Intermediate microeconomics. Lecture 3: Production theory. Varian, chapters 19-24

Transcription

1 Intermediate microeconomics Lecture 3: Production theory. Varian, chapters 19-24

2 Part 1: Profit maximization 1. Technology a) Production quantity and production function b) Marginal product and technical rate of substitution c) Short run and long run d) Returns to scale 2. Profit maximization a) Profit b) Profit maximization in the short run and in the long run c) Profit maximization and returns to scale Adam Jacobsson, Department of Economics 2

3 Part 2: Cost minimization and supply 3. Cost minimization 4. Cost functions and returns to scale 5. Sunk costs 6. Cost curves 7. Firm supply in the short run 8. Profit and producer surplus 9. Firm supply in the long run 10. Market supply Adam Jacobsson, Department of Economics 3

4 1. Technology Inputs (factors of production): land, labour, capital (physical and financial). The technological constraints are described by the production set: Definition 1: The production set contains all possible combinations of outputs and inputs. Definition 2: The production function f measures the maximum possible output of good y for any given amount of inputs (x 1, x 2 ): y=f(x 1, x 2 ) Adam Jacobsson, Department of Economics 4

5 Production set and production function with one input Output Production function y=f(x) Production set Input Adam Jacobsson, Department of Economics 5

6 Assumptions about the technology s properties Monotonicity: If the amount of one input increases, output will increase or remain unchanged. Free disposal: The firm can costlessly dispose of any inputs. Convexity: If it is possible to produce y with inputs X=(x 1, x 2 ) or Z=(z 1, z 2 ), then it is possible to produce y with inputs H= λx + 1 λ Z, λ [0,1] Adam Jacobsson, Department of Economics 6

7 An isoquant consists of all combinations of inputs that are just sufficient to produce a given quantity of output. Input 2 x 2 X Convexity implies that using inputs H=(h 1, h 2 ) leads to an output at least as large as y. h 2 z 2 x 1 h 1 z 1 Z Isoquant y >y Isoquant y Input Adam Jacobsson, Department of Economics 7

8 Marginal product (MP) For any given combination of inputs, the MP measures how much output changes in relation to an change in the amount of input i: MP i x 1, x 2 = y xi = f(x 1, x 2 ) xi Assumption about decreasing MP: MP for one input decreases as the amount of this input increases, given that the amounts of all other inputs remain constant Adam Jacobsson, Department of Economics 8

9 Technical rate of substitution (TRS) For a given combination of inputs: Which change in the amounts of inputs is consistent with an unchanged output level? dy = f(x 1, x 2 ) x 1 dx 1 + f(x 1, x 2 ) x 2 dx 2 = 0 = MP 1 dx 1 + MP 2 dx 2 = Adam Jacobsson, Department of Economics 9

10 TRS, continued TRS x 1, x 2 = dx 2 dx 1 = MP 1 x 1, x 2 MP 2 x 1, x 2 The TRS equals the slope of the isoquant, that is, how much less of input 2 is needed if the firm uses one more unit of input 1 and output is fixed. Assumption about decreasing TRS: the slope of the isoquant decreases in absolute terms (that is, it becomes less negative) Adam Jacobsson, Department of Economics 10

11 Short run and long run Fixed factor: can only be used in a fixed amount. The firm cannot abstain from the input even if nothing is produced. Variable factor: can be used in different amounts. The firm can abstain from the input if nothing is produced. Quasifixed factor: is needed in a fixed amount independent of how much is produced, but if nothing is produced the firm can abstain from this input Adam Jacobsson, Department of Economics 11

12 Short and long run, continued Short run: One or more inputs are fixed (for example physical capital like factory buildings). Long run: All inputs can be varied freely, for example by setting up new production facilities (factories for example). The firm can also choose zero inputs to produce zero output Adam Jacobsson, Department of Economics 12

13 Returns to scale If the amounts of all inputs are scaled up by a factor t>1, by how much does output increase? Constant returns to scale (CRS): Output is also scaled up by a factor t: f(tx 1, tx 2 )=tf(x 1, x 2 ) Increasing returns to scale (IRS): Example! Output is scaled up by more than a factor t: f(tx 1, tx 2 )>tf(x 1, x 2 ) Decreasing returns to scale (DRS): Output is scaled up by less than a factor t: f(tx 1, tx 2 )<tf(x 1, x 2 ) Adam Jacobsson, Department of Economics 13

