1. Intermediate Microeconomics WESS16
Cost and cost minimization
Laura Sochat

2. Sunk versus non sunk costs
Another important concept for the firm’s decision making. It will enable the firm to assess the costs that the decision will actually affect.
Sunk costs refer to the costs which the firm has already incurred. They therefore cannot be avoided by the firm.
Non sunk costs refer to the costs which would only be incurred should a particular decision was taken. They can therefore be avoided by not taking such a decision.

3. Cost minimization in the long run
Let’s go back to an important decision the firm needs to take- Out of all possible combinations of inputs possible to produce a given amount of output, which one will be the cost minimizing combination?
The long run here refers to the situation where the firm is able to vary the quantities of all the inputs it uses. What can we say of the costs associated with a long run decision for the firm?
The short run will refer to a situation where the firm faces some constraints as it will not be able to vary the quantities of all inputs.
Suppose the firm uses both Labor and capital . The price of labor is the wage rate 𝑤 and the price of capital is 𝑟. The firm has to produce amount 𝑄 of output over the next year. The firms costs of production are such that:
𝑇𝐶=𝑤𝐿+𝑟𝐾

4. Isocost lines
An isocost line represents a set of combination of labor and capital that have the same total cost for the firm.
K, Capital
L, Labor
𝑇𝐶 2 𝑟
𝑇𝐶 1 𝑟
𝑇𝐶 3 𝑟
𝑇𝐶 1 𝑤
𝑇𝐶 2 𝑤
𝑇𝐶 3 𝑤
For a cost level of £1m, and assuming that the price of labor, 𝑤=£10 and the price of capital is such that 𝑟=£20.
What is the equation describing the £1m isocost line?
More generally, what is the equation for an isocost line.
Slope: − 𝑤 𝑟

5. Graphical representation of the long run cost minimization problem
K, Capital
L, Labor
𝑇𝐶 1 𝑟
𝑇𝐶 2 𝑟
𝑇𝐶 1 𝑤
𝑇𝐶 2 𝑤 D A B C
𝑄 isoquant
Point D is off the isoquant
Point B and C are technically efficient, but not cost minimizing
Point A is technically efficient and cost minimizing
At point A: The slope of the isoquant = The slope of the isocost.
𝑀𝑅𝑇𝑆 𝐿,𝐾 = 𝑤 𝑟
𝑀𝑃 𝐿 𝑀𝑃 𝐾 = 𝑤 𝑟

6. Cost minimization using Lagrangean method
We can re write the cost minimization problem of the firm chosing Labor and Capital to minimize its cost such as:
min 𝐿,𝐾 𝑤𝐿+𝑟𝐾
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:𝑓 𝐿,𝐾 =𝑄
Defining the Lagrangean function such as
Λ 𝐿,𝐾,𝜆 =𝑤𝐿+𝑟𝐾−𝜆(𝑓 𝐿,𝐾 −𝑄)
𝑀𝑃 𝐿 𝑀𝑃 𝐾 = 𝑤 𝑟 ; f L,K =Q

7. Corner point solutions
The cost minimizing input combination happens at a point where the firm uses no Capital:
K, Capital
L, Labor B A C
𝑄 isoquant
𝑀𝑃 𝐿 𝑀𝑃 𝐾 = 𝑤 𝑟 does not hold at any point along the isoquant. More particularly, the isocost line is flatter than the isoquant at every point.
The situation is such that:
𝑀𝑃 𝐿 𝑀𝑃 𝐾 > 𝑤 𝑟
𝑀𝑃 𝐿 𝑤 > 𝑀𝑃 𝐾 𝑟
Every dollar spent on labor is more productive than every dollar spent on Capital.

8. Comparative statics: A change in input prices
Suppose that the price of capital, 𝑟 , and the quantity of output, 𝑄, are both held fixed. The figure below shows the effect of an increase in the price of labor, 𝑤.
K, Capital
L, Labor A B
𝐼 1
𝐼 2
𝐾 2
𝐾 1
𝐿 2
𝐿 1
𝑄 isoquant
As the price of labor increases, the isocost line becomes steeper.
With diminishing marginal rate of technical substitution of labor for capital, the new optimal point is farther up the isoquant.
The firm now uses more capital and less labor, as the relative price of labor has increased.
Note two important assumptions needed for those results:
The firm was already using positive amounts of both inputs
The isoquants are smooth

9. Comparative statics: A change in output
Suppose that now the price of both inputs stays unchanged but that the level of output the firm wants to produce first changes to 𝑄=200, from an initial level of 𝑄=100 and subsequently increase further, to 𝑄=300.
K, Capital
L, Labor A B C
𝐾 3
𝐾 2
𝐾 1
𝐿 2
𝐿 1
𝑄=100
𝐿 3
𝑄=200
𝑄=300
Expansion path
The optimal combinations of inputs move north east as the quantity of output increases.
The line linking the three different optimal bunbles is called the expansion path.
Both capital and labor are normal goods.

10. Using comparative statics to derive input demand curvesB B’ C’
L, Labor
K, Capital
𝑤, dollar per unit of Labor £1 £2
The top graph shows both the effect of a change in the price of labor, and the effect of a change in output
The bottom graph summarises the implications of the re optimisation of the firm: The labor demand curve
As the quantity of output changes (and both the price of labor and capital stay unchanged), the labor demand curve shifts upwards.

11. Costs in the short run: Fixed and variable costs
In the short run, the firm faces constraints in its ability to vary the quantity of some inputs. We will consider a case where the amount of capital the firm can use is fixed in the short run. We can rewrite the firm’s total cost such that:
𝑇𝐶=𝑤𝐿+𝑟 𝐾
Where 𝐾 represents the fixed amount of capital.
The cost of labor constitues the firm’s total variable cost- It will vary as the firm chooses to produce more or less output.
The cost of capital constitutes the firm’s total fixed cost- It will not vary as the firm produces more or less output.

12. Cost minimization in the short run
The firm’s only technically efficient combinaition of inputs occurs at point B, where it uses the minimum amount of labor which allows the firm to produce an amount 𝑄 of output, in combination to the fixed quantity of capital.
K, Capital
L, Labor A 𝐾
𝑄 Isoquant B
In the short run, the firm cannot substitute between the two inputs: The optimal combination of inputs does not involve a tangency condition.

13. Cost minimization in the short run
The short run combination of inputs usually differs from the combination of inputs used in the long run: The firm typically operates at higher costs in the short run.
K, Capital
L, Labor A B C 𝐾
𝑄 1 Isoquant
𝑄 2 Isoquant
𝑄 3 Isoquant
Short run expansion path D E
Consider point B in the figure to the left. In the long run, the firm will choose to produce at point B (it chooses freely between Labor and Capital). In the short run, the firm can only use an amount 𝐾 of capital to produce a level 𝑄 2 of output: In this case, the firm incurs the same cost in the long run and in the short run.