Cost Minimisation

Question 1

The firm’s profit maximisation problem can be split into 2 parts:

Answer 1

Must operate at lowest cost for any output

Must choose profit maximising quantity

Question 2

Firm has inputs X₁ X₂, with their factor prices W₁ W₂.

What would the cost minisation expression look like?

Answer 2

Min w₁x₁+w₂x₂

Question 3

Cost minimisation function expression

Answer 3

C(w₁,w₂,y)

Shows minimum cost of producing output y given factor prices W.

Question 4

2 components to illustrate this cost minimisation problem

Answer 4

Isocosts and isoquants

Question 5

Isocosts

Answer 5

Shows all combinations of inputs that have the same given cost (budget constraints essentially)

Question 6

Isocosts equation

Answer 6

w₁x₁ + w₂x₂ = C

Rearrange to make it suitable in graph form (x₂ on y axis)
x₂ = c/w₂ -w₁/w₂ x₁

So isocosts are straight lines with slope -w₁/w₂.

Question 7

How to solve cost minimisation algebraically

(Remember final equation)

Answer 7

Lagrange method.

And we will end up with
w₁/w₂ = MP₁/MP₂

Which is factor price ratio (slope of isocost) = TRS (slope of isoquant) , which is the tangency between isocost and isoquant

Question 8

Fixed proportion for cost minimisation

Answer 8

Recall fixed proportion isoquant:
f(x₁,x₂) = min{x₁,x₂}

Cost minimising way of producing y units is to use y units of each input. (I.e Y labour and Y machines needed)

So:
c(w₁,w₂,y) = (w₁ + w₂)y
LHS just means cost minimisation for y amount of output.

Question 9

What does perfect substitutes isocost look like graphically (pg 11)

Answer 9

Isocot has to touch the isoquant once at the y or x intercept. (They meet at either these points because its means that only one F.O.P is used to produce the output) (similar to green line in cournot duopoly

Question 10

So what is the cost function for perfect substitutes

Answer 10

c(w₁,w₂,y) = min {w₁x₁,w₂x₂} = min {w₁,w₂}y

Question 11

Cost minimisation for Cobb Douglas technology : how do we solve (2 methods)

Answer 11

Min w₁x₁ + w₂x₂ such that
y = x₁ to power a x₂ to power b

1. Solve by TRS=Price ratio (which is tangency between isoquant and isocost)
2. Solve by lagrangean
   Structure should look like this, in order to do F.O.C
   L=w₁x₁ + w₂x₂ - λ(x₁ to the a x₂ to the b - y)

Question 12

method 1: TRS = Price ratio example on slides is simple.

Method 2: lagrane is also a good example work through.

Question 13

So what were the 3 types of cost minimisation

Answer 13

Fixed proportions
Perfect substitutes
Cobb douglas

Question 14

Recall short run costs assumption

Answer 14

One factor is fixed (we let X₂ be fixed so Xbar₂) , so a firm can only choose to adjust the quantity of X₁.

Question 15

Short run cost minimisation expression?

Answer 15

Min w₁x₁ +w₂xbar₂

Such that f(x₁,xbar₂)=y

So w₂Xbar₂ is fixed cost. w₁x₁ is variable cost

Question 16

What is the fixed and variable cost in this expression

Answer 16

So w₂Xbar₂ is fixed cost. w₁x₁ is variable cost

Question 17

Short run cost function (hint: factor demand- not that important )

Answer 17

We look for the smallest quantity of x₁ to produce y.
X₁=Xs₁ (w₁,w₂,Xbar₂,y)

Factor demand depends on factor prices, output, and the fixed amount of the other factor

Cs (y,Xbar₂) =w₁xs₁ (w₁,w₂,Xbar₂,y) +w₂Xbar₂

Question 18

Now move on to cost curves. What assumptions do we make now?

Answer 18

W1 and W2 are fixed, and rewrite cost function as c(y)

Question 19

Total cost expression

Answer 19

c(y) = cv(y) + F

Cv(y) is variable costs
F is fixed cost.

Question 20

What happens to AFC as output increases

Answer 20

Falls as spreads across more units.

Question 21

What happens to AVC as output rises

Answer 21

Could fall initially as more productive, but then diminishing marginal product (gets crowded etc less productive)

Question 22

Marginal cost formulas (2)

Answer 22

Differentiate cost

Or differentiate variable cost (since fixed cost is a parameter which will disappear when differentiated so essentially it is just differentiating only variable costs!)

Question 23

1. Relationship between VC and MC,
2. Relationship between AVC and MC
3. relationship between AC and mC

Answer 23

1.Variable cost is the sum of marginal costs of all units.

2 and 3.
If MC>AVC AVC is increasing (since average goes up)
If MC<AVC AVC is falling (since average goes down)
(Same logic for AC)

MC touches through bottom of AVC and AC curve

Question 24

What happens to costs in long run

Answer 24

All costs are variable in LR so fixed costs dont exist.

Question 25

How does LRAC curve look

Answer 25

Envelope of SR AC curves

Could correspond to a firm choosing a diff number of factories. We pick the curve (optimal factory size) that produces the desired quantity at the lowest AC. (Pg18)

Question 26

Relationship between LRAC and SAC

Answer 26

For any level of output we can find an optimal factor size - corresponding to one SAC curve. (As mentioned last card)

Everywhere else on that same SAC curve corresponds to a different level of output, where a different factory size will be chosen, so each SAC curve only touches LAC once.

Question 27

Can the LRAC join at the lowest point of other SAC curves?

Answer 27

No, it must be at tangency.

Only exception if there is CRS so LRAC would be a straight horizontal line where it touches the lowest points of the SAC’s

Question 28

Draw short run and long run diagram.