Competitive Input Markets

1. Competitive Input Markets Chapter 16

2. Introduction • In general, firms will employ inputs (factors of production) in production of an output with objective of maximizing profit • To determine profit-maximizing level of an input, associated cost per unit (implicit or explicit) of each input is required • In a free market economy, this cost per unit is determined by supply and demand for inputs • For example, perfect competition in a factor market is characterized by intersection of factor market supply and demand curves (Figure 16.1) • X is total market quantity of an input • v is per-unit price for input X • Intersection of market supply and demand curves yields equilibrium input price ve and quantity of input Xe • Inputs are supplied by resource owners • Agents who either own land and capital or supply their labor as inputs into production • We assume input supply curve is upward sloping • An increase in input price, v, results in an increase in quantity supplied of input, X • In contrast, input demand curve would then be downward sloping • A decrease in v yields an increase in quantity demanded for X • This demand curve is a derived demand based on firms’ objective of maximizing profit • We assume that firms hire land, labor, and capital to produce a profit-maximizing output and that quantities hired depend on level of output

3. Figure 16.1 Factor market under perfect competition

4. Introduction • In this chapter, we investigate theoretically intuitive discussion of market demand and supply for inputs, particularly for labor market • We derive input market supply curve for labor as horizontal summation of individual workers’ supply curves • We evaluate this input market supply curve for various inputs in terms of surplus benefits (called economic rent) it provides to suppliers of the input • Based on this economic rent, we investigate Henry George’s single-tax scheme • We then develop Ricardian rent as an important predecessor to marginal analysis based on profit maximization • We derive comparative statics of an input demand curve in terms of substitution and output effects • We illustrate this responsiveness of input demand to input price changes with minimum wage • Based on aggregation of individual firms’ demand for an input, we develop competitive input market equilibrium • Given this market equilibrium price, we determine firms’ optimal profit-maximizing level of the input

5. Introduction • Our aim in this chapter is to investigate perfectly competitive input market • Economists use this market as standard for measuring input market efficiency • Any monopoly power in input market is judged against Pareto-efficient allocation resulting from a perfectly competitive input market

6. Market Supply Curve for Labor • In perfectly competitive markets, determination of wage rates and employment levels depend directly on labor market supply curve • Based on individual workers’ labor supply curves (Chapter 4), we can derive labor market supply curve • Specifically, this curve is horizontal summation of individual workers’ supply curves • Illustrated in Figure 16.2 for two workers, where ℓ1 and ℓ2 are worker 1’s and worker 2’s supply, respectively • Wage rate is w, and X is total amount of labor supplied in market • Both workers are facing same wage rate • Assumed they take this wage rate as given • Individual labor supply curves will have a positive slope when income effect does not fully offset substitution effect • Otherwise, an individual labor supply curve will be backward bending (generally only at high wage rates)

7. Market Supply Curve for Labor • When the individual labor supply curves have positive slopes, at a given wage rate, w' • Each worker is willing and able to supply a given level of labor services • As illustrated in Figure 16.2, at w' workers 1 and 2 are willing to supply 8 and 10 units of labor, respectively • Market supply curve for this labor is the sum of the hours (8 + 10 = 18) • As wage rate increases, each worker is willing to supply more labor services • Sum of labor supplied will increase • Even if some workers have backward-bending labor supply curves • Market supply will likely still be positively sloping • However, if a substantial number of workers have a backward-bending supply curve, then market supply curve may also be backward bending • In early 20th century, as wages increased average work week declined to around 40 hours per week • Mathematically, labor market supply function for two workers is sum of each worker’s individual labor supply function • ℓS1 = ℓ1(w) and ℓS2 = ℓ2(w) • Total labor market supply is sum of amounts supplied by the two workers • XS = ℓS1 + ℓS2

8. Figure 16.2 Market supply curve for labor

9. Economic Rent • Rent is a naturally occurring surplus • Potential return arising solely from use of a particular site • Anyone who has use of that site has access to its economic rent • Cannot be abolished by any law or destroyed by agreements between landlords and tenants • Most that can occur is that potential is not tapped • A deadweight loss • Specifically, concept of economic rent, based on factor supply, is defined as • Portion of total payments to a factor that is in excess of what is required to keep factor in its current occupation • Economic rent is same concept as producer surplus except surplus accrues to the factor • From Figure 16.3, total dollar amount necessary to retain level Xe in this occupation is given by area 0ABXe • If firms could perfectly discriminate among factor suppliers, total payment would be 0ABXe • Perfectly competitive markets, however, do not work in this manner • All similar inputs are paid the same price • However, some factor owners would settle for less • Leads to economic rent

