65 terms

Econ 4011 - Firms and Production

Blackbox technology that takes inputs and creates outputs
Function from a set of inputs as X and outputs as Y, technology f:

f: X → Y
(3) objects used to describe a firm
(1) output it produces
(2) inputs it requires for that ouput
(3) technology it uses to change inputs into outputs
(2) production input/factors generally focused on by economists
(1) Capital: K
(2) Labor: L

A firm is technology f: (K,L) → Y
Payment to factor
firm uses inputs X = (x₁, x₂, ....xn) and have to pay pi per unit of factor i

Payment to factor i is pi(xi) --- price of factor i times input: the total amount you pay for a certain factor

Total payment to factors is ∑pi(xi)
2 factor economy of capital and labor: payment to factors
(1) Rent - capital factor cost, denoted by r
Payments to capital are rK

(2) Wages - labor factor cost, denoted by w
Payments to labor are wL

Total payments to factors in two-factor economy = rK + wL
Factor share of factor i
Payment to that factor divied by total payments to factors:

si = pj(xj) / ∑pi(xi)
Labor share
Total US production: Y ≈ wL + rK

Labor share s subscript L = wL/Y
Capital Share
Total US production: Y ≈ wL + rK

Labor share s subscript K = rK/Y

Total factors sum to 1 --- capital share = 1 - labor share
Factor ratio of labor and capital
Factor ratio of labor and capital = K/L

reflects how much capital per worker there is, i.e. how many machines per person

larger ratio in one firm or industry identified as more capital intensive
Trends of capital-labor ratio and labor share in the US economy over time
(1) Capital labor share has risen over time - indicating there are now more machine hours per work hour in the US economy

(2) However labor share (wL/Y) has remained constant over time, indicating that the percentage share of output paid to labor is constant. Thus costs of machines must have reduced over time
Production function
another name for firm's technology

mapping from quantities of inputs as X = {x₁, x₂, xn} to the quantity of outputs as Y, technology f:

f: X^n → Y
Perfect substitutes in production

(1) General case

(2) Two-input case

(3) What if one input was twice as productive as another?
(1) f(x₁, x₂, xn) = f(ax₁ + bx₂, ...xn)

Where a and b are scaling factors so that each input is in the same units

(2) Two inputs of labor --- L₁ and L₂

f(L₁, L₂) = L₁ + L₂

(3) What if L₁ workers are twice as productive as L₂ in performing the same tasks?

f(L₁, L₂) = 2L₁ + L₂

i.e. you get twice as much output from L₁ as L₂
Perfect complements in production

(1) General case

(2) Two-input case

(3) What if one input was half as productive as another?
(1) f(x₁, x₂, xn) = f( min{ax₁ + bx₂} ...xn)

Where a and b are scaling factors so that each input is in the same units

(2) Suppose two types of machinery exist that make same output per working hour of machinery:

f(K₁, K₂) = min {K₁ + K₂}

(3) What if K₁ is half as fast as K₂?

f(K₁, K₂) = min {K₁/2 + K₂}
Ex: y = x₁ + min {(x₂ + 3x₃), x₄}

Identify all perfect complements and perfect substitutes
(1) x₂ and x₃ are perfect substitutes

(2) x₄ and choice of bundle (x₂, x₃) are perfect complements

(3) x₁ and bundle of [min {(x₂ + 3x₃), x₄}] are perfect complements

(1) General definition

(2) Formal equation
(1) Curve that shows all the input combinations that lead to the same amount of output

Similar to an indifference curve in utility theory, but for the case of production

(2) Given production function Y = f(X) and target level of production y₀, isoquant is

I(y₀) = {X | f(X) = y₀}

In other words, all combination of inputs X that give you the same level of output y₀

Note: with production functions you cannot use linear transformations since we care about the ultimate output - must leave production functions as they are originally written
Isoquant curves on graph

(1) Perfect Complements

(2) Perfect substitutes
(1) Downward sloping straight line - solve for one input in terms of the desired level of output and the other input