14 2. Profit maximization Profit = revenues costs. In economics the concept of profit implies that all inputs and outputs are valued according to their opportunity costs. Hence, an input has to be valued according to its best alternative use instead of being valued according to its acquisition value. The cost of a machine is measured in terms of what it would cost to rent during the time it is used. If there is no well-functioning machine market: the cost of use is then the price of the machine at the beginning of the production minus the machine s selling price after production Adam Jacobsson, Department of Economics 14

15 A firm that produces n goods by using m inputs makes the following profit: O U Π = M p i y i PQR M w j x j VQR Where p i is the price of good i and w j is the price of input j and y i = f(x 1, x 2,, x U[R, xm) Adam Jacobsson, Department of Economics 15

16 Short run profit maximization Assume two inputs, where input 2 is fixed, i.e. x 2 =x] 2. The optimization problem: max Π s = pf(x 1, x] 2 ) w 1 x 1 w 2 x] 2 _ R FOC: Π s = p f(x R, x] 2 ) w x 1 x 1 = 0 1 pmp 1 (x R, x] 2 ) = w 1 The market value of the marginal product of the input has to equal the price of this input (assuming decreasing MP 1 ) Adam Jacobsson, Department of Economics 16

17 Isoprofit lines Profit is given by: Π c = py w 1 x 1 w 2 x] 2 By solving for y we obtain isoprofit lines: y = Πc p + w 2 p x] 2 + w 1 p x 1 Add w 1 x 1 + w 2 x] 2 to LHS & RHS and divide by p! Slope The slope can also be obtained in the following way: dπ c = pdy w 1 dx 1 = 0 dy e fg dx h Qi = w 1 R p The slope of the isoprofit lines express how much output changes in response to a change in the amount of input, given that profit remains constant Adam Jacobsson, Department of Economics 17

18 Profit maximization in the short run Output, y Isoprofit line with slope l R m Production function y=f(x 1, x] 2 ) Π c Π c y* p + w 2 p x] 2 Profit max Production set At the profit max point the following is true: l R m = MP 1(x R, x] 2 ) x 1 * Input Adam Jacobsson, Department of Economics 18

19 Long run profit maximization No input is fixed now! The optimization problem: max _ R,_ o Π = pf(x 1, x 2 ) w 1 x 1 w 2 x 2 FOC: pg,_ s ) = p pq(_ r w p_ R p_ 1 = 0 (1) R pg Rearrange (1) & (2)!,_ s ) = p pq(_ r w p_ o p_ 2 = 0 (2) o pmp 1 (x R, x o ) = w 1 & pmp 2 (x R, x o ) = w Adam Jacobsson, Department of Economics 19

20 Hence, the value of the marginal product of each input should equal its price. From conditions (1) and (2) the optimal solutions x R and x o can be obtained. By varying p, w 1 and w 2 we obtain the factor demand functions x R (p, w 1, w 2 ) and x o p, w 1, w 2! Adam Jacobsson, Department of Economics 20

21 The inverse factor demand curve of input 1 measures what the price of input 1 must be for a given quantity of input 1 to be demanded, given the optimal choice of input 2 (x o ). pmp 1 (x R, x o ) = w 1 Factor price input 1 (=w 1 ) pmp 1 (x 1, x o ) w 1 Factor demand curve for input 1 x R Input Adam Jacobsson, Department of Economics 21

22 3. Cost minimization An isocost curve consists of inputs 1 and 2, x 1 and x 2, for which costs are constant (=C). w 1 x 1 + w 2 x 2 =C Or (solving for x 2 ) : x 2 = l o l R l o x 1 Slope Adam Jacobsson, Department of Economics 22

23 The cost minimization problem, continued Input 2 x 2 = C w 1 x w 2 w 1 2 Isocost lines with slope - l R l o At the optimum: - l R l o = TRS x o Isoquant f(x 1, x 2 )= y} with slope TRS = ~ R _ R,_ o ~ o _ R,_ o x R Input Adam Jacobsson, Department of Economics 23

24 The cost minimization problem, continued Minimize costs to attain a given production level y] min w 1 x 1 + w 2 x s. t. f x 2 1, x 2 = y] _ R,_ o Set up the Lagrangian: FOC: M x 1, x 2, μ = w 1 x 1 + w 2 x 2 μ f x 1, x 2 p~,_ s y] = w p_ 1 μ pq _ r = 0 (i) r p_ r p~,_ s = w p_ 2 μ pq _ r = 0 (ii) s p_ s p~ = f x pƒ R, x o y] = 0 (iii) Adam Jacobsson, Department of Economics 24