10. Economic Rent • For example, in terms of labor, a worker receiving a wage rate of $25 per hour may be willing to work for only $20 per hour • $5 difference is a per-hour surplus (economic rent) accruing to worker • In Figure 16.3, total factor payments are 0veBXe • Subtracting area necessary to retain level Xe in this occupation, 0ABXe, from total factor payments, 0veBXe • Results in economic rent, AveB

11. Figure 16.3 Economic rent

12. Economic Rent and Opportunity Cost • Any factor of production that has many alternative uses will have a very elastic supply curve for one type of employment • This factor can receive almost as high a price elsewhere • Quantity supplied will be reduced sharply for a small decline in factor price • As illustrated in Figure 16.4, economic rent is small for factors with this very elastic supply • Factor earns only slightly in excess of what it might earn elsewhere (opportunity cost) • Opportunity cost is represented by area 0ABXe, leaving very little economic rent, area AveB • For labor market, wage rate is just above wage at which a worker would just be willing to supply his or her labor services (called reservation wage) • Thus, a worker’s opportunity cost is high, resulting in low economic rent • Results in supply of workers being very responsive to a change in wages • A decline in wages can result in this opportunity cost exceeding income from working • Results in a decline in number of workers

13. Figure 16.4 Low level of economic rent associated with an elastic …

14. Economic Rent and Opportunity Cost • In the extreme, a perfectly elastic supply curve results in zero economic rent accruing to factor • So reservation wage is equal to wage rate • Opportunity cost is equal to total factor payments • Making a factor owner indifferent between supplying the factor or not supplying the factor • For example, secretaries have many opportunities at approximately the same wage • Their opportunity cost for working at a particular place is large relative to their wage rate • In contrast, professional football players generally have limited opportunities at approximately the same wage • Their wage rate is substantially above wage at which they would just be willing to supply their labor services (reservation wage) • Their economic rent is relatively large (Figure 16.5) • A decrease in a football player’s wage will have limited impact on decreasing supply (relatively inelastic supply) • Some football players would be willing to play football for free • Alternative earning possibilities (opportunity cost) of football players are generally quite low, so a large part of their wage is economic rent • In Figure 16.5, opportunity cost is ABXe, with shaded area representing economic rent • As labor supply curve becomes more inelastic, opportunity cost declines with an associated increase in economic rent

15. Figure 16.5 Economic rent associated with an inelastic supply curve

16. Land Rent • Henry George applied idea of large economic rents accruing to factor owners with highly inelastic supply curves to land • He assumed that land is in fixed supply (perfectly inelastic) • In Figure 16.6, no matter what the level of demand, supply of land is fixed at M° • Given demand curve MD, a return (economic rent) to landowners is 0voAM° • With demand curve MD', return is 0v1A'M° • Thus, an increase in demand for land has no effect other than to enrich landowners • Henry George proposed that those rents accruing to such fortunate landowners be taxed at a very high level • Because this taxation would have no effect on quantity of land provided • He assumed a zero supply response, so a tax on land would not create inefficiencies • Given no deadweight loss, some proponents of this Henry George Theory even suggested this should be the only method of tax collection • May be worth considering in an agrarian economy where all land is of the same type yielding the same productivity • When land has only one main use, opportunity cost is near zero • Resulting in a highly inelastic supply curve

17. Figure 16.6 Land rent

18. Land Rent • However, for most economies there are multiple uses for land • Such as for residential, commercial, or industrial development • Thus, an opportunity cost exists for using land in a particular activity • Which creates inefficiencies associated with single-tax scheme • Might be feasible to tax other factors used in a production activity where alternative uses are slight • For example, a high tax rate on professional sports players would have little or no effect on number and quality of professional players • Such a tax would not greatly distort market allocations (there would be little if any deadweight loss) • Plus disadvantaged youth would not see sports as a substitute for education for achieving success • Major League Baseball Commission has considered taxing players’ salaries in an effort to reduce these salaries • It would then use tax revenue to support ball clubs with relatively fewer resources

