Lecture 4: Production and Costs Reading: Chs 6, 7 in Begg et al or…

1. Lecture 4: Production and Costs Reading: Chs 6, 7 in Begg et al or… Ch 5 in Sloman and Wride AND Notes Ec111, Lecture 4

2. Outline Our “black box” Model of the firm Production (short run – at least one input fixed): Describing production: the production function in algebra and graphs Standard shapes of production functions Costs (short run – at least one input fixed) Describing costs: the cost function in algebra, graphs and tables Production and Costs in the Long Run (all inputs variable) Returns to Scale and Economies of Scale Minimum Efficient Scale Solving for the Cost minimising input combination

3. Our model of the firm inputs outputs production function transforms inputs into outputs technology determines “how”

4. Production in the Short run • Production functions • factors of production (inputs) • labour • land and raw materials • capital • the relationship between inputs and output • TPP = Q = ƒ(F1, F2, F3, … Fn)

5. Short Run Production Functions Example: Q = KL APL = Q/L = KL/L = K; APK = Q/K = KL/K = L MPL = ΔQ/ΔL = ΔKL/ΔL = KΔL/ΔL = K(1) = K; MPK = ΔQ/ΔK = ΔKL/ΔK = LΔK/ΔK = L(1) = L; In each case, one input is fixed!

6. The marginal product of labour The marginal product of labour is the increase in output obtained by adding 1 unit of the variable factor but holding constant the inputs of all other factors. Labour is often assumed to be the variable factor with capital fixed.

7. The law of diminishing returns Holding all factors constant except one, the law of diminishing returns says that: beyond some value of the variable input, further increases in the variable input lead to steadily decreasing marginal product of that input. E.g. trying to increase labour input without also increasing capital will bring diminishing returns.

8. Single Input Production Functions – Total Physical Product Number of workers 0 1 2 3 4 5 6 7 8 TPP 0 3 10 24 36 40 42 42 40 Tonnes of wheat produced per year Number of farm workers

9. Maximum output Diminishing returns set in here Single Input Production Functions – Total Physical Product d TPP Tonnes of wheat produced per year b Number of farm workers

10. APP = TPP / L Total, Average and Marginal Products TPP Tonnes of wheat per year Number of farm workers (L) Tonnes of wheat per year APP Number of farm workers (L) MPP

11. b Diminishing returns set in here b Total, Average and Marginal Products TPP Tonnes of wheat per year Number of farm workers (L) Tonnes of wheat per year APP Number of farm workers (L) MPP

12. Total, Average and Marginal Products c Slope = TPP / L = APP c d TPP Tonnes of wheat per year b Number of farm workers (L) b Tonnes of wheat per year APP d Number of farm workers (L) MPP

13. Our model of the firm Behavioural Assumption: For the moment, cost minimisation – whatever the target output level we choose, we will select inputs to achieve this at least cost. inputs outputs Q0 L0, K0 production function transforms inputs into outputs costs determine which technologically feasible set of inputs to choose technology determines “how”

14. The Meaning of Costs • Opportunity costs • opportunity cost = value of a resource in its best alternative use • The relevant concept for a model of choice • Measuring a firm’s opportunity costs: • factors not owned by the firm: explicit costs • factors already owned by the firm: implicit costs • historic costs – irrelevant • replacement costs – relevant only if replacing something

15. Costs in the Short run • Fixed costs and variable costs • Total costs • total fixed cost (TFC) – does not vary with output • total variable cost (TVC) – does vary with output • TVC and the law of diminishing returns • total cost (TC = TFC + TVC) TVC = 50Q TC=50+50Q Example: TC = 50 + 5Q TFC = 50 TVC = 5Q £ 50 TFC=50 Q

16. Total and Variable Costs £ Output (Q) 0 1 2 3 4 5 6 7 TVC (£) 0 10 16 21 28 40 60 91 TC (£) 12 22 28 33 40 52 72 103 TFC (£) 12 12 12 12 12 12 12 12 TC TVC TFC output

17. Diminishing marginal returns set in here £ Total and Variable Costs TC TVC TFC output

18. Costs in the Short run • Average cost • average fixed cost (TFC/Q) • average variable cost (TVC/Q) • average (total) cost (TC/Q) = AFC + AVC • Marginal cost = TC/ Q = dTC/dQ Example: TC = 50 + 5Q; TFC = 50; TVC = 5Q AC = [50+5Q]/Q = 50/Q + 5; AFC = 50/Q; AVC = 5 MC = Δ(50+5Q)/ΔQ = 0+5 = 5

19. Total, Fixed, Average, and Variable Costs

20. MC Diminishing marginal returns set in here x Average and Marginal Costs Costs (£) Output (Q)

21. Short-run Costs • Average and marginal cost curves • marginal cost • relationship between average and marginal cost curves

22. MC AC AVC z y x AFC Average and Marginal Costs Costs (£) Output (Q)

23. Some Maths – An Example An example of a short-run total cost function: Where SFC=F and SVC = cQ+ dQ2 and Thus the short-run average fixed cost decreases steadily as Q increases.

