Microeconomics Course E John Hey. This week: The Firm Tuesday Chapter 11: Cost minimisation and the demand for factors. Wednesday Chapter 12: Cost curves.

MicroeconomicsCourse E

John Hey

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.

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

• 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.

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.

Returns to scale with Cobb-Douglas technology : examples

• Case 1: f(q1,q2) = q10.4 q2

0.6

• Constant returns to scale.

• Case 2: f(q1,q2) = q10.3 q2

0.45

• Decreasing returns to scale.

• Case 3: f(q1,q2) = q10.6 q2

0.9

• 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 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.)

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.

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 • Let’s go to Maple…

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.

Factor demands with CD technology

q1 y

1a b

a w2

b w1

ba b

q2 y

1a b

b w1

a w2

aa b

Factor demands with CRS C-D

• The production function:

• y= q1a q2

b where a + b =1

• The factor demands:

• q1 = y (aw2/bw1)b

• q2 = y (bw1/aw2 )a

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.

Compito a casa/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

• 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!