19. Ricardian Rent • Even agricultural land parcels range from very fertile (low cost of production) to rather poor quality (high cost) land • There is a supply response associated with land that restricts application of an efficient single tax on land • Based on this observation, David Ricardo made one of the most important conclusions in classical economics • More fertile land tends to command a higher rent • Ricardo’s analysis assumed many parcels of land of varying productive quality for growing wheat • Resulting in a range of production costs for firms • As an example, in Figure 16.7, three levels of firms’ SATC and SMC curves are illustrated, along with market demand and supply curves for wheat • Market for wheat determines equilibrium price for wheat • At this equilibrium, an owner of a low-cost land parcel earns a relatively large pure profit • p > SATC

20. Table 16.1 Optimal Tax Rates by Sector

21. Figure 16.7 Ricardian rent

22. Ricardian Rent • Considering this profit as a return to land, low-cost firm is earning relatively high rents (Ricardian rent) for its land • A medium-cost firm earns less profit (Ricardian rent) • Price is still greater than SATC, but not as great as for low-cost firm • In contrast, marginal firm is earning a zero pure profit (Ricardian rent), p = SATC • Any additional parcels of land brought into wheat production will result in a loss • No incentive for these parcels to be brought into production • Presence or absence of Ricardian rent in a market works toward allocating resources to most efficient use • Ricardo’s analysis indicates how demand for land is a demand derived from output market • Level of market demand curve for output determines how much land can be profitably cultivated and how much profit in the form of Ricardian rent will be generated • Theory explains why some firms earn a pure profit in competitive markets • When managerial ability, location, or land fertility differ

23. Ricardian Rent • For example, a favorably situated store (firm) will earn positive pure profits while stores at margin earn only normal profits • But it is not store’s cost of production that determines store’s output prices • Are determined by market demand and supply curves for these outputs • Those prices, in turn, determine profit (Ricardian rent) • In a perfectly competitive output market, it is not true that a store can offer lower prices because it does not have to pay “high downtown rents” • If its rent is lower than downtown, store may earn a short-run pure profit • But in long-run, store will only experience a normal profit • Any pure profit gets capitalized into the firm’s costs • Thus store’s prices may be less, but it is not because its rent is less

24. Marginal Productivity Theory of Factor Demand • Ricardian Rent Theory was an important predecessor to development of economic theory based on marginal analysis derived from profit maximization • Particularly true in terms of theory associated with factor demand • A firm’s demand for a factor is based on firm’s attempt to maximize profits • Differences in a firm’s demand for factors determine at what proportions these factors are used in production

25. One Variable Input • Let’s consider a production function with only labor, L, as the variable input, q = f(L) • Assume output is sold in a perfectly competitive market at a price per unit p and firm can hire all the labor it wants at prevailing wage rate, w • Because we assume perfect competition in output market, firm’s output market demand curve is perfectly elastic • Firm has no control over output price • Because firm is able to hire all the labor it wants at a wage rate of w, it is also facing a perfectly elastic labor supply curve • Firm takes wage rate as given • Firm’s profit-maximizing objective is • Incorporating production function into firm’s profit-maximizing objective function yields • Where pf(L) is total revenue as a function of level of labor, L, employed • wL is firm’s total variable cost (wage bill)

26. One Variable Input • F. O. C. is • d/dL = pdf(L)/dL – w = 0, or p(MPL) = w • Recall that df(L)/dL = MPL, marginal product of labor • Here, pMPL is called value of marginal product of L, VMPL • In a perfectly competitive output market, VMPL is additional revenue received by hiring an additional unit of L • Marginal product measures additional output from employing an additional unit of an input • How much this additional unit is worth (valued) is determined by multiplying this additional output by a measure of its per-unit value • In a perfectly competitive output market, output price is measurement for per-unit value • Thus, price  marginal product is value of marginal product • F. O. C. for profit maximization results in a tangency, point A in Figure 16.8, between an isoprofit line and production function • Where VMPL = w