Production function: f(K, L) = aK + bL

K = (y₀ - bL)/ a

(2) "L" shape

Production function: f(K, L) = min {aK + bL}

y₀ = aK if aK < bL
y₀ = bL if aK > bL
Cobb Douglas in production

(1) General case

(2) Two-input case
(1) f(x₁, x₂, xn) = Ax₁^(α₁)x₂^(α₂)*x₂^(α₂)*, ...*xn^(αn))

(2) f(K, L) = AK^α *L^β

Note: in utility theory we could eliminate the "A" coefficient due to monotonic transformation, but in production we care about the units of output so this is not possible

Note: we do not require the share parameters α to sum to one in production theory
Total Factor Production (Cobb Douglas)
Production Function: f(K, L) = AK^α *L^β

A is the total factor productivity (TFP) representing the improvement/efficiency of technology over time

i.e. larger A results in larger input mix
Returns to scale

(1) Constant returns to scale

(2) Increasing returns to scale

(3) Decreasing returns to scale
(1) Constant returns to scale if, ∀λ > 0
f(λx₁, λx₂,....λxn) = λ f(x₁, x₂, ....xn)

i.e. increasing inputs by multiple of λ increasing your output by multiple of λ

(2) Increasing returns to scale if ∀λ > 1
f(λx₁, λx₂,....λxn) > λ f(x₁, x₂, ....xn)

i.e. increasing inputs by multiple of λ increases your output by more than multiple of λ
i.e. doubling your inputs results in more than double your output

(3) Decreasing returns to scale if ∀λ > 1
f(λx₁, λx₂,....λxn) < λ f(x₁, x₂, ....xn)

i.e. increasing inputs by multiple of λ increases your output by less than multiple of λ
i.e. doubling your inputs results in less than double your output
Returns to scale:

(1) Perfect substitutes

(2) Perfect complements
(1) Production function: f(K, L) = aK + bL
Determine return to scale by plugging in λ*input for each:
f(λK, λL) = a(λK) + b(λL)
f(λK, λL) = λ(aK + bL)
f(λK, λL) = λf(K, L)

Constant returns to scale

(2) Production function: f(K, L) = min {aK + bL}
Determine return to scale by plugging in λ*input for each:
f(λK, λL) = min {a(λK) + b(λL)}
f(λK, λL) = λ min {aK + bL}
f(λK, λL) = λ f(K, L)

Constant returns to scale
Returns to scale

Cobb Douglas
Production Function: f(K, L) = AK^α *L^β
Determine return to scale by plugging in λ*input for each:
f(λK, λL) = A(λK)^α *(λL)^β
f(λK, λL) = λ^(α+β) * f(K, L)

If α+β > 1, increasing returns to scale
If α+β < 1, decreasing returns to scale
If α+β = 1, constant returns to scale

Note: typically we like working with the case of constant returns to scale where α+β = 1
What does returns to scale mean? (3)
(1) If returns to scale are decreasing, optimal for firm to be small

(2) If returns to scale are increasing, optimal for firm to be very large

(3) If returns to scale are constant, the size of the firm doesn't matter
Marginal product of factor:

(1) Definition
(2) How to compute?
(1) the additional amount of output a small increase in that factor leads to

(2) Given Production function f(x₁, x₂, xn)

Marginal productivity of factor x₁ =

MP₁ = ∂f/∂x₁

i.e. partial derivative with respect to x₁
Marginal product of factor:

Cobb Douglas - two input case
Production Function: f(K, L) = AK^α *L^β

MPK = α*(Y/K)
MPL = β*(Y/L)

where Y is the total output

Y/K and Y/L are the average products of capital and labor
Marginal product of factor:

(1) Perfect substitutes

(2) Perfect Complements
(1) Production function: f(K, L) = aK + bL
MPK = a, MPL = b

(2) Production function: f(K, L) = min {aK + bL}

Three potential cases for each:
(a) MPK = a if aK < bL
(b) MPK = 0 if aK > bL
(c) MPK = 0 if aK = bL
Marginal Rate of Technical Substitution: two input case
We will think of the rate at which labor can be substituted for capital


i.e. by taking away one unit of capital, how much labor do you need to be indifferent
MRTS for Cobb Douglas
Production Function: f(K, L) = AK^α *L^β

MPK = α*(Y/K)
MPL = β*(Y/L)

