Lecture 7: Technology, Cost, Profit
ECOS2001 Intermediate Microeconomics
Stephen L. Cheung
The University of Sydney
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 1 / 36
Technology
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 2 / 36
Technology and production function
Consider a firm that uses 2 inputs x1 and x2 to produce a single
output q. We want to characterise the feasible methods for turning
inputs into output. This is what we mean by the technology.
An input bundle (x1, x2) specifies a particular combination of the
inputs, analogous to a consumption bundle for a consumer.
A production function f (x1, x2) specifies the maximum amount
of output that can be feasibly produced from the inputs (x1, x2).
In some ways analogous to a consumer's utility function.
Since a firm wants to get the most it can from its inputs, we usually
just write q = f (x1, x2), where it is implicit that q is the maximum.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 3 / 36
Isoquants
An isoquant is the set of all input combinations that yield the
same output. It is analogous to a consumer's indifference curve.
There will be a family of such curves; ones further from the origin
represent larger values of q.
x1
x2
increasing q
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 4 / 36
Example: Cobb-Douglas production function
f (x1, x2) = Ax
a
1x
b
2
This generates isoquants similar to the ones on the previous slide.
Notice that since q is a physical quantity of output, we cannot
arbitrarily transform a production function like we did with utility.
It is no longer invariant to positive monotonic transformations.
We cannot normalise the coefficients a and b to sum to one.
(See returns to scale, below.)
We also need an extra scale parameter A. Think of this as
determining the units in which q is measured.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 5 / 36
Marginal product
The marginal product of input i is the rate of increase in output
in response to an increase in xi, holding other inputs constant.
MPi (x1, x2) =
∂f
∂xi
For the two-input Cobb-Douglas example, f (x1, x2) = Ax
a
1x
b
2:
MP1 (x1, x2) =
∂f
∂x1
= Aaxa−11 x
b
2 MP2 (x1, x2) =
∂f
∂x2
= Abxa1x
b−1
2
Thus notice that the marginal product of one input may depend on
the level at which the other input is being held constant.
For a < 1, MP1 diminishes with x1, but it still increases in x2.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 6 / 36
Technical rate of substitution
The slope of an isoquant is known as the technical rate of
substitution (TRS), and is analogous to a consumer's MRS:
TRS =
dx2
dx1
∣∣∣∣
dq=0
= −∂f/∂x1
∂f/∂x2
= −MP1
MP2
Interpretation: the TRS measures the amount of x2 that can be
freed up by employing an extra unit of x1, holding output constant.
(What was the analogous interpretation of MRS for a consumer?)
Just as MRS was (the negative of) the ratio of marginal util-
ities, the TRS is (the ve of) the ratio of marginal products.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 7 / 36
Technical rate of substitution
For the two-input Cobb-Douglas example:
TRS = −MP1
MP2
= −Aax
a−1
1 x
b
2
Abxa1x
b−1
2
= −a
b
x2
x1
Thus when x2 is large relative to x1 the isoquant is relatively steep,
indicating that the firm is able to substitute a large amount of x2
for one unit of x1. Conversely when x2 is small relative to x1.
This is called diminishing technical rate of substitution.
This generates the convex, well-behaved shape of isoquants that
we assumed for a consumer's indifference curves.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 8 / 36
Long run and short run
In the long run, the firm has full flexibility to vary all its inputs.
On the other hand, in the short run some inputs are fixed.
With some inputs fixed, the short-run production function spec-
ifies maximum feasible output as a function of the variable inputs,
conditional on the quantities of the fixed inputs.
For example, for two inputs, with x2 fixed in the short run, we have
q = f (x1, x¯2), where x¯2 denotes the level at which x2 is fixed.
Note this is a function of a single variable (x¯2 is held constant).
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 9 / 36
Short-run production function
For each fixed level of x¯2, there is a different short-run function:
q
x1
increasing x2
The slope of the short-run production function is just the marginal
product of the variable input x1. The idea of diminishing marginal
product is thus a short-run concept.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 10 / 36
Returns to scale
Returns to scale is a long-run concept. It describes what happens
to output when all inputs increase jointly, in the same proportion.
If when all inputs are scaled up by a factor t > 1, output increases
by the same factor t, we say there are constant returns to scale:
f (tx1, tx2) = tf (x1, x2)
If output increases more than proportionately, we say that there are
increasing returns to scale:
f (tx1, tx2) > tf (x1, x2)
Finally, there would be decreasing returns to scale if:
f (tx1, tx2) < tf (x1, x2)
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 11 / 36
Returns to scale
For the two-input Cobb-Douglas example, f (x1, x2) = Ax
a
1x
b
2:
f (tx1, tx2) = A (tx1)
a (tx2)
b
= A (taxa1)
(
tbxb2
)
= ta+bAxa1x
b
2
= ta+bf (x1, x2)
This technology thus exhibits:
Constant returns to scale if a+ b = 1;
Increasing returns to scale if a+ b > 1;
Decreasing returns to scale if a+ b < 1.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 12 / 36
Cost minimisation
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Profit maximisation and cost minimisation
We usually assume that the ultimate goal of a firm is to maximise
profit. To do this, it must choose both its level of output q, and
the combination of inputs (x1, x2) it uses to achieve that output.
