Transcript Lecture 11: Minimisation of cost and demand for factors

Microeconomics 2
John Hey
Supply: the firm
• Firms have to decide (at least) two things:
• (1) How to produce and (2) How much to
produce. (If a monopolist or an oligopolist also
the price – chapters 28 and 31).
• How to produce: Chapter 11 - Cost minimisation
and the demand for factors.
• Chapter 12: Cost curves (preliminary to chapter
13)
• How much to produce: Chapter 13 – Supply.
• Note that there is some ‘homework’ at the end of this presentation.
Rating the lectures
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Lecture 10 (technology): no stars
Lecture 11 (how to produce): *
Lecture 12 (cost curves): **
Lecture 13 (supply): ***
Lecture 14 (ppfs): ****
Lecture 15 (prodn & exchange): *****
Lecture 16 (empirics): ****** but tough
Chapter 11
• In Chapter 10 we introduced the idea of an
isoquant – the locus of the points (in the
space (q1,q2) of the quantities of the
inputs) for which the output is constant.
• Also the production function:
• y = f (q1,q2) where y denotes the output.
• An isoquant is given by:
• y = f (q1,q2) = constant.
Particular cases of technology
• Perfect substitutes 1 to a: isoquants are
straight lines with slope a.
• Perfect complements 1 with a: isoquants are
L-shaped and the line joining the corners has
slope a.
• Cobb-Douglas with parameter a: isoquants
are smoothly convex everywhere.
• CES with parameters c1, c2, ρ and s: isoquants
are smoothly convex everywhere.
Two dimensions
• The shape of the isoquants: depends on
the substitution between the two inputs.
(We call the magnitude of the slope of an
isoquant the marginal rate of substitution
between the inputs).
• The way in which the output changes from
one isoquant to another – depends on the
returns to scale.
Returns to scale with CobbDouglas technology : examples
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Case 1: f(q1,q2) = q10.4 q20.6
Constant returns to scale.
Case 2: f(q1,q2) = q10.3 q20.45
Decreasing returns to scale.
Case 3: f(q1,q2) = q10.6 q20.9
Increasing returns to scale.
Note: the ratio of the exponents is the same –
hence the shape of the isoquants is the same –
but they have different returns to scale.
Chapters 11, 12 and 13
• We assume that a firm wants to maximise
its profits.
• We start with a small firm that has to take
the price of its output and those of its
inputs as given and fixed.
• Given these prices, the firm must choose
the optimal quantity of its output and the
optimum quantities of its inputs.
Chapters 11, 12 and 13
• We do the analysis in stages…
• …in Chapter 11 we find the optimal
quantities of the inputs – given a level of
output.
• …in Chapter 12 we develop the idea of a
cost function, which enables us...
• ...in Chapter 13 to find the optimal quantity
of output.
• (Recall that we are assuming that all prices are given.)
Chapter 11
• So today we are finding the cheapest way of
producing a given level of output at given factor
(input) prices.
• This implies demands for the two factors...
• ... which are obviously dependent on the ‘givens’
– namely the level of output and the factor
prices.
• If we vary these ‘givens’ we are doing
comparative static exercises.
• The way that input demands vary depends upon
the technology.
Notation
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We use the following notation:
y for the level of the output.
p for the price of the output.
w1 and w2 for the prices of the inputs.
q1 and q2 for the quantities of the inputs.
• We define an isocost by
• w1q1 + w2q2 = constant
• …this is a line with slope –w1/w2
• Let’s go to Maple…
How to produce
• The optimal combination of the
inputs is given by the condition:
• The slope of the isoquant at the
optimal point must be equal to to
the relative prices of the two
inputs. (MRS=w1/w2)
• This assumes that the isoquants are strictly convex.
• Rather obviously, the output must be equal to the
desired output.
Conclusions
• The demand curve for an input is a
function of the prices of the inputs and
the desired output.
• The shape of the demand function
depends upon the technology.
• From the demand functions we can
infer the technology of the firm.
• Before you go, some homework...
Homework
• CES technology with parameters c1=0.4, c2=0.5,
ρ=0.9 and s=1.0.
• The production function:
• y = ((0.4q1-0.9)+(0.5q2-0.9))-1/0.9
• In the next slide I have inserted the isoquant for
output = 40 (and also that for output=60).
• I have inserted the lowest isocost at the prices
w1 = 1 and w2 = 1 for the inputs.
• The optimal combination: q1 = 33.38 q2 = 37.54
• and the cost = 33.58+37.54 = 70.92.
What you should do
• Find the optimal combination (either graphically
or otherwise) and the (minimum) cost to produce
the output for the following:
• w1 = 2 w2 = 1 y=40
• w1 = 3 w2 = 1 y=40
• w1 = 1 w2 = 1 y=60
• w1 = 2 w2 = 1 y=60
• w1 = 3 w2 = 1 y=60
• Put the results in a table.
Chapter 11
• Goodbye!