Microeconomics 2

John Hey

Health Warning

• There is a LOT of detail in the Maple html file.• Do not be swamped by this detail.• Stand back and try and understand intuitively

the properties and relationships that we are exploring.

• You will not be examined on the detail, nor do you have to remember the detail...

• ... but you are expected to understand the properties and relationships.

Properties and relationships

Cost functions

•In all cases when we talk about ‘cost’ (of producing any level of output) we mean the minimum cost (Chapter 11).•This lecture talks about the properties of the entries, and the relationships between the rows and the columns.•The long run is when all inputs (factors) are variable and the short run is when one factor (‘capital’) is fixed.

Long run Short run

Total cost ● ●

Marginal cost ● ●

Average cost ● ●

What you should take away from this lecture

• We are going to talk about cost functions.• There are two total cost functions (the minimum total cost of

producing a given output) – one in the long run (when both factors are variable) and one in the short run (when one factor is fixed). These are the two columns in the table above.

• For each of these we can define two other cost functions: the marginal cost function and the average cost function. So we have three altogether. These are the three rows in the table above.

• We can derive the marginal and the average from the total.• We can derive the total from the marginal or the average.• There are important envelope properties relating the various

functions.• The shape of the total long run cost function depends on returns

to scale and the technology. But we are not going to be specific.

Why are we doing this?

• Recall “HOW?” and “HOW MUCH?” ?• …in Chapter 11 we found the optimal quantities of

the inputs – given a level of output. (the “how?”)• …in Chapter 13 we will find the optimal quantity of

the output.... (the “how much?”)• ... the key to which is the cost function – which is the

thing which we explore today – is a function of the level of output.

• The total cost function tells us the cheapest cost of producing that output.

• And all of this will tell us about how to empirically measure surpluses/profits.

Powerpoint and Maple html

• In this lecture, I start with the PowerPoint presentation - which largely contains notation and definitions of various kinds of cost functions that we will need later in the lecture.

• In the Maple html file I present a large number of examples showing the form of, and relationships between, these various cost functions. You should develop intuition about these. You do not need to remember them.

• We start with notation and definitions.

What we know from Chapter 11

• The demand for the factors (inputs) depends on the technology...

• ... an increase in a factor price leads to a decrease in the demand for that factor.

• We also know that the total (minimum) cost for producing a given level of output y is increasing in y.

• Call this function C(y). This is the (total) cost function.

What follows from Chapter 11

• Denote this cost function C(y).

• C(y) is an increasing function of y.

• C(0) = 0. (in the long run)

• We also know that its shape depends upon the returns to scale.

• C(y) is linear with constant returns.

• C(y) is convex with decreasing returns.

• C(y) is concave with increasing returns.

Chapter 12: Cost curves

• We have been talking about THE TOTAL (minimum) COST C(y).

• So far we have been assuming that the firm can choose the amounts of both factors.

• We call this the LONG RUN.

• However in the SHORT RUN one of the factors (capital) is fixed.

The short run

• One of the factors (input 2 – capital) is fixed.

• We can still talk about the cost of producing a given level of output C(y).

• The firm has no choice – with q2 fixed there is only one value of q1 that produces y.

• The function C(y) is necessarily increasing and convex (with decreasing returns to a factor) and clearly C(0) is equal to the cost of the fixed factor (the fixed cost).

• This is the short run total cost curve.

Three kinds of cost – in both the long and the short run

• TOTAL (minimum) COST – already discussed.• MARGINAL COST – the rate at which total cost

increases with output – hence the slope of the total cost curve.

• AVERAGE COST – the cost for every unit produced: hence the slope of the line from the origin to the total cost curve.

• We have all three kinds of cost curves in both the long and the short run.

• Let us go to the Maple html file...

From the total cost curve to the marginal cost curve and back

• The marginal cost curve is the slope of the total cost curve...

• ... hence the total cost curve is the area under the marginal cost curve.

• or using mathematical jargon

• The marginal cost curve is the derivative of the total cost curve...

• ... hence the total cost curve is the integral of the marginal cost curve.

Fact 1: long run total costs and returns to scale

Fact 2: the long run total cost curve is the envelope of the short run total cost curves

Fact 3: long run average cost curve is the envelope of the short run average cost curves

Fact 4: relationship of the short run average cost curve and the marginal cost curve

Chapter 12

• Goodbye!