1. Microeconomics Course 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 (q 1,q 2 ) of the quantities of the inputs) for which the output is constant. Also the production function: y = f (q 1,q 2 ) where y denotes the output. An isoquant is given by: y = f (q 1,q 2 ) = 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(q 1,q 2 ) = q 1 0.4 q 2 0.6 Constant returns to scale. Case 2: f(q 1,q 2 ) = q 1 0.3 q 2 0.45 Decreasing returns to scale. Case 3: f(q 1,q 2 ) = q 1 0.6 q 2 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.

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. w 1 and w 2 for the prices of the inputs. q 1 and q 2 for the quantities of the inputs. We define an isocost by w 1 q 1 + w 2 q 2 = constant …a line with slope –w 1 /w 2 Let’s 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= q 1 a q 2 b where a + b =1 The factor demands: q 1 = y (aw 2 /bw 1 ) b q 2 = y (bw 1 /aw 2 ) 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 c 1 =0.4, c 2 =0.5, ρ=0.9 and s=1.0. The production function: y = ((0.4q 1 -0.9 )+(0.5q 2 -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 w 1 = 1 and w 2 = 1 for the inputs. The optimal combination: q 1 = 33.38 q 2 = 37.54 and the cost = 33.58+37.54 = 70.92.

20. What you should do Find the optimal combination (either graphically or otherwise) and the (minimum) cost to produce the output for the following: w 1 = 2 w 2 = 1 y=40 w 1 = 3 w 2 = 1 y=40 w 1 = 1 w 2 = 1 y=60 w 1 = 2 w 2 = 1 y=60 w 1 = 3 w 2 = 1 y=60 Put the results in a table.

21. Chapter 11 Goodbye!