ECON2101 Intermediate Microeconomics
Technology
Aleksandra Balyanova
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The supply side of the market
Having studied consumer choice (under certainty and uncertainty) and derived
demand, we now turn to the firm side of the market. We will
1.
2.
3.
4.
Model firm technology (now)
Derive from the firm’s technology its cost function, i.e. the cheapest way for it
to produce any given output level (next)
Derive from its cost function its supply function, i.e. how much it will supply to
the market for any given price of its output (later)
Aggregate individual supply to study market supply and long run equilibrium in
a perfectly competitive market (even later)
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Firms
Firms are complex and heterogenous organisations.
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They employ different types of resources to produce a variety of goods and
services sold to consumers in different markets.
They employ a wide variety of incentive schemes and compensation structures
(to varying effect!) to induce lower-level employees to perform in a way desired
by higher level management
In modelling the firm, we abstract from all of these details. Our bare-bones model of
the firm depicts it only as a “black box” that transforms inputs (like labour and
capital) into output. The firm’s technology is the process by which inputs are turned
into outputs.
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Technology
Example 1:
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You are a firm that transforms the input “hours spent studying” into the output
“Micro 2 mark”. The technology by which you do so can be modeled by the
production function
√
f (h) = h
i.e. your mark increases rapidly at the beginning, but as you spend more and
more time studying, your mark increases more slowly (the marginal product of h
is decreasing - more on this later). Notice that if you study for 9 hours, you can
get a maximum mark of 3, but you can also “waste” some of your effort and get
a mark of 2, or even a mark of 0.
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Technology
When the technology is
modeled by f (z), for any
input quantity z, f (z)
describes the maximum
possible output.
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From 1 input to N inputs
A single, price-taking firm operating in a market economy uses N inputs (capital,
labour, raw materials, etc.) and transforms them into output q.
Denote inputs by zn ≥ 0. An input vector is z = (z1 , . . . , zN )
The N + 1-dimensional input-output vector (z, q) is a feasible plan for the firm if q
can be produced given z, given the firm’s available technology.
Given an input vector (z1 , . . . , zN ), the maximum output that can be produced is
f (z1 , . . . , zN ). The function f : RN
+ → R+ is the firm’s production function.
A plan is feasible if
q ≤ f (z1 , . . . , zN )
The collection of all feasible plans is the production set.
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From N to 2 inputs
Much like with the consumer optimisation problem, we will focus on the two-input
case.
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Similar reasoning as for consumer problem: two inputs is the minimum needed
to capture the fact that there may be choices to be made about how to produce
a given output
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From N to 2 inputs
Example 2:
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1
1
Suppose the production function of the firm is q = f (z1 , z2 ) = 2z13 z23
The firm can produce q = 8 using (z1 , z2 ) = (8, 8)
But there are other ways - e.g.
z10 , z20 = (64, 1) and z100 , z200 = (1, 64)
In fact, any (z1 , z2 ) such that z1 × z2 = 64 can be used.
Isoquants
The q-output isoquant is the collection of all input vectors that yield output level q
using the firm’s available technology.
The map of isoquants is an equivalent way to describe the production function.
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Isoquants
Isoquants are to the
production function what
indifference curves are to
the utility function:
mappings of combinations
of inputs (of the function)
that yield the same
output (of the function).
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Marginal production
Marginal production
The production function f : RN
+ → R+ represents the maximum amount of output
that can be produced using an input vector z = (z1 , . . . , zN ). Isoquants are
representations of different combinations of inputs that yield the same output level.
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This does not mean that all input combinations are economically equivalent.
A profit maximising firm asks what is the most cost effective way to produce q.
Marginal Products
Given a differentiable production function f : RN
+ → R+ , the marginal product of
input n = 1, . . . , N at b
z is the rate of change of output as the quantity used of input n
changes slightly
∂f bz
for n = 1, . . . , N
MPn b
z =
∂zn
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Technical rate of substitution
To be able to eventually determine what is the most cost effective way to produce an
output q, we explore the trade-offs
involved in input use. Fix an input vector b
z at
which the output level is f b
z = q. Totally differentiating this expression gives
∂f b
z
∂z1
∂f
dz1 + · · · +
b
b
z
∂zN
dzN = 0
When focusing on changes in two inputs only, say input l and input k, we have
dzj = 0 for all j 6= k, l. Rearranging, we see that
∂f b
z /∂zl
MPl b
z
dzk
=−
=−
dzl
∂f b
z /∂zk
MPk b
z
This expression captures the rate at which the firm has to decrease input k to
marginally increase its use of input l while keeping output constant.
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Technical rate of substitution
Given a differentiable
production function
f : RN
+ → R+ „ the
technical rate of
substitution between
input l and input k at b
z is
TRSl,k
MPl b
z
bz =
MPk b
z
For inputs 1 and 2, the
TRS1,2 at zb1 , zb2 is the
absolute value of the
slope of the isoquant at
that input vector.
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Commonly used production
functions
Perfect substitutes
Let f : R2+ → R+ be given by
f (z1 , z2 ) = az1 + bz2
This production function
represents perfect substitution
between inputs 1 and 2.
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Perfect substitutes
To employ one extra unit of input
1 without changing the production
level, the firm gives up ba units of
input 2.
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Perfect complements/fixed proportions
Let u : R2+ → R be given by
f (z1 , z2 ) = min {az1 , bz2 }
This production function
represents perfect
complementarity between inputs
1 and 2.
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Perfect complements/fixed proportions
The firm uses at least ba units of
input 2 for each unit of input 1.
When inputs are in the “ideal”
ratio, additional units of either do
not affect output (if the other
input is fixed).
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Cobb Douglas
Any production function of the
form
f (z1 , z2 ) = Cz1a z2b
with C , a, b > 0 is called a Cobb
Douglas production function.
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Cobb Douglas
e.g. Example 2,
1
1
f (z1 , z2 ) = 2z13 z23
Isoquants are hyperbolic,
approaching but never touching
either axis
CD technology captures smooth
substitutability between inputs
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Technology: exercise 1
Firm 1 has technology given by
f1 (z1 , z2 ) = 2z1 + z2
Firm 2 has technology given by
f2 (z1 , z2 ) = z1 + 2z2
The two firms merge. As a result, the new firm has access to the production methods
that were available to both firms. The joint firm’s production function is then
fj (z1 , z2 ) = max {2z1 + z2 , z1 + 2z2 }
Depict the joint firm’s isoquant for q = 8. Label the TRS along the isoquant.
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Technology: exercise 1
The joint firm’s production function is then
fj (z1 , z2 ) = max {2z1 + z2 , z1 + 2z2 }
Depict the joint firm’s isoquant for q = 8. Label the TRS along the isoquant.
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When 2z1 + z2 > z1 + 2z2 , i.e. when z1 > z2 , Firm 1’s method is more
productive: q = 2z1 + z2 . Substituting in q = 8 and rearranging, we have that
below the 45◦ line, the isoquant is given by z2 = 8 − 2z1
When z1 < z2 , Firm 2’s method is more productive: q = z1 + 2z2 . Substituting
in q = 8 and rearranging, we have that above the 45◦ line, the isoquant is given
by z2 = 4 − 21 z1
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Technology: exercise 1
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Below the kink when z1 > z2 - the
joint firm is using
Firm 1’s method
and the isoquant is
given by
z2 = 8 − 2z1
Above the kink when z2 < z2 - the
joint firm is using
Firm 2’s method
z2 = 4 − 12 z1
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