27. One Variable Input • An isoprofit line represents a locus of points where level of profit is the same • For movements along an isoprofit line, profit remains constant • An upward shift in an isoprofit line represents an increase in profit • We develop isoprofit line from isoprofit equation for a given level of profit • * = pq – wL - TFC • Where * represents some constant level of profit • Solving for q yields • q = (* + TFC)/p + (w/p)L • As illustrated in Figure 16.8, slope of isoprofit line is dq/dL = w/p • At tangency point A, slopes of isoprofit line and production function are equal • Since slope of production function is MPL = dq/dL, at this tangency w/p = MPL • Multiplying through by p yields F.O.C. for profit maximization • w = pMPL = VMPL

28. Figure 16.8 Isoprofit lines and production functions

29. One Variable Input • F.O.C. for profit maximization states that isoprofit line tangent to production function will maximize profit subject to production function • Termed price efficiency • Requires both allocative and scale efficiency • Allocative efficiency occurs where ratio of marginal products of inputs equals ratio of input prices • Scale efficiency is where marginal cost equals output price • Overall economic efficiency for a firm is established when both price efficiency and technological efficiency exist (Figure 16.9) • A firm is technologically efficient when it is using the current technology for producing its output • At point B in Figure 16.8, firm is technologically efficient but not price efficient • Moving up along production function, firm shifts to a higher isoprofit line • Representing a higher level of profit • At tangency point A, firm can no longer remain on production function constraint and further increase profit • Firm has reached the highest isoprofit line possible and maximizes profits for this technology • At this point A, firm is also price efficient

30. Figure 16.9 Flowchart illustrating the different types of efficiencies for a firm

31. One Variable Input • As illustrated in Figure 16.10, at equilibrium wage w*, value of marginal product of labor, VMPL, equals wage rate • If only L' workers are hired instead, profit could be enhanced by increasing amount of labor • Increasing labor from L' to L* results in additional revenue of L'ABL* with associated cost of L'CBL* • Additional revenue is greater than additional cost, so profit increases by CAB • Alternatively, at L", decreasing amount of labor to L* results in revenue falling by L*BL" with cost declining even more, L*BDL" • Reduction in cost is more than loss in revenue, so profit will increase by area BDL" • At point B, where VMP = w*, firm maximizes profits • For case of one variable input, VMPL curve is labor demand curve • Solving F.O.C., w = VMPL, for L results in firm’s input demand function for labor, L = L(p, w) • This demand for labor is directly derived from F.O.C. • Output price, p, is a determinant of this input demand

32. Figure 16.10 First-order condition for profit maximization …

33. Figure 16.10 First-order condition for profit maximization …

34. Two Variable Inputs • Let’s extend analysis to two variable inputs by allowing both capital, K, and labor, L, to vary • Production function with these two variable inputs is q = f(K, L) • Where q, K, and L are all traded in perfectly competitive markets at prices p, v, and w, respectively • Problem facing a profit-maximizing firm with this production function is • F.O.C.s are then • ∂/∂L = pMPL – w = 0 and ∂/∂K = pMPK – v = 0 • From these F.O.C.s, v = pMPK = VMPK and w = pMPL = VMPL

35. Two Variable Inputs • F.O.C.for labor is illustrated in Figure 16.10 • At L', VMPL > w* • Addition to revenue for an increase in labor is greater than additional cost, so profit is enhanced by increasing labor • At L", VMPL < w* • Loss in revenue for a decrease in labor is less than loss in cost, so profit is enhanced by decreasing labor • At VMPL = w*, point B, profits are maximized • Similarly, changing capital around optimal level will result in a decline in profit • For both variable inputs firm will equate the VMP for variable input to associated input price as a necessary condition for profit maximization • Can generalize this result for k inputs, where for each input F.O.C. for profit maximization is to set VMP for an input equal to its associated price • Specifically, VMPj = vj, j = 1, …, k, where vj denotes input price for input xj • Solving these F.O.C.s simultaneously for k inputs yields input demand functions, xj = xj(p, v1, … , vk) • In contrast to one-input case, with two or more inputs VMP curves are not input demand curves • For example, in two-input case, solving w = VMPL for L yields L = fL(p,w,K), which is not input demand function for labor • Obtain input demand function for labor by solving simultaneously F.O.C.s, resulting in L = L( p, w, v)