We can simplify this by breaking profit maximisation into two steps:
First, taking as given some desired output q, how should the firm
choose (x1, x2) to produce q at lowest cost?
Cost minimisation is a necessary condition of profit maximising.
We summarise the solution to this problem in a cost function.
Second, given the cost function and market structure, analyse the
profit maximising choice of q separately (later lectures).
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 14 / 36
Cost minimisation
Let (w1, w2) be the prices of the inputs. We assume the firm treats
these as given, i.e. it is a price taker in the markets for its inputs.
The production function is f (x1, x2), and the desired output is q.
We take q as given, and worry about how it is chosen later.
The cost minimisation problem is to choose the bundle of inputs
(x1, x2) to minimise total cost, subject to attaining an output q:
min
x1,x2
w1x1 + w2x2
s.t. f (x1, x2) = q
Since we allow the firm to vary both inputs, this is the long-run
version of the cost minimisation problem. The parameters of this
problem are the input prices w1, w2, and the desired output q.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 15 / 36
Cost minimisation
min
x1,x2
w1x1 + w2x2
s.t. f (x1, x2) = q
The solution to this problem is a set of conditional input demand
functions, x1 (w1, w2, q) and x2 (w1, w2, q). These express the cost
minimising choices of x1 and x2 as functions of the parameters.
They are conditional demands because they take q as given.
If we substitute back into the objective function w1x1 + w2x2, we
get an expression for the minimised value of total cost, as a function
of the parameters. This is the (long-run) cost function of the firm:
c (w1, w2, q) = w1x1 (w1, w2, q) + w2x2 (w1, w2, q)
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 16 / 36
Graphical solution
Recall that an isoquant is the set of (x1, x2) combinations that
yield the same output, and is analogous to an indifference curve.
Define an isocost as the set of (x1, x2) combinations that have the
same total cost. This is roughly analogous to a budget line. Let C
be some fixed level of total cost. Then the equation of an isocost is:
w1x1 + w2x2 = C
x2 =
C
w2
− w1
w2
x1
So the slope of an isocost is (the ve of) the ratio of input prices.
Of course, there will be a family of isocost lines, with ones closer to
the origin associated with lower total costs.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 17 / 36
Graphical solution
In utility maximisation, the goal was to reach the highest possible
indifference curve, subject to not going outside the budget line.
In cost minimisation, the goal is to reach the lowest possible isocost
line, subject to not going below the isoquant (desired output level).
Despite this important difference, the solution is again characterised
by a tangency: where the lowest isocost just touches the isoquant.
So at a cost minimum, the technical rate of substitution equals the
ratio of relative input prices:
MP1 (x1, x2)
MP2 (x1, x2)
=
w1
w2
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 18 / 36
Graphical solution
x1
x2
Isoquant (blue)
slope = –MP1/MP2
Isocost lines (green)
slope = –w1/w2
decreasing cost
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 19 / 36
Graphical solution
If the isoquant were steeper than the isocost, we would have:
MP1
MP2
>
w1
w2
⇒ MP1
w1
>
MP2
w2
This says that the MP of spending a dollar on x1 is greater than
for x2. In that case, the firm ought to use more of x1 and less of
x2. (Conversely if the isoquant were flatter.)
So another way to interpret tangency is that the marginal product
of a dollar spent on each input should be the same.
As usual, tangency will not hold at a corner solution (e.g. perfect
substitutes), or if the isoquant is kinked (e.g. perfect complements).
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 20 / 36
Calculus solution: Example
Suppose the production fn is f (x1, x2) = x1x2. This is not chosen
to be realistic; rather it is particularly simple to differentiate. So
you can focus on the logic instead of getting bogged down in math.
The technical rate of substitution is:
TRS = −∂f/∂x1
∂f/∂x2
= −x2
x1
So the tangency condition is:
x2
x1
=
w1
w2
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 21 / 36
Calculus solution: Example
Rearrange the tangency condition as x2 =
w1
w2
x1, and use this to
substitute for x2 in the production function to get q =
w1
w2
x21.
Solve this to express x1 as a function of the parameters w1, w2, q.
This is the conditional input demand function for x1:
x1 (w1, w2, q) =
√
w2
w1
q
Thus notice that the demand for x1 is decreasing in its own price
w1, and increasing in the price of the other input w2 as well as the
desired output q. All of which makes good sense. Similarly:
x2 (w1, w2, q) =
√
w1
w2
q
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 22 / 36
Calculus solution: Example
x1 (w1, w2, q) =
√
w2
w1
q x2 (w1, w2, q) =
√
w1
w2
q
Finally, substitute back in the objective function w1x1 + w2x2 to
get the minimised value of total cost as a function of w1, w2, q:
w1
√
w2
w1
q + w2
√
w1
w2
q =
√
w1w2q +
√
w1w2q
So the cost function is:
c (w1, w2, q) = 2
√
w1w2q
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 23 / 36
Example: Perfect complements
f (x1, x2) = min {ax1, bx2}
This technology has L-shaped isoquants, kinked where ax1 = bx2.
