Production & Cost

Question 1

What is the problem a firm solves?

Answer 1

(select quantity of factors to maximise) profit = revenue - cost

Question 2

What does a production function represent?

Answer 2

How much output a firm can produce with a given amount of inputs (not ordinal), capturing their technology (ability to convert inputs to output)

Question 3

How can a technology be classified by its returns to scale?

Answer 3

f(tx, ty) < tf(x, y) for all t > 1: DRS
f(tx, ty) = tf(x, y) for all t > 1: CRS
f(tx, ty) > tf(x, y) for all t > 1: IRS
Can be defined globally or marginally
Would still expect f to be weakly increasing in each input

Question 4

What is the marginal product?

Answer 4

How much a single factor contributes to output
MP₁(x₁, x₂) = ∂f(x₁, x₂) = f₁(x₁, x₂)

Question 5

What is the usual behaviour of marginal product?

Answer 5

Diminishes beyond a certain level of production
f₁₁(x₁, x₂) < 0

Question 6

What can be said about the Cobb-Douglas production function f(x, y) = x^ay^b with a, b > 0?

Answer 6

Returns to scale constant when a + b = 1, decreasing when <, increasing when >
Marginal product of x is diminishing when a < 1 (even under IRS)

Question 7

What is the relationship between the marginal products and unit prices of factors at an optimal interior solution?

Answer 7

MP₁/w₁ = MP₂/w₂

Question 8

What is the q-isoquant of f?

Answer 8

The set of all input bundle (x, y) which would produce exactly q units of output
{(x, y) | f(x, y) = q}

Question 9

What is the TRS?

Answer 9

The Technical Rate of Substitution captures how much of y must be substituted for x to still produce q
The TRS is also the slope of the isoquant
TRS_{1, 2}(x₁, x₂) = limit of Δ₂/Δ₁ as Δ₁ –> 0 = -MP₁/MP₂ = -w₁/w₂

Question 10

Does the condition TRS_{1, 2}(x, y) = -w₁/w₂ find the cheapest bundle of inputs to produce f(x, y)?

Answer 10

Not always, there may be corner solutions and it could find the most expensive bundle or neither most nor least expensive bundle

Question 11

What is the profit maximising choice of inputs?

Answer 11

Maximising profit means maximising revenue - cost so maximising pf(x, y) - (w₁x + w₂y)
Interior solution requires pf_x = w, otherwise profit could be increased

Question 12

What is the cost function?

Answer 12

The function which finds the lowest possible cost of producing q units of output
c(w₁, w₂, q) solves min w₁x₁ + w₂x₂ wrt w₁, w₂ such that f(x₁, x₂) = q

Question 13

What are the steps for profit maximising?

Answer 13

Using knowlegde of the technology and factor prices, minimise costs, i.e. find c(q), then using knowledge of demand find the optimal level of production q*
max_q pq - c(q) so interior optimum will have p = MR = MC

Question 14

What is the relationship between the cost function and production function?

Answer 14

A CRS technology will have c(q) = qc(1), i.e. a linear cost function, AC = c(1)
IRS: c(q) < qc(1), i.e. cost function grows slower than linearly in output, AC decreasing
DRS: c(q) > qc(1) i.e. cost function grows faster than linearly, AC increasing

Question 15

How are the costs a firm pays broken up?

Answer 15

Total Cost = Fixed Cost + Variable Cost
Fixed costs need to be paid regardless of how much is produced
c(q) = c_v(q) + F

Question 16

What are marginal and average cost and how do they relate?

Answer 16

Marginal cost if the additional cost of producing an extra unit: MC(q) = c’(q) = c_v‘(q)
Average cost is the cost per-unit: AC(q) = c(q)/q
At the minimum of AC, AC = MC
At the minimum of AVC, AVC = MC (also true at q = 0)

Question 17

What is the difference between the short and long run?

Answer 17

In the short run only some factors are variable, in the long run all factors can be varied

Question 18

What is the relationship between short and long run AC curves?

Answer 18

Varying the fixed factor (usually K) continuously, the LAC curve traces out the minima of the SAC curves