36. Two Variable Inputs • Figure 16.11 illustrates this difference in demand curve for an input and associated VMP curves for two variable inputs labor and capital • VMPL depends on level of capital also employed • An increase in amount of capital employed will enhance productivity of labor • VMPL curve will shift upward • Illustrated in figure by a shift in VMPL from VMPL|K° to VMPL|K', given an increase in capital from K° to K' • A decrease in wage rate from w° to w' results in VMPL|K° > w‘ • Firm will hire more workers • Will result in VMPK increasing, so VMP|K > v • Firm will purchase more capital, which shifts VMPL curve upward • New equilibrium level of labor L' associated with w' is where w‘ = VMPL|K' • Demand curve for labor then intersects initial wage/labor level (w°, L°) and new level (w', L') • This labor demand curve is more elastic than VMPL curves because level of capital is allowed to vary along labor demand curve • In contrast, for a given VMPL curve, capital is fixed • Only where demand curve intersects a VMPL will this fixed level of capital for a given VMPL curve correspond with optimal level

37. Figure 16.11 Labor demand curve with variable capital

38. Perfectly Competitive Equilibrium in the Factor Market • Market for secretaries in a large city with many secretarial positions is characteristic of a perfectly competitive factor market • Illustrated in Figure 16.12, a perfectly competitive factor market is characterized by many buyers and many sellers of the input, labor (secretaries) • No single employer or employee can influence wage rate, we • we is determined through free interaction of supply and demand • Each firm can hire all the labor it wants at market wage • Representative firm is facing a horizontal labor supply curve (perfectly elastic supply curve, S = ) • Perfect competition in output market results in p = MR • Thus, pMPL = MR(MPL) • Where pMPL is VMPL and MR(MPL) is defined as marginal revenue product for labor, MRPL • Which is change in total revenue resulting from a unit change in labor • Here, MRPL is additional revenue received from increasing labor and measures how much this increase in labor is worth to the firm • ∂TR/∂L = (∂TR/∂q)(∂q/∂L) = MR(MPL) = MRPL

39. Figure 16.12 Perfect competition in both the factor and output market

40. Perfectly Competitive Equilibrium in the Factor Market • As illustrated in Figure 16.13, if there is imperfect competition in output market, then p > MR = SMC and pMPL = VMPL > MRPL = MR(MPL) • Specifically, recalling that MR may be expressed in terms of elasticity of demand, D, we have • MR = p[1 +(1/D)] • Then • MRPL = p[1 + (1/D)]MPL = [1 + (1/D)]pMPL = [1 + (1/D)]VMPL • Given that a profit-maximizing firm only operates in elastic region of demand curve • Then D < -1 resulting in 1 ≥ [1 + (1/D)] > 0 • As elasticity of demand tends toward negative infinity, 1/D will approach zero where MRPL = VMPL • Otherwise, for any firm facing a downward-sloping market demand curve (indicating at least some monopoly power) MRPL < VMPL

41. Figure 16.13 Value of the marginal product and marginal revenue product …

42. Perfectly Competitive Equilibrium in the Factor Market • In general, profit-maximizing problem for a firm facing a competitive wage rate is • Where pf(L) is total revenue as a function of level of labor employed and wL is firm’s total variable cost (wage bill) • F.O.C. is then • d/dL = MR(MPL) – we = 0 = MRPL – we = 0 • As illustrated in Figure 16.12, if firm is also in a perfectly competitive market for its output, then p = MR, resulting in MRPL = VMPL • In contrast, as illustrated in Figure 16.13, if firm has some monopoly power in its output market, then p > MR, yielding MRPL < VMPL • In both cases, market for labor is assumed to be perfectly competitive • With wage rate determined by intersection of market demand and supply curves for labor • Firm will take this competitive wage rate as given and equate it to its MRPL

43. Perfectly Competitive Equilibrium in the Factor Market • As indicated in Figure 16.12, if the firm is facing a competitive output price, then it will hire Le workers at we • Instead, as indicated in Figure 16.13, if firm has some monopoly power in output market, by restricting its output it will hire less labor, Le' • Horizontal supply curve for labor is called average input