24. Some Maths – An Example Short run average variable cost is: And short run average total cost:

25. Short Run and Long Run Production Compared In the long run, all factors variable…

26. Production in the Long run • Returns to scale/Economies of scale • specialisation & division of labour • indivisibilities • container principle • greater efficiency of large machines • multi-stage production • organisational & administrative economies • Economies of scope - by-products

27. Production in the Long run • All factors variable in long run • The scale of production: • constant returns to scale • increasing returns to scale • decreasing returns to scale • Returns to scale and economies and diseconomies of scale Economies of scale Diseconomies of scale

28. Costs in the Long run • Long-run costs • no fixed costs; law of diminishing returns does not apply – but economies/diseconomies of scale may • shape of the LRAC curve • assumptions behind the curve • factor prices fixed at each level of output • technology does not change • firms operate efficiently: our LRAC curve does not incorporate “waste”

29. LRAC Alternative long-run average cost curves Economies of Scale Costs O Output

30. LRAC Alternative long-run average cost curves Diseconomies of Scale Costs O Output

31. LRAC Alternative long-run average cost curves Constant costs Costs O Output

32. LRMC Long Run Average and Marginal Costs Initial economies of scale, then diseconomies of scale LRAC Costs O Output

33. SRAC5 SRAC1 SRAC2 SRAC4 SRAC3 5 factories 4 factories Deriving Long Run Average Cost Curves – Factories of Fixed Size 1 factory Costs 2 factories 3 factories O Output

34. Deriving Long Run Average Cost Curves: Factories of Fixed Size SRAC5 SRAC1 SRAC2 SRAC4 SRAC3 LRAC Costs O Output

35. Deriving Long Run Average Cost Curves – Factories of Fixed Size LRAC Costs O Output

36. Minimum Efficient Scale LRAC Costs “Minimum Efficient Scale” O Output

37. Minimum Efficient Scale Varies by Industry

38. Some Maths…Cost Minimising Input Combination Take input prices w, the wage, and r, the rental price of capital as given. TC = wL + rK TC0 = wL + rK  rK = TC0 – wL  K = TC0/r – (w/r)L For higher total cost, TC1, K = TC1/r – (w/r)L K TC1/r TC0/r The “isocost” -w/r L

39. Some Maths…Cost Minimising Input Combination Now, consider all inputs, K and L, that generate output level Q0 All L, K such that Q0 = f(L, K) Suppose that more inputs are required to generate more outputs AND Inputs are somewhat substitutable, but variety of inputs is more efficient at generating output than using lots of a single input. K TC1/r The “isoquant” TC0/r Q0 = f(L, K) -w/r L The slope of the isoquant is the “marginal rate of technical substitution”

40. Some Maths…Cost Function What is the smallest possible cost at which we can produce output Q0? We choose the lowest isocost line that just touches the desired isoquant line. L0, K0 is our cost minimising choice of inputs to generate desired output level Q0. The cost function is the cost of producing output at least cost, so we have C(Q0) = wL0 + rK0 . Isocost line corresponding to TC2 K TC1/r TC0/r K0 Q0 = f(L, K) -w/r L L0

41. Some Maths…Conditional Input Demands Isocost line corresponding to TC2 K TC1/r TC0/r K0 Q0 = f(L, K) -w/r L L0 =- MPL/MPK Or 0 = MPLΔL + MPKΔK

42. So we use the equations to solve for the optimal (cost minimising) input combination to reach desired output Q0: Qo = f(K, L) w/r = MPL/MPK (= MRTS) to obtain demands K0, L0 for any output level, Q0. Substituting these into the cost equation TC = wL + rK we can obtain the cost for any output level Q0.

43. Summary We have a “black box” model of the firm Production and costs are modelled in the short and long term Short term production has diminishing returns, in general, and is characterised by marginal, average, and total products. Short term costs are characterised by marginal, average, and total costs. Due to diminishing returns, marginal costs tend to be U-shaped. Marginal and average quantities are systematically related. Long term production can have various returns to scale, which give rise to different shapes of long term average cost . Minimum efficient scale is the scale of operation we would expect an industry to move to. Conditional factor demands can be derived from a choice problem that is the “mirror image” of the consumer’s choice problem.