1. Chapter 3 Profit and costs1 CHAPTER 3 Profit maximisation, input demand, output supply and duality

2. Chapter 3 Profit and costs2 Before going into detail 1.Producers/firms make decisions that fit them best. That also holds for producers in rural areas. 2.Decisions result in output supply, input demand and reward for the quasi-fixed inputs (labour, capital, land) 3.If producers realize relative low rewards for quasi-fixed inputs, then it is to be expected that those quasi-fixed inputs shift to other production processes (may take time!).

3. Chapter 3 Profit and costs3 Overview and basic concepts Cost minimisation (output given) –Cost function –Compensated demand functions –(Marginal cost function, supply and producer surplus) Profit maximisation –Profit function –Uncompensated supply and demand functions

4. Chapter 3 Profit and costs4 Profit maximisation Profits = total revenues of output – total costs of variable inputs Marginal revenue = Marginal costs

5. Chapter 3 Profit and costs5 Profit maximisation: overview Multiple outputs (1,..,m) Multiple inputs: −Variable inputs 1,..,r −Quasi fixed inputs 1,..,s Short run optimisation of profits under technology constraint (transformation function) given level of quasi-fixed inputs:

6. Chapter 3 Profit and costs6 Cost minimisation Cost minimisation: subject to:

7. Chapter 3 Profit and costs7 Cost minimisation Optimisation under restrictions (Lagrange): Graphical analysis (next slide)

8. Chapter 3 Profit and costs8 Effect of a price increase of input x 1 Optimal input changes from (x 1 *,x 2 * ) to (x 1 ’,x 2 ’ ) Cost minimisation

9. Chapter 3 Profit and costs9 Cost minimisation Compensated demand function: Restrictions of demand functions –Homogeneous of degree zero in prices –Decreasing in own price –Symmetry restriction

10. Chapter 3 Profit and costs10 Cost function Minimum cost to produce a certain level of y, given input prices and quasi fixed inputs

11. Chapter 3 Profit and costs11 Cost function Minimum cost to produce a certain level of y, given input prices and quasi fixed inputs Characteristics cost function Increasing in input prices Linear homogeneous in input prices

12. Chapter 3 Profit and costs12 Cost function Concave in input prices Increasing in output Symmetry condition Shephard’s Lemma: (downward sloping input demand!!)

13. Chapter 3 Profit and costs13 Profit maximisation with a cost function Apply the principle marginal costs = marginal revenue (=price)

14. Chapter 3 Profit and costs14 Profit maximisation with cost function Solve for y (inverse supply curve): Marginal cost function = inverse supply Area below supply curve: Total variable costs Area between supply curve and price line: producer surplus (also called restricted or variable profit)

15. Chapter 3 Profit and costs15 Profit maximisation with cost function Illustration by means of a figure

16. Chapter 3 Profit and costs16 One stage profit maximisation subject to: Uncompensated demand and supply functions:

17. Chapter 3 Profit and costs17 Profit function Substitution of demand and supply equation in profit equation leads to profit function: is maximum profits, given prices and quasi fixed inputs

18. Chapter 3 Profit and costs18 Profit function Characteristics Convex in all prices Increasing in output prices Decreasing in input prices

19. Chapter 3 Profit and costs19 Profit function Linear homogeneous in all prices Symmetry conditions

20. Chapter 3 Profit and costs20 Profit function Hotelling’s Lemma The shadow prices of quasi-fixed inputs

21. Chapter 3 Profit and costs21 Dual approach profit function It is also possible to START with a specification of a profit function that meets the theoretical requirements, and subsequently derive demand and supply equations. This is known as the DUAL APPROACH. (= uncompensated supply)

22. Chapter 3 Profit and costs22 Dual approach profit function (= uncompensated input demand) (upward sloping supply curve) (downward sloping demand curve) shadow price quasi fixed input k

23. Chapter 3 Profit and costs23 Shadow prices of quasi-fixed input (VMP L ) Land –Agriculture –Forestry –Nature preservation –Recreation –Residential area How to approach? Labour –Agriculture –Handicraft work –Non-farm work –Migration Same framework?

24. Chapter 3 Profit and costs24 Dual approach profit function Continuing with shadow prices (with two processes with profit functions  1 and  2 ): and

25. Chapter 3 Profit and costs25 Dual approach profit function This provides information on the directions where quasi-fixed inputs move between processes. Direct calculation of welfare effects:

26. Chapter 3 Profit and costs26 Conclusions Cost and profit functions are powerful tools They contain a lot of information about the behaviour (not about location!) Several functions can be derived from the cost function or the profit function: they make sense in economic analysis Theory provides many restrictions which should hold under the assumptions of cost minimisation or profit maximisation Shadow prices might be the most relevant concept; in particular of land

27. Chapter 3 Profit and costs27 Main Characteristics Production function increasing in inputs concave in inputs Cost function Input demand increasing in input prices decreasing in input prices concave in input prices increasing in output linear homogeneous in input prices homogeneous of degree zero in prices symmetry Profit function Input demand Output supply decreasing in input prices convex in all prices increasing in output prices linear homogeneous in all prices homogeneous of degree zero in prices symmetry