Firm Theory

Rafayal Ahmed

NSU SBE

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What is a Firm (in Microeconomic Theory)?

▶ Think of a firm as a machine that transforms a number of

inputs into output(s).
▶ Inputs can be labor, capital, potatoes, tomatoes, pencils,
whiteboards, markers etc.
▶ Both the inputs and outputs are consumption goods, we will
assume these goods have positive market prices.
▶ For now, we will analyze firms that are price-takers in both
the input and output markets.
▶ This can mean the firm is very small compared to the market,
so its decisions do not affect market price.
▶ Also, it can mean it is not strategic, and does not consider
how its decisions will affect market prices.

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Production Function

▶ The firm can produce a certain amount of output using a

given amount of inputs.
▶ We will only consider firms that produce a single output good.
▶ These amounts are best described using a production
function, f : RL−1
+ →R
▶ The total number of goods in this economy is L.
▶ We will often write y = f (x ), where y ∈ R+ is the output
amount and x ∈ RL−1+ is the input vector.
▶ The function f is a description of the firm’s technology (what
it can produce using a given combination of inputs).
▶ In some cases, I might insert an additional parameter to allow
different production capabilities based on some outside
parameter.
▶ e.g. y = f (x , T ), where T is this month’s average
temperature which affects production of cauliflowers.

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Production Technology

▶ We will assume that more input =⇒ more output.

▶ We will also assume that the production function is
differentiable.

Definition
∂f (x )
The marginal product of input ℓ at input bundle x is MPℓ = ∂xℓ

Assumption
∂f (x )
(Non-negative marginal product) For every input ℓ, ∂xℓ ≥ 0.

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Substitution of Inputs

▶ Often it is possible to use a different input bundle to produce

the same amount of output.
▶ e.g. more AI robots and fewer human workers to produce cars.
▶ The technology (production function) tells us the rate of
substitution.
▶ This is an obvious analog to marginal rate of substitution
(MRS) for consumers.
▶ For a firm’s production function, we will call this the marginal
rate of technical substitution (MRTS).

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MRTS

Definition
For any output quantity y , the isoquant associated with y is the
set Q (y ) = {x : f (x ) = y }
▶ The isoquant is the set of all different input combinations that
produce output amount y .
▶ Notice the obvious analog to indifference curves.

Definition
At some input bundle x , the marginal rate of technical substitution
between good i and good j is the ratio of infinitesimal changes
dxj
dxi such that df = 0 due to only these changes.
▶ Due to the same reasons as MRS, we have
∂f (x )
∂xi MPi
MRTSi,j = ∂f (x ) = MPj .
∂xj

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Returns to Scale

▶ We assumed more inputs means more output, but by how

much?
▶ Does doubling all input amounts:
1. Doubles the output amount?
2. Increases output but to less than double the amount?
3. More than doubles?
▶ Based on the answer, we have different returns to scale.

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Returns to Scale

Definitions
At input bundle x , for any λ > 1, the production function f
exhibits:
1. Constant Returns to Scale (CRS) if f (λx ) = λf (x ).
2. Decreasing Returns to Scale (DRS) if f (λx ) < λf (x ).
3. Increasing Returns to Scale (IRS) if f (λx ) > λf (x ).

Remark
Notice this is a different concept from marginal product. Here we
are increasing the amount of all inputs together, so this is different
from partial derivatives.

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The Firm’s Profit Maximization Problem (PMP)

▶ Let p be the price of the output, and let w = (w1 , ..., wL−1 )
be the input price vector.
▶ The competitive firm takes p and w as given parameters.
▶ It knows the values of p and w but cannot change them.
▶ For given p and w , the firm solves its PMP.
▶ In this part of the course, we will focus on comparative statics
of the firm’s PMP.
▶ That is, how the outcomes change if p or some wi changes.

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Analyzing the PMP

▶ Taking y = f (x ), the firm’s profit maximization problem is:

max pf (x ) − w .x
x

▶ This is maximizing total revenue minus total cost. Notice that

P
w .x = wi xi is the firm’s total cost of buying all the inputs.
i
▶ Let’s say x ∗ is a solution to this problem. That is,
x ∗ ∈ arg max pf (x ) − w .x .
x
▶ For input i, the necessary first order condition (FOC) is:

∂ ∂f (x ∗ )
[pf (x ∗ ) − w .x ∗ ] = p − wi ≤ 0
∂xi ∂xi
▶ Otherwise, increase the amount xi and profit increases.

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Analyzing the PMP

▶ In case xi∗ > 0 (interior solution for input i), we must have
(x ∗ )
p ∂f∂x i
− wi = 0.
∗
▶ Rewriting this, we get ∂f∂x
(x )
= wpi .
i
▶ In words, marginal product of input i = price of input i
price of output
∗
▶ As should be obvious to you, if xi > 0, and either wi or p
changes, we would expect xi∗ to change as well.
▶ For any two inputs i and j where xi∗ > 0 and xj∗ > 0, we will
wi
have MRTSi,j = w j
.

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BIG Definitions

Definition
The unconditional factor demand for input i is xi∗ = xi (p, w ).
Definition
The unconditional factor demand is the bundle x ∗ = x (p, w ).