MRTS (l for k) = (α/β) * (K/L)

by taking away units of capital, how much more labor do you need to be indifferent

when you have more capital you prefer to substitute labor for capital
Firm profits =
= Revenues - Expenditures

= pY - ∑pii*xi for each factor
Optimal profit function

π*(py, p₁, p₂...) =

py is subscript y - price of output Y
π*(py, p₁, p₂...) = max x₁, x₂ π(Y, x₁, x₂.... xn, py, p₁, p₂...)

s.t. Y = f(x₁, x₂.... xn)


= max x₁, x₂, ...xn pyf(x₁, x₂.... xn) - ∑pi) - ∑pi*xi

- Firms choose inputs so that it makes as much profit as possible, given a set of prices and its production function
- We can predict what the choices of inputs and the amount of output would be after price changes
How to write out optimal profit function and constraint:

Example of a car manufacturer needing 1 engine and 4 wheels for each car

pc is price of the output car, pe is the price of the engine, pw is the price of the wheel
π(pe, pw) = max e,w pcC - pee - pw*w

s.t. C = min{e, ¼w}
With constant returns to scale, what happens if we double the size of the firm?
New profit max problem:
π(py, p₁, p₂..) = max x₁, x₂ py x₂ py* f(2x₁, 2x₂.... 2xn) - ∑pi*2xi

With CRS, the 2 factors out:
π(py, p₁, p₂..) = 2 max x₁, x₂ py x₂ py* f(x₁, x₂.... xn) - ∑pi*xi

So double the set of inputs results in twice as much profit
Optimal Profit: Perfect substitutes: K is capital, r is cost of capital, L is labor, w is wage for labor

(1) Profit function

(2) Intuitive solution
(1) π(r, w) = max K, L (aK + bL) - r*K - w*L

where a and b are constants for production weighting factors

(2) You want to produce more, conditional on the cost of the cheaper good

Re-write maximization problem as follows:

π(r, w) = max K, L (a-r)*K + (b-w)*L

Whichever input is more productive relative to its cost, use all of that input and none of the other since they are perfect substitutes

Optimal factor ratios:

K/L (r,w) =
(1) 0 if a - r < b - w ---- K = 0, L = + for all production
(2) ∞ if a - r > b - w ---- K = +, LL* = 0 for all production
Optimal Profit: Perfect complements: K is capital, r is cost of capital, L is labor, w is wage for labor

(1) Profit function

(2) Intuitive solution
(1) π(r, w) = max K, L min{aK, bL} - r*K - w*L

(2) Getting more of one product without more of the other is wasteful for production --- so you don't want aK > bL or bL > aK

So optimal factor ratio must be:

aK/bL = 1 ---- K/L (r, w) = b/a

Factor ratio is constant - always investing in fixed proportion of inputs, otherwise waste exists
General rule of profit maximization
Factors are paid their marginal products

This means that the "last" unit of capital and labor are productive enough to exactly offset their rental and wage respectively
Profit Maximization: Cobb Douglas

(1) Profit function to maximize

(2) Marginal product of each factor
(1) π(r, w) = max K, L AK^α L^(1-α) - r r*K - w*L

(2) MPK = Aα K^(α-1) L^(1-α) = Aα*(L/K)^(1-α)

MPL = A(1-α) K^α L(-α) = A(1-α)*(K/L)^α
General rule of profit maximization: Cobb Douglas

π*(r, w) = max K, L AK^α L^(1-α) - r*K - w*L
- Decreasing marginal product of each factor --- i.e. production for each product decreases as the input level increases

- Marginal cost of each factor stays the same

- Optimum equates the marginal cost and the price of each input:

In the two input case: MPK = r and MPL = w

or MPK/MPL = r/w

Based on Cobb Douglas:
MPK = Aα(L/K*)^(1-α)
MPL = A(1-α) (K/L*)^α

MPK/MPL = α/(1-α) {L/KK*}

------ set equal to the price ratio ratio and solve for the optimal factor ratio:
K/L (w, r) = α/(1-α) {w/r}

Interpretation: if labor is relatively expensive versus capital - should use more capital
Factor shares for Cobb Douglas

π*(r, w) = max K, L [A* K^α * L^(1-α) ] - r*K - w*L
We know K/L(w,r) = α/(1-α) {w/r} based on profit maximization problem of Cobb Douglas

Re-arrange to determine:
rK/wL = α/(1-α)
------ this is total amount paid to capital / total amount paid to labor

so the share factor of Labor:
s* = wL / (wL + rK) = 1 / [wL + (rK/wL)] = 1 - α

Share of income going to each factor is the same regardless of the prices:

Share of labor: s*L = (1-α)
Share of capital: s*K = α
In an exchange economy, how do we solve for market-clearing prices?
(1) Start with preferences and initial endowments to calculate income of each individual

(2) Preferences allow you to solve for the optimal demand for each good (and individual) as a function of prices and income

(3) Find the prices where the sum of individual demand for one of the goods equals the total market endowment

This will give you the appropriate ratio of prices in equilibrium
(1) General Equilibrium

(2) Exchange economy

(3) Endowment and compensation choices

(4) Rule of equilibrium
(1) Approach to modeling an economy with a distinct number of interrelated markets - whenever anything changes in one section on one side of the market, it changes everything else (i.e. how demand in the consumer sector affects the notebook market)

(2) Consumes trade their goods at fixed prices --- total amount of goods in the economy does not change - interested in reallocation

(3) Each individual starts with a certain endowment of goods

Denote endowment by ωig = (wi₁, wi₂)

Compensation choices by xig = (xi₁, xi₂)

Consumers i = 2 (in this example)
Commodities g = 1, 2

(4) Agents are optimizing and the market clears
Definition of General Equilibrium (4)
An equilibrium of an economy is a set of:
- prices
- allocations for each agent

such that
- all agents allocations are optimal at those prices
- markets all clear
What do we need to do to check an equilibrium?
Need to friend prices and allocations such that the markets clear

Markets clearing implies there is no excess supply or demand for each good in the market
Income of each consumer in general equilibrium
Income is based on initial endowment of each good ---

Price of each good times the endowment of that good
Optimal demand of a good with Cobb Douglas preference
optimal demand = share parameter = [Income / price of good]
Pareto Set
Efficient allocation - want to make one agent better off without making another worse off

This implies

Comparative Statics
Analysis of what happens with small changes to endowments or preferences
Equilibrium: what happens when the ratio of prices (py/px) increases?
We know net positions of both individuals in equilibrium must be zero (i.e. cannot be net buyer of both goods because you're endowment won't allow it)

px(x-x₀) + py(y-y₀) = 0
so, dividing both sides by px:
(x-x₀) + py/px (y-y₀) = 0

So with increase in the price ratio py/px:
(1) Individuals who are net buyers of good y are made worse off
(2) Individuals who are net buyers of good x are better off

Price increases can help people in equilibrium if they are going to sell that good
Transfer paradox
Circumstances in which giving an individual more money in equilibrium can make them worse off

Assume given some initial endowment there is an equilibrium. Consider giving some of Bob's endowment to Annie. This shift will cause a change in the relative prices, and Annie can be made worse off, depending on her net positions of the goods

This implies that it is possible for someone to improve their position by destroying some of their initial endowment
Utility maximization problem for Robinson Crusoe
for individual consumption
Wants to maximize his utility conditional on the number of coconuts he can gather

max u (c,l) subject to c = f(24 - l)

where c is coconuts for consumption
l is hours of leisure
24-l is hours of labor (not leisure
Equilibrium in Robinson Crusoe economy with producer and consumer
(1) wage w (prices)
(2) choice of consumption c and leisure l by the "consumer" and labor demanded z* by the "producer" (allocations)
such that
(3) given the wage rate, both the consumer and the producer are making optimal choices
(4) Markets clear:
24 - l = z and c = f(z)
Optimal decision problem of producer Robinson
Producer Robinson takes wages (w) as given and maximizes profits:

π(w) = max z ≥ 0 f(z) - w w*z

We know that the producer will choose labor so that its paid its marginal product in equilibrium:

w = f ' (z*)
Consumer Robinson: Budget Constraint
(1) Left hand side of the budget constraint
(2) 2 parts of right hand side of the budget constraint
(1) Amount of coconuts to purchase at normalized price of 1