Since TRS is not defined at the kink, we cannot solve by tangency.
But we know it can only be cost minimising to operate at the kink:
Whenever ax1 6= bx2, we can reduce one of the inputs and thus
reduce total cost without causing the min, and thus q, to fall!
At a kink, the amounts of x1 and x2 needed to reach an output q
are simply x1 = q/a, and x2 = q/b. These are the cost-minimising
conditional input demands. They depend only on q, not w1 or w2.
By definition, the cost function is the minimised value of total cost:
c (w1, w2, q) = w1
q
a
+ w2
q
b
=
(w1
a
+
w2
b
)
q
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 24 / 36
Example: Perfect complements
0
q
x2
x1q
a
q
b
a/b
Orange: isoquant; Blue: isocost. At kink point, ax1 = bx2 = q.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 25 / 36
Example: Perfect substitutes
f (x1, x2) = ax1 + bx2
This technology has linear isoquants, with TRS = −a/b. Now we
cannot rely on tangency because we typically have corner solutions.
If w1/w2 < a/b we have x2 = 0 and x1 = q/a, with cost w1q/a.
If w1/w2 > a/b we have x1 = 0 and x2 = q/b, with cost w2q/b.
Remember there is also a third case where w1/w2 = a/b. In that
case, all input combinations on the isoquant have the same cost.
All three cases can be summarised by writing the cost function as:
c (w1, w2, q) = min
{w1q
a
,
w2q
b
}
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 26 / 36
Example: Perfect substitutes
0
x1
x2
0
x1
x2
w2
w1 TRS>w2
w1 TRS=
0
x1
x2
w2
w1 TRS<
q
a
q
a
q
b
q
b
TRS is the same in all three panels, while w1/w2 varies.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 27 / 36
Short-run cost minimisation
Suppose that in the short run, x2 is fixed at x¯2.
Since x2 is fixed, the solution is generally no longer at a tangency.
As the firm has less flexibility in the short run, its cost can never be
lower in the short run than in the long run.
In the special case that x2 just happens to be fixed at the level that
would be optimal to produce q in the long run (x¯2 = x
∗
2), then the
long- and short-run conditional input demands for x1 will coincide.
At that point, the short-run cost is just equal to long-run cost.
Everywhere else, short-run cost is greater than long-run cost.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 28 / 36
Short-run cost minimisation
x1
x2
f (x1, x2) = q
x1
*
x2
*
_
x2
x1
s
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 29 / 36
Short-run cost minimisation
c(w1, w2, q)
cs(w1, w2, q; x2)
c
q
_
FC = w2x2
_
q
_
x¯2 is long-run optimal for producing output of q¯. At q¯, short- and
long-run costs are equal. Everywhere else, short-run cost is greater.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 30 / 36
Profit maximisation
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 31 / 36
Profit maximising input demand
To derive the cost function, we assumed the firm took the prices of
its inputs as given.
If the firm also takes the price p of its output as given, we have the
case of a competitive firm that is a price taker in all markets.
The firm uses two inputs (x1, x2), with technology q = f (x1, x2).
In terms of input demand, the profit maximisation problem is:
max
x1,x2
pi (x1, x2) = p · f (x1, x2)− w1x1 − w2x2
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 32 / 36
Profit maximising input demand
max
x1,x2
pi (x1, x2) = p · f (x1, x2)− w1x1 − w2x2
The first-order conditions are:
∂pi
∂x1
= p · ∂f∂x1 − w1 = 0
∂pi
∂x2
= p · ∂f∂x2 − w2 = 0
This says that the firm should use each input up to the point where
the value of the marginal product of the input is equal to its price.
Dividing the first-order condition for x1 by the one for x2 gives the
tangency condition for cost minimisation. This reaffirms that cost
minimisation is a necessary condition for profit maximisation.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 33 / 36
Graphical solution
Consider the profit-maximising choice of one of the inputs, say x1.
The partial relation between x1 and q is represented by a short-run
production function holding x2 constant. Its slope is given by MP1.
Consider how profit varies with changes in x1 (and hence q) holding
x2 constant. To illustrate this graphically, define an isoprofit line
as the combinations of x1 and q that result in the same total profit:
pq − w1x1 − w2x¯2 = p¯i
or in intercept-slope form:
q =
p¯i + w2x¯2
p
+
w1
p
x1
This is positively sloped, with profit increasing to the north-west.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 34 / 36
Graphical solution
Given x¯2, the profit maximising choice of x1 is where the highest
isoprofit line just touches the production function.
q
x1
_
q = f (x1, x2)
slope = MP1
x1
*
q*
Isoprot lines
slope = w1/p
At this point, the slopes of the two lines are equal:
∂f
∂x1
= w1p , or
p · ∂f∂x1 = w1. This is the same FOC we derived previously for x1.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 35 / 36
Textbook sections relevant to this lecture
Varian (9th edition);
Chapter 19: all Sections.
Chapter 20: Sections 20.520.8.
Chapter 21: Sections 21.1, 21.4.
Stephen L. Cheung Lecture 7: Technology, Cost, Profit 36 / 36

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