUCTION COSTS
Total, Average and Marginal Costs
. But based on our understanding from
production, we know that cost is a multivariable
function, that is, it is determined by many
factors.
Thus, in the short-run,
),,,(
KPTXfC f
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PRODUCTION COSTS
Total, Average and Marginal Costs
Where C = total costs; X = output; Pf=
prices of factors; T = technology; and
= fixed factors.
In the long-run
K
),,( fPTXfC
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PRODUCTION COSTS
Total, Average and Marginal Costs
We have earlier made the assumption that the
firm is a competitive firm.
Thus, it seeks to be efficient in production,
aiming to produce at the minimum cost of
production for any given output level, Q, when
factor prices are Pf .
If we also assume that firms are price takers in
the factor markets, then Pf is fixed.
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PRODUCTION COSTS
Total, Average and Marginal Costs
Thus, we can write our cost function as
dependent on output, Q, alone. This can be written
as
Hence total costs can be written as
)(QcC
FCQcC )(
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PRODUCTION COSTS
Total, Average and Marginal Costs
The marginal cost function can be written as
Q
FC
Q
Qc
Q
QcQMC v
)()()(
VCQ
QcQMC v
)()(
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PRODUCTION COSTS
Total, Average and Marginal Costs
But note that the marginal cost measures the rate of change, hence we can define the marginal cost function as
Refresh your memory on the relationship between MC, AVC and AC.
dQ
Qcd
dQ
dTCQMC
)]([)(
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PRODUCTION COSTS
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PRODUCTION COSTS
Total, Average and Marginal Costs
The marginal cost curve, average variable
cost curve and average total cost curves are
generally U-shaped.
The U-shape in the short run is attributed to
increasing and diminishing returns from a
fixed-size plant, because the size of the plant is
not variable in the short run.
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PRODUCTION COSTS
Total, Average and Marginal Costs
The marginal cost and average cost curves are
related:
When MC exceeds AC, average cost must
be rising
When MC is less than AC, average cost
must be falling
This relationship explains why marginal cost
curves always intersect average cost curves at
the minimum of the average cost curve.
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PRODUCTION DECISION
Optimal Input Combinations
How will a competitive firm combine inputs to produce a given quantity of output?
As a first approximation, we assume that firms are out-and-out profit maximizers; that is to maximize the difference between revenue (R) and (economic) costs (C) incurred.
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PRODUCTION DECISION
Optimal Input Combinations
Thus, our typical firm seeks to maximize profits;
The assumption of profit maximization implies
that a firm will seek to minimize the costs of
producing a given output or seek to maximize
the output derived from a given level of cost.
CR max
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PRODUCTION DECISION
Optimal Input Combinations
Also remember the assumption of perfect
factor markets, such that firms are price takers
in the input markets.
Thus, if we suppose there are two inputs,
labour (L) and capital (K), then what
combinations of L and K should the firm
choose?
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PRODUCTION DECISION
Optimal Input Combinations
If the wage rate (w) is the cost of labour and
the rental rate (r) is the cost of capital, then the
total cost outlay, C, is given by:
Lr
w
r
CK
rKwLC
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PRODUCTION DECISION
Optimal Input Combinations
Thus, the various combinations of L and K
that can be purchased, given input prices and
total outlays can be represented by a straight
line. K
LC/w
C/rSlope = (w/r)
0
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PRODUCTION DECISION
Optimal Input Combinations
A family of Isocost lines can be illustrated
belowK
L0
C(3)
C(2)C(1)
C(3) > C(2) > C(1)
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Maximization of output for given cost
L
K
0
100
200
300R
L*
K*
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PRODUCTION DECISION
Optimization Condition
The firm maximizes output at point R, by
choosing L* and K* of labour and capital
respectively .
At point R, the isoquant is tangent to the
isoquant. Thus,
r
w
MP
MPMRTS
K
LKL ,
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PRODUCTION DECISION
Optimization Condition
It follows that the optimal combination of inputs
is where.
