Lecture # 12a Costs and Cost Minimization Lecturer: Martin Paredes

1. Lecture # 12a Costs and Cost Minimization Lecturer: Martin Paredes

2. Outline • Long-Run Cost Minimization (cont.) • The constrained minimization problem • Comparative statics • Input Demands • Short Run Cost Minimization

3. Corner Solution Example: Linear Production Function • Suppose : Q(L,K) = 10L + 2K • Suppose: Q0 = 200 w = € 5 r = € 2 • Which is the cost-minimising choice for the firm?

4. Example (cont.): • Tangency condition • MRTSL,K = MPL = 10 = 5 MPK 2 • w = 5 r 2 • So the tangency condition is not satisfied

5. Example: Cost Minimisation: Corner Solution K Isoquant Q = Q0 L

6. Example: Cost Minimisation: Corner Solution K Isoquant Isocost line L

7. Example: Cost Minimisation: Corner Solution K Direction of decrease in total cost L

8. Example: Cost Minimisation: Corner Solution K Cost-minimising choice A • L

9. Changes in Input Prices • A change in the relative price of inputs changes the slope of the isocost line. • Assuming a diminishing marginal rate of substitution, if there is an increase in the price of an input: • The cost-minimising quantity of that input will decrease • The cost-minimising quantity of any other input may increase or remain constant

10. Changes in Input Prices • If only two inputs are used, capital and labour, and with a diminishing MRTSL,K : • An increase in the wage rate must: • Decrease the cost-minimising quantity of labor • Increase the cost-minimising quantity of capital. • An increase in the price of capital must: • Decrease the cost-minimising quantity of capital • Increase the cost-minimising quantity of labor.

11. K Example: Change in the wage rate Q0 0 L

12. K Example: Change in the wage rate A • Q0 -w0/r 0 L

13. K Example: Change in the wage rate B • A • Q0 -w1/r -w0/r 0 L

14. Changes in Output • A change in output moves the isoquant constraint outwards. • Definition: An expansion path is the line that connects the cost-minimising input combinations as output varies, holding input prices constant

15. K Example: Expansion Path with Normal Inputs TC0/r • Q0 L TC0/w

16. K Example: Expansion Path with Normal Inputs TC1/r • TC0/r • Q1 Q0 L TC0/w TC1/w

17. K Example: Expansion Path with Normal Inputs TC2/r TC1/r • • TC0/r Q2 • Q1 Q0 L TC0/w TC1/w TC2/w

18. K Example: Expansion Path with Normal Inputs TC2/r Expansion path TC1/r • • TC0/r Q2 • Q1 Q0 L TC0/w TC1/w TC2/w

19. Changes in Output • As output increases, the quantity of input used may increase or decrease Definitions: • If the cost-minimising quantities of labour and capital rise as output rises, labour and capital are normal inputs • If the cost-minimising quantity of an input decreases as the firm produces more output, the input is an inferior input

20. K Example: Labour as an Inferior Input TC0/r • Q0 L TC0/w

21. K Example: Labour as an Inferior Input TC1/r • TC0/r • Q1 Q0 L TC0/w TC1/w

22. K Example: Labour as an Inferior Input Expansion path TC1/r • TC0/r • Q1 Q0 L TC0/w TC1/w

23. Input Demand Functions Definition: The input demand functions show the cost-minimising quantity of every input for various levels of output and input prices. L = L*(Q,w,r) K = K*(Q,w,r)

24. Input Demand Curves Definition: The input demand curve shows the cost-minimising quantity of that input for various levels of its own price. L = L*(Q0,w,r0) K = K*(Q0,w0,r)

25. K Example: Labor Demand Q = Q0 0 L w L

26. K Example: Labor Demand • Q = Q0 w1/r 0 L w • w1 L L1

27. K Example: Labor Demand • • Q = Q0 w2/r w1/r 0 L w • w2 • w1 L L2 L1

28. K Example: Labor Demand • • • Q = Q0 w3/r w2/r w1/r 0 L w • w3 • w2 • w1 L L3L2 L1

29. K Example: Labor Demand • • • Q = Q0 w3/r w2/r w1/r 0 L w • w3 • w2 • w1 L*(Q0,w,r0) L L3 L2 L1

30. Input Demand Functions Example: • Suppose : Q(L,K) = 50L0.5K0.5 • Tangency condition • MRTSL,K = MPL = K = w MPK L r => K = w . L r => This is the equation for expansion path

31. Input Demand Functions Example (cont.): • Isoquant Constraint: • 50L0.5K0.5 = Q0 => 50L0.5(wL/r)0.5 = Q0 => L*(Q,w,r) = Q . r 0.5 50 w K*(Q,w,r) = Q . w 0.5 50 r ( ) ( )

32. Input Demand Functions Example (cont.): • So, for a Cobb-Douglas production function: • Labor and capital are both normal inputs • Each input is a decreasing function of its own price. • Each input is an increasing function of the price of the other input

33. Short-Run Cost Minimisation Problem Definition: The firm’s short run cost minimization problem is to choose quantities of the variable inputs so as to minimize total costs… • given that the firm wants to produce an output level Q0 • under the constraintthat the quantities of some factors are fixed (i.e. cannot be changed).

34. Short-Run Cost Minimisation Problem • Cost minimisation problem in the short run: • Min TC = rK0 + wL + mM subject to: Q0=F(L,M,K0) • L,M • where: M stands for raw materials • m is the price of raw materials • Notes: • L,M are the variable inputs. • wL+mM is the total variable cost. • K0 is the fixed input • rK0 is the total fixed cost

35. Short-Run Cost Minimisation Problem • Solution based on: • Tangency Condition: MPL = MPM w m • Isoquant constraint: Q0=F(L,M,K0) • The demand functions are the solutions to the short run cost minimization problem: Ls = L(Q,K0,w,m) Ms = M(Q,K0,w,m)

36. Short-Run Cost Minimisation Problem • Hence, the short-run input demands depends on plant size (K0). • Suppose K0 is the long-run cost minimizing level of capital for output level Q0… … then, when the firm produces Q0, the short-run input demands must yield the long run cost minimizing levels of both variable inputs.

37. Summary Opportunity costs are the relevant notion of costs for economic analysis of cost. The input demand functions show how the cost minimizing quantities of inputs vary with the quantity of the output and the input prices. Duality allows us to back out the production function from the input demands.

38. Summary The short run cost minimization problem can be solved to obtain the short run input demands. The short run input demands also yield the long run optimal quantities demanded when the fixed factors are at their long run optimal levels.