▶ We will usually assume the firm’s PMP has a unique solution, so we can
say these are functions (not correspondences).
▶ However, if many solutions exist, you should be aware that the
comparative statics results we will show are true for all solutions.

Definition
The firm’s supply function is y (p, w ) = f (x (p, w )).
Definition
The firm’s profit function is π (p, w ) = pf (x (p, w )) − w .x (p, w ).

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Properties of Factor Demands

1. Negative effect of own price on factor demand:

∂xi (p, w )
≤0
∂wi
2. Positive effect of output price:

∂xi (p, w )
≥0
∂p
3. Law of supply:
∂y (p, w )
>0
∂p

Proof.
You will prove these.

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More Properties

Theorem
Both x (p, w ) and y (p, w ) are homogenous of degree zero.
Proof.
If you increase both input and output prices by some factor λ > 0
then the PMP becomes:

max (λp) f (x ) − (λw ) .x = max λ [pf (x ) − w .x ]

x x

which clearly is maximized by the same bundle x (p, w ) as the

original problem:
max pf (x ) − w .x
x

Therefore,
y (λp, λw ) = f (x (λp, λw )) = f (x (p, w )) = y (p, w ).

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More Properties

Theorem
The profit function is homogenous of degree 1.
Proof.
For any λ > 0, using the fact that x (λp, λw ) = x (p, w ); we can
write

π (λp, λw ) = (λp) f (x (λp, λw )) − (λw ) .x (λp, λw )

= λpf (x (p, w )) − λw .x (p, w )
= λ [pf (x (p, w )) − w .x (p, w )]
= λπ (p, w )

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Hotelling’s Lemma

Lemma
∂π(p,w ) ∂π(p,w )
∂p = y (p, w ) ≥ 0 and ∂wi = −xi (p, w ) ≤ 0

Proof.
This is a straightforward application of the envelope theorem to
the value function

π (p, w ) = max pf (x ) − w .x
x

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Cost Minimization Problem

▶ The profit maximization problem we have seen is sometimes

called the 1-step profit maximization.
▶ Profit maximization is sometimes better understood in 2
steps.
1. For any output amount y and input price vector w , find the
optimal (cost-minimizing) input bundle. This is the
cost-minimization problem:

min w .x
x
subject to f (x ) ≥ y

2. For a given cost function and output price p, solve

max py − C (w , y )
y

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Cost Minimization Problem

▶ Step 1 is the cost minimization problem:

min w .x
x
subject to f (x ) ≥ y

▶ Notice that w and y are exogenous parameters in this

problem. Output price p is irrelevant here.
▶ For any solution x ∗ to this problem, for any two inputs i and j
where xi∗ > 0 and xj∗ > 0, we will have:

∂f (x ∗ )
∂xi wi
∂f (x ∗ )
=
wj
∂xj

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Cost Function

▶ Solving to the cost-minimization problem gives us:

Definition
The conditional (to y ) factor demand z (w , y ) = x ∗ = arg min w .x
x
subject to f (x ) ≥ y .
Definition
The cost function C (w , y ) is the minimized cost:
C (w , y ) = min w .x subject to f (x ) ≥ y .
x

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Cost Function

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 ▶ Even though we have the cost function as C (w , y ), we think
of the cost function as primarily a function of y .

Definition
For a given cost function C (w , y ) at a given input price vector w ,
the marginal cost is MC (y ) = ∂C ∂y(w ,y )
.

Definition
For a given cost function C (w , y ) at a given input price vector w ,
the average cost is AC (y ) = C (wy ,y ) .

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Profit Maximization

▶ After obtaining the cost function, the second step is to choose

the profit-maximizing output quantity:

max py − C (w , y )
y

▶ If the optimal y ∗ > 0, the necessary FOC for this problem is:

∂C (w , y )
p− =0
∂y
▶ In words, this is the familiar p = MC from ECO101.
▶ However, we have said nothing about the existence of an
optimal y ∗ .

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Second Order Condition

▶ Consider again the profit maximization problem:

max py − C (w , y )
y

▶ In order for y ∗ to be the optimal quantity, the SOC must hold

as well.
2 2
▶ The SOC for this problem is: − ∂ C∂y(w2 ,y ) < 0, or ∂ C∂y(w2 ,y ) > 0.
▶ This means, for the FOC to give us a maximum, the cost
function must be convex in y .
▶ If the cost function is concave, the FOC gives us a minimum.
▶ In fact, if C (w , y ) is concave in y , there is no solution
(optimal to increase y forever).
▶ So in this competitive firm setting, we must have convex cost
function.

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Convex Cost Functions

2
▶ Notice that ∂ C∂y(w2 ,y ) > 0 means the marginal cost, ∂C ∂y
(w ,y )
is
increasing in y .
▶ If the marginal cost is decreasing, then p = MC gives us the
worst outcome for the firm.
▶ It is quite easy to show that if the production function f (x ) is
concave, then the cost function C (w , y ) is convex in y .
▶ In fact, you will prove this yourself.
▶ You should also know that convex cost function results from
decreasing returns to scale.
▶ However, the proof of that requires Euler’s homogenous
function theorem, and is a bit convoluted.

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