Download Lecture 12: Cost curves - User Web Areas at the University of York

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
Long run
Short run
Total cost
●
●
Marginal cost
●
●
Average cost
●
●
• 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.
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!