Ch6 Production

Production function

Maximum output from given inputs, given current technology

Short run (production)

Period in which at least one input is fixed

Long run (production)

Period long enough that every input can be varied

Fixed input

Input that cannot practically be varied in the short run

Variable input

Input that can easily be varied within the period

Most likely fixed input: paper, librarians, bookshelves or maintenance workers?

Bookshelves (durable capital)

Marginal product of labour (MP_L)

ΔQ/ΔL: extra output from one more worker

Average product of labour (AP_L)

Q/L: output per worker

If MP_L is above AP_L, AP_L is...

Rising: the marginal worker pulls the average up

MP_L crosses AP_L where?

At the maximum of AP_L

Law of diminishing marginal returns

Adding one input, others fixed, its marginal product eventually falls

6 workers make 40 units; the 7th adds 2. AP_L at 7 workers?

6

q = 0.2L: MP_L and AP_L?

Both 0.2 and constant

Isoquant

All input combinations that produce the same output

Higher-output isoquants lie...

Further northeast

Why isoquants cannot cross

One input bundle would give two different maximum outputs

Why isoquants slope down

Using less of one productive input needs more of the other

Convex isoquant means...

|MRTS| falls as labour increases

Straight-line isoquants

Perfect substitutes, constant MRTS

L-shaped isoquants

Perfect complements, fixed input proportions

MRTS formula

ΔK/ΔL = −MP_L/MP_K

Why MRTS = −MP_L/MP_K

Along an isoquant ΔQ = 0, so MP_L·ΔL = −MP_K·ΔK

Partial derivative: how to treat the other input

As a constant

Partial derivative: added vs multiplied constants

Added constants drop out; multiplied constants stay

Power rule: derivative of L^a

a·L^(a−1)

Derivative of L^0.8

0.8·L^(−0.2)

MRTS shortcut for q = A·L^a·K^b

−(a/b)(K/L)

q = 200·L^0.5·K^0.5, L = 4, K = 16. MRTS?

−4

f = 3K + L: MP_L and MP_K

1 and 3

f = 3K + L: MRTS

−1/3

f = K·L: MP_L and MP_K

K and L

f = K·L: MRTS

−K/L

f = K^0.5·L^0.8: MRTS

−1.6K/L

Returns to scale test

Compare f(λL, λK) with λ·f(L, K)

Cobb-Douglas exponents sum above 1

Increasing returns: doubling inputs more than doubles output

Cobb-Douglas exponents sum equal to 1

Constant returns: doubling inputs exactly doubles output

Cobb-Douglas exponents sum below 1

Decreasing returns: doubling inputs less than doubles output

q = 10·L^0.5·K^0.3: returns to scale?

Decreasing (0.8 < 1)

Inputs double, exponents sum to 0.8: output rises by?

About 74% (2^0.8 ≈ 1.74)

f = K·L: returns to scale?

Increasing (output scales by λ²)

f = 3K + L: returns to scale?

Constant: f(λK, λL) = λ(3K + L)

Returns to scale: short run or long run?

Long run only, since all inputs change

Three sectors of the economy

Private, public, non-profit

What all three sectors need

Revenue to cover costs or fund their mission

Firm lifecycle

Start, growth and maturity, decline

Profit

Total revenue − total cost

Productive (technological) efficiency

Maximum output from given inputs

Allocative efficiency

Best distribution of resources across uses

Is technological efficiency enough for profit maximisation?

No: necessary but not sufficient

Three input categories

Capital, labour, materials

Capital (as an input)

Durable tools, equipment, machinery

Materials (as an input)

Raw inputs, intermediate goods, consumables such as gasoline

Electricity is usually treated as which input?

Materials

AI as an input: agreed classification?

None; capital, labour or material depending on use

Cobb-Douglas production function

Q = A·L^α·K^β

A in Cobb-Douglas

Total factor productivity

α and β in Cobb-Douglas

Output elasticities of labour and capital

Isoquants bunch closer at higher output

Increasing returns to scale

Isoquants spread apart at higher output

Decreasing returns to scale

Typical returns pattern as a firm grows

Increasing, then constant, then decreasing

Beer, tobacco, fabricated metals: returns to scale in that order

Constant, decreasing, increasing

Why fabricated metals show increasing returns

Capital intensity lets output grow faster than inputs

Why tobacco shows decreasing returns

Logistics, disease spread and land limits

Perfect substitute inputs: example

Delivery worker and an equally productive robot

Perfect complement inputs: example

Truck and driver

Curvature of a convex isoquant shows...

How complementary labour and capital are

Where on an isoquant is |MRTS| large?

High capital, low labour

Neutral technical change

Output rises with the capital/labour ratio unchanged

Capital-biased technical change

Production becomes more capital intensive

Organisational change

Shifts the production function, like new technology