25 Rearrange (i) and (ii): Divide (i) by (ii): w 1 = μ pq _ r,_ s p_ r w 2 = μ pq _ r,_ s w 1 w 2 = p_ s f x R, x o x R f x R, x o x o = MP 1 MP 2 (i) (ii) [ _ r,_ s By adding μ pq _ r,_ s and μ pq _ r,_ s p_ s p_ r to both RHS and LHS respectively Adam Jacobsson, Department of Economics 25

26 The optimal solutions x R w 1, w 2, y and x o w 1, w 2, y are the conditional factor demand equations. Note the difference between these demand equations and the ones we got from profit maximization: x R w 1, w 2, p and x o w 1, w 2, p. The cost function: c w 1, w 2, y = w 1 x R w 1, w 2, y + w 2 x o w 1, w 2, y measures the minimal cost to produce y given factor prices w 1 and w Adam Jacobsson, Department of Economics 26

27 4. Cost functions and returns to scale Assume constant returns to scale (CRS). Solve the cost minimization problem for y=1. We then obtain the unit cost function c(w 1, w 2, 1). If we produce y>1 units, CRS implies that we have to scale up the amounts of inputs by y. Thus, costs will be scaled up by y: c w 1, w 2, y = yc w 1, w 2,1 That is, costs are proportional to y when we have CRS Adam Jacobsson, Department of Economics 27

28 If we have IRS, costs increase less than proportionately: c w 1, w 2, y < yc w 1, w 2,1 If we have DRS, costs increase more than proportionately: c w 1, w 2, y > yc w 1, w 2,1 Cost of producing the first unit Define average costs: AC y = c w 1, w 2, y y For y>1: AC y > c w 1, w 2,1 AC y = c w 1, w 2,1 AC y < c w 1, w 2,1 if DRS if CRS if IRS Adam Jacobsson, Department of Economics 28

29 5. Sunk costs Definition 1. A sunk cost is a payment that cannot be recovered. Example: A firm uses SEK to purchase furniture. At the end of the year the furniture can be sold at a price of The sunk cost is the reduction in value, that is, Adam Jacobsson, Department of Economics 29

30 6. Cost curves The total cost for producing y is given by: c y = cv y + F Where c v (y) is the variable cost for producing y, and F is the fixed cost. The average cost is given by: AC y = c y y = c v y y + F y ( ) ( ) Adam Jacobsson, Department of Economics 30

31 The marginal cost is given by: MC y = c y y = c v y y For y=0 we have: MC 0 = AVC(0) Adam Jacobsson, Department of Economics 31

32 How is the average variable cost affected by changes in the scale of production? AVC(y) ( c v y y ) = = y y Remember the following rule: if we have f(x)g(x), then (f x g x ) = f x g x + f x g x x Also, can be written as c v y y š [R (_) q( ) = c v y y y[r c v y y [o = = c v y 1 y y cv y 1 y = Adam Jacobsson, Department of Economics 32

33 Factor out R : = 1 y c v y y c v y y = = 1 y MC(y) AVC(y) For a given y the following applies: If MC<AVC: AVC decreases. If MC=AVC: AVC is constant. If MC>AVC: AVC increases Adam Jacobsson, Department of Economics 33

34 How is the average cost affected by changes in the scale of production? AC y = c v(y) y + F y dac(y) dy = AVC y + AFC y = = = c v y y ( c v y y ) ( F y ) + y y = 1 y cv y 1 y F 2 y = Adam Jacobsson, Department of Economics 34

35 Factor out R : = 1 y c v y y c v y + F y = 1 y MC y AC(y) ~ ž For a given y the following applies: If MC<AC: AC decreases. If MC=AC: AC is constant. If MC>AC: AC increases Adam Jacobsson, Department of Economics 35

36 MC, AVC and AC MC, AVC, AC MC AC AVC AVC(0)= MC(0) The MC curve crosses the AVC and AC curves at their lowest points! See previous conditions relating MC, AVC and AC! y Adam Jacobsson, Department of Economics 36