Or
r
w
MP
MP
K
L
r
MP
w
MP KL
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PRODUCTION DECISION
Optimization Condition
More generally, a firm will choose an input
combination such that.
Where MPa, MPb, ... ..., MPn are the marginal
products of inputs, a, b, ..., n, and Pa, Pb, ... ..., Pn
are input prices.
n
n
b
b
a
a
P
MP
P
MP
P
MP .......
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Minimization of Cost for a Given
Output Level
L
K
0
400
Z
L*
K*
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PRODUCTION DECISION
Optimization Condition
To minimize the cost of producing the output level, Q=400, the firm chooses point Z. Here too, the firm must equate the MRTS to the ratio of input prices
r
w
MP
MPMRTS
K
LKL ,
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PRODUCTION DECISIONConstrained Optimization
The firm’s decision, as with the case of consumers, can be represented as a constrained optimization problem.
We first consider the case of constrained output maximization.
Thus, given Q = f(K, L) and C = rK + wL, we can set up the Lagrangian function:
)(),( CwLrKLKfZ
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PRODUCTION DECISION
Constrained Optimization
Z symbolises the Lagrangian function. Here
is an undetermined Lagrange multiplier
Also note that another formulation of the
Lagrangian function is:
)(),( wLrKCLKfZ
0
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PRODUCTION DECISION
Constrained Optimization
We set the first-order conditions, which are to
set to the partial derivatives of K, L, and λ equal to
zero.
)1(0
rf
K
Zk
)2(0
wf
L
Zl
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PRODUCTION DECISION
Constrained Optimization
From (1) and (2): moving the price terms to the
right and dividing (2) by (1):
)3(0
CwLrK
Z
)4(r
w
f
f
k
l
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PRODUCTION DECISION
Constrained Optimization
From (4) the first order conditions state that the
ratio of marginal products must be equated with
their price ratios.
Solving (1) and (2) for λ, yields:
)5(w
f
r
f lk
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PRODUCTION DECISION
Constrained Optimization
(5) states that the contribution to output
of the last money outlay expended on each
input must equal λ.
The multiplier, λ, is the derivate of
output with respect.
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PRODUCTION DECISION
Constrained Cost Minimization
The firm may desire to minimize the cost of
producing a prescribed level of output.
As with our earlier analysis, we form the
Lagrangian function, and set the partial
derivatives to zero for K, L, and λ.
),(( 0 LKfQwLrKZ
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PRODUCTION DECISION
Constrained Cost Minimization
An alternative formulation of the Lagrangian
function is:
In what follows, we set the various partial
derivatives to zero and obtain the optimal
conditions for input combination in production
)),(( 0QLKfwLrKZ
42Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Constrained Cost Minimization
)'1(0
kfr
K
Z
)'2(0
lfw
L
Z
)'3(0),(0
LKfQ
Z
43Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Constrained Cost Minimization
Since r and w are both positive, moving the
price terms and dividing (2’) by (1’), we obtain:
The first order conditions for the minimization
of cost subject to an output constraint are similar
to those for the maximization of output subject to
a cost constraint.
MRTSf
f
r
w
k
l
44Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Second-Order Conditions
We shall leave out the details of second-order
conditions. However, more generally, for output
maximization subject to a given cost, and for cost
minimization subject to a given output level, the
slope of the marginal product curves for the two
inputs must be negative.
or: 0;02
2
2
2
L
Q
K
Q 0;0 llkk ff
45Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Second-Order Conditions
The second-order conditions ensure that we
are satisfied that the isoquants are convex to
the origin.
If the isoquant is concave, then we have a
corner solution.
46Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Second-Order Conditions
e
e1
e2 L
K
0
Q=230
47Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Second-Order Conditions
In the diagram above, output, Q=230, can be
produced at points e, e1 and e2.
This indicates different costs of producing the same level of output.
The lowest cost point is given by e2.
48Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISIONProfit Maximization: A Formal Analysis
A firm is usually free to vary the levels of both cost and output, with the ultimate objective being to maximize profits rather than a solution to a constrained maximum or constrained minimum problems.
The firm we have been analysing is assumed to operate in a competitive market. Its total revenue is given by the number of units of Q sold multiplied by the fixed unit price it receives.
49Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
The firm’s profit is thus defined as:
Given Q = f(K, L) and C = rK + wL, the firm’s profit function is given by
П = Pf(K, L) – rK - wL
Profit is a function of K and L and is thus maximised with respect to these variables.
CQP .
50Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
Differentiating the profit function with
respect to capital and labour, gives:
Moving input price items to the right, we
have:
)6(0 rPfkk
)7(0 wPfll
51Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
Thus, Pfk and Pfl are the values of the
marginal product of K and L respectively, and
they represent the rate at which the firm’s
revenue would increase with further increases
in K or L.
rPfk wPfl
52Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
Profit maximization requires that each input be utilised up to the point at which the value of its marginal product equals its price.
Profits can be increases as long as Pfk > rand Pfl > w.
That is, as long as the addition to revenue from employing an additional unit of input exceeds its cost.
53Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
The second-order conditions for profit maximization require that:
These suggest that profits must be decreasing with respect to further increases in L and K.
02
2
llPf
L0
2
2
kkPf
K
54Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Profit Maximization: A Formal Analysis
Because P > 0, this suggests that the marginal
product of both L and K must be decreasing.
The conditions for first- and second-order
profit maximization require that the isoquant be
strictly concave in the neighbourhood of the a
point at which the first-order conditions are
satisfied with non-negative levels of inputs (K
and L).55Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Optimal Expansion Path in the Short-Run
In the short-run, K is fixed, the firm is therefore
forced to expand along a straight line parallel to
the horizontal axis.
With prices of factors constant, the firm does not
maximise profits in the short-run, due to the
constraint of given capital.
The optimal path would be along OA, but the
firm can only expand along in the short run.__
KK
56Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Optimal Expansion Path in the Short-Run
0
_
K_
K
A
K
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PRODUCTION DECISION
Optimal Expansion Path in the Long-Run
In the long-run all factors are variable.
Output can therefore be expanded without
limitation.
As is always the case, the firm’s objective of
profit maximization, this means it chooses the
least-cost combination of inputs, which is
represented by the points of tangency between
the isocosts curves and isoquants.
58Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Optimal Expansion Path in the Long-Run
0
E
K
L
15012080
59Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISIONOptimal Expansion Path in the Long-Run
The expansion path indicates how, as output rate changes (but input prices remain fixed),the quantity of each input changes.
If the production function is homogeneous, the expansion path will be a straight line through the origin, whose slope depends on the ratio of factor prices.
With only two inputs in production, it is also easy to derive the long-run cost function from the expansion path.
60Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Optimal Expansion Path in the Long-Run
It is also worth noting that the maximum
profit-input combination lies on the expansion
path.
Given Pfl = w and Pfk = r, we note that this is a
special case of the constrained output
maximization discussed earlier.
61Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Optimal Expansion Path in the Long-Run
That is, along the long-run expansion path, the
condition is satisfied.
Also the implies that profit is also maximised,
that is,
r
w
f
f
k
l
r
w
Pf
Pf
k
l
62Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Input Demand Functions
The firm’s input demands are derived from the
underlying demand for the goods and services it
produces.
Thus, the firm’s input demand functions are
obtained by solving the first-order conditions for
profit maximization for L and K, as functions of
input prices and output price.
63Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Input Demand Functions
Therefore, more generally, given a production
function of the Cobb-Douglas form, we can obtain
the firm’s input demand functions
1:0,, KALQ
rKwLKPAL
64Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe
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PRODUCTION DECISION
Input Demand Functions
Solving for L and K, we obtain:
01 wKALPl
01 rKALPk
),,(;),,( ** prwfKprwfL
0;0;0 prw fff65Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe