production analysis

input in the long run

all factors of production can be flexible

inputs in the short run

only one input may be flexible

production function

Q = F(L,K)

cobb douglas production function

Q = F(L,K) = L^α K^β

inputs have a degree of substitutability

linear production functions

Q=F(L,K)=αL+βK

inputs are perfect substitutes

leontief production function

Q(L,K)=min⁡{αL,βK}

inputs must be used in fixed portions (perfect compliments)

short run production function

only one input is changing so Q=F(L,K ̅ )

average production

AP_L = Q(L,K)/L

marginal product

MP_L = ΔQ/ΔL

as one input changes it will impact the MP of both goods if L increases MPL will decrease and MPK will increase

marginal product of labour

dQ/dL

diminishing marginal returns

the next unit will not be as productive as the last

profit maximising input

occurs when the value of the marginal product is equal to the cost of the input

value of marginal product

P * MP_L

keep hiring until the benefit is equal to the cost

assumptions and issues of production functions

• full efficiency is assumed, but this is not realistic

• estimations are hard

• need inputs to make outputs

isoquants

like indifference curves, shows the efficient combinations of labour and capital that can produce the same level of output, the slope is the marginal rate of technical substitution

marginal rate of technical substitution

MRTS = MP_L / MP_K

how much of one input needs to increase to decrease the other input by one, maintaining the same output

isoquant - cobb douglas Q=L^α K^β

isoquant - perfect substitutes Q= αL+βK

isoquant - perfect compliments Q(L,K)=min⁡{αL,βK}

returns to scale

increasing → increasing by x% output increase more than x%

decreasing → increasing by x% output increases but less than x%

constant → increase by x% means an equal output increase by x%

isocosts

shows every combination of inputs that yield the same cost

C (L,K) = wL + rK

r → rental

w → wage

cost minimisation

firms produce in the cheapest way possible

min wL + rK (subject to) Q = f(L,K)

tangency point between the lowest isocost and isoquant

the point where

MRTS = w/r

MPL / MPK = w/r

cost of an input causes isocost to

pivot and will no longer hold MPL/MPK = w/r

what happens to change in input costs if inputs are perfect substitute

• inputs must be able to be used interchangeably eg. short and long wood or humans and robots

• must find cheapest possible output

• happy to have corner solutions

corner solutions

to minimise costs will only use one of the substitutable inputs and ignore the other

change in costs with perfect compliments

• need to have a given ration eg recipes

• need to find the minimum combination of goods that can achieve the output

types of costs

• total

• variable → vary with output

• fixed costs → do not vary with output

• marginal costs → how much do costs increase to increase output by 1

cost functions

how much does it cost to produce at a given amount

eg TC (Q) = 2Q² + Q + 5

VC = 2Q² + Q

FC = 5

deriving the cost function

1. find tangency conditions

2. make an input the subject and sub into output

3. rearrange for output

4. find both inputs in terms of output

5. sub to the TC function

multiple product cost functions

C(Q_1,Q_2 )=α+βQ_1 Q_2+δQ_1^2+γQ_2^2

C(Q_1,0)+C(0,Q_2 )>C(Q_1,Q_2 )

• some inputs can be used in both goods

cost complementarity

the cost of producing good 1 declines with and increase in production of good 2

• take derivative with respect to one output then take that with respect to the other output