37 Cost in the short and long run Let k (a fixed input like capital previously we called this x] 2 ) be fixed in the short run. The cost function in the short run is given by c s y, k. The cost function in the long run is given by c y. The cost of producing y in the short run is at least as large as the cost of producing y in the long run, since k can always be adjusted in the long run: c y c s y, k Let k = k(y )be the optimal value of k for y (i.e. for some given value of y). Hence, for k we have c y = c s y, k Adam Jacobsson, Department of Economics 37

38 AC and MC in the short run SAC, SMC One bakery Two bakeries Three bakeries Four bakeries Five bakeries SAC 1 SAC 5 SMC 1 SAC 2 SMC 2 SAC 3 SMC 4 SAC 4 SMC 5 SMC 3 y cr y co y c y c y c y Adam Jacobsson, Department of Economics 38

39 AC and MC in the short and long run SAC, SMC, LAC, LMC LMC LAC SAC 3 SMC 3 y Adam Jacobsson, Department of Economics 39

40 7. Firm supply in the short run The firm s decision about how much to produce is constrained by: Technology (the cost function) Market conditions Assume perfect competition (many firms): Price is taken as given (does not depend on the firm s choice of output). No strategic interaction! Adam Jacobsson, Department of Economics 40

41 The firm s output decision in a market with perfect competition The optimization problem: FOC: max Π(y) = R(y) c y dπ(y) dy = R(y ) y ~ ( ) c y = 0, or y ~ MR y = MC(y ) Since we have R(y)=py in a market with perfect competition: R(y ) = p y The FOC under perfect competition can thus be expressed as: p = MC(y ) Adam Jacobsson, Department of Economics 41

42 If p>mc, the firm can increase profits by increasing supply. If p<mc, the firm can increase profits by decreasing supply. Note that p=mc is a necessary, but not a sufficient condition for profit maximization. It has to be profitable to produce something! If p=mc<avc, the firm cannot cover its variable costs. Therefore this part of the MCcurve is not part of the firm s supply curve. The firm s supply curve is thus given by the part of the MC-curve that lies above AVC, i.e. where MC AVC Adam Jacobsson, Department of Economics 42

43 The firm s supply curve in the short run MC, AVC, AC MC AC AVC AVC(0)= MC(0) Short run supply curve y Adam Jacobsson, Department of Economics 43

44 8. Profit and producer surplus Profit = Revenue minus total costs Π = py c v y F Producer surplus = revenues minus variable costs PS = py cv y Production should cease if the PS is negative, i.e. if variable costs exceed revenues: PS < 0 py cv y < 0 py < c v y Since c v y = AVC y y we thus obtain the following shutdown condition: py < AVC y y, or p < AVC y Adam Jacobsson, Department of Economics 44

45 If the market price of output is lower than the average variable cost, production ceases. However, there is an interval of prices, AVC y p < AC y, for which producer surplus is positive, but profit is negative. In this case production is not shut down despite the fact that a loss has occured, because revenues exceed variable costs. The producer gets some revenue to pay at least a part of the fixed costs Adam Jacobsson, Department of Economics 45

46 9. Firm supply in the long run In the long run all inputs can be varied. In the long run it is also possible to shut down production. Profits must therefore be non-negative: Π = py c y 0, or p c y = LAC y y i.e. price must be at least as large as long run average costs. The firm s long run supply curve is therefore given by the section of the LMC that lies above the LAC Adam Jacobsson, Department of Economics 46

47 The firm s supply curve in the long run LMC, LAC LMC LAC ---- Long run supply curve LAC min =LAC(y min ) Level of production with minimal long run unit cost, LAC min. y min y Adam Jacobsson, Department of Economics 47

48 10. Market supply (Industry supply) Market supply with n firms is given by: O S p = M S P (p) PQR Where S P (p) is firm i s supply at output price p. In the short run, market supply consists both of firms that make a loss and of firms that make profits. In the long run, however, firms can adjust fixed inputs. Firms making a loss will quit the market. In the long run, firms that use the technology of profitable firms will enter the market, given free entry, putting downward pressure on the market price. If there are sufficiently many firms in the long run, the equilibrium market price will be close to the minimal unit cost, LAC min. Profits of firms will then go to zero Adam Jacobsson, Department of Economics 48

49 Market supply curve in the long run Price S 1 Supply of firm 1 Supply of firms 1 & 2 S 1 + S 2 Supply of firms 1, 2 & 3 S 1 + S 1 +S 1 LAC min y min 2y min 3y min 4y min Quantity Adam Jacobsson, Department of Economics 49