Microeconomics Course E

1
MicroeconomicsCourse E

• John Hey

2
This week The Firm

• Tuesday
• Chapter 11 Cost minimisation and the demand for
  factors.
• Wednesday
• Chapter 12 Cost curves.
• Thursday
• Exercise 4 A mathematical exercise on profit
  maximisation.

3
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.

4
Particular cases

• 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.

5
Two dimensions

• The shape of the isoquants depends on the
  substitution between the two inputs. (We call 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.

6
Returns to scale with Cobb-Douglas technology
examples

• 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.

7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
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.

11
Chapters 11, 12 and 13

• We will do the analysis in two stages
• in Chapter 11 we find the optimal quantities of
  the inputs given a level of output.
• in Chapters 12 and 13 we will find the optimal
  quantity of the output.
• (Recall that we are assuming that all prices are
  given.)

12
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.

13
Chapter 11

• 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
• a line with slope w1/w2
• Lets go to Maple

14
Chapter 11

• The optimal combination of the inputs is given by
  the conditions
• The slope of the isoquant at the optimal point
  must be equal to to the relative prices of the
  two inputs.
• (this assumes that the isoquants are strictly
  convex)
• The output must be equal to the desired output.

15
Factor demands with CD technology
16
Factor demands with CRS C-D

• The production function
• y q1a q2b where a b 1
• The factor demands
• q1 y (aw2/bw1)b
• q2 y (bw1/aw2 )a

17
Chapter 11

• What do we note?
• The demand curve for an input is a function of
  the prices of the inputs and the desired output.
• The shape of the function depends upon the
  technology.
• From the demands we can infer the technology of
  the firm.

18
Compito a casa/Homework

• CES technology with parameters c10.4, c20.5,
  ?0.9 and s1.0.
• The production function
• y ((0.4q1-0.9)(0.5q2-0.9))-1/0.9
• I have inserted the isoquant for output 40 (and
  also that for output60).
• 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 33.5837.54 70.92.

19
(No Transcript)
20
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 y40
• w1 3 w2 1 y40
• w1 1 w2 1 y60
• w1 2 w2 1 y60
• w1 3 w2 1 y60
• Put the results in a table.

21
Chapter 11

• Goodbye!

22
(No Transcript)