(2) Two parts: Labor income and firm profits paid as dividends

- Labor income = labor hours worked times wage rate
= w*z ------- z = 24-l

Labor income can be written as a decreasing function of leisure w*(24-l)

- Firm profits: consumer Robinson owns full stock in the firm and will receive a dividend equal to the firm's profits

Firm profits = π*(w) --- this is lump sum transfer the consumer can't affect

Final Budget Constraint:

c = w(24 - l) + π*(w)

Increase in w leads to more labor income but less firm profits
Consumer Robinson utility max problem
Consumer Robinson gets utility from consumption c and leisure l:

max U (c,l) subject to c = w(24 - l) + π*(w)

Consumer Robinson can't influence wage rate, so wage rate and firm profits are fixed from his perspective

Re-arrange constraint so that left hand side are things under his control

c + wl = 24w + π*(w)
Social Planner
Individual who can make simultaneous choices to make society the best off
- Best social planner's solution is often called the first-best
- Equilibrium is what happens when prices dictate where resources are allocated
Robinson Crusoe outcome: social planner versus competitive equilibrium
In Robinson Crusoe economy the social planner's solution coincides wit competitive equilibrium outcome

Theorem: in some economies, a competitive price system delivers the first-best outcome (even with no coordination between parties other than optimal price)
Hayek and prices
Concept that being a social planner requires an unbelievable amount of both information and calculation

But in some situations, a small number of prices can convey all the information that any market participants need
Role of prices in Robinson Crusoe economy
- Only price is wage in the economy
- Wage conveys opportunity cost of more unit of leisure for the worker
- Also is the marginal product of one more hour of labor given to the firm
- Prices are fully informative and markets reach first best outcome
Welfare: can governments achieve first-best?
Outcome depends on what the government, know, calculates, and can force people to do

If government knows it can enforce a tax to achieve the appropriate price level to reach the first-best outcome it can achieve

But this requires that the government has more information about appropriate prices than the people making the decisions (producers and consumers)

there is no general principle is economics that indicates whether "markets work best" or "governments work best"

Will be possible for governments to achieve first-best, but very difficult and likely impracticable
(3) Steps for decentralized solution to Robinson Crusoe
(1) Solve the producer's problem for labor demand and profits as a function of the wage

(2) Solve the consumer's problem for labor supply as a function of the wage

(3) Find the wage that sets (1) and (2) equal in equilibrium
Definition of equilibrium in decentralized Robinson Crusoe problem
(1) wages w* (ratio to output price that maximizes): set of prices
(2) Allocations (z, Y) and (c, l) such that each producer and consumer chooses their optimal level of each

such that

(3) Agents are optimizing
(4) Market clears: no excess supply or demand
c*(w*) = Y(w) --- production of firm (Y) equals consumption o(w*) = Y*(w*) --- production of firm (Y) equals consumption of consumer (c)
z = (24 - l) --- labor demanded equals leisure demanded by the consumer
Producers Economy - multiple producers and agents ---- what is equilibrium?

(1) wage rate w (price)
(2) individual consumption ci and aggregate output Y and individual labor supply choices of individuals li and aggregate labor demand L such that
(3) agents are optimizing given this price bundle
(4) markets clear:
Consumption = production --- ∑ci = Y
Labor supply = labor demand --- ∑li = L
How to solve for equilibrium in multiple producers economy
(1) First solve for the firm's decision and the worker's choice given arbitrary prices

(2) Find the prices that solve the market-clearing requirements
Malthus hypothesis regarding relation of capital and wages (starting with a fixed supply of capital that is fixed)
As capital stock remains the same and level supply increases, what happens to marginal product of labor?

Marginal product of labor is decreasing - marginal product of labor is equal to the wage rate

So wages will fall as the population (labor supply) grows with fixed capital
Summary of population growth and technological growth
- Mathus showed that fixed factors make long-term growth impossible in the standard setting, because per-capita profits increase too slowly to make up for decreases in welfare due to falling wages

- Technological growth or human capital investment both provide ways to explain why some countries have grown over time (i.e. increasing TFP A)
Cobb Douglas: f(K, L) = AK^α *L^(1-α)

Optimal factor ratio?
K / L = [ α/(1-α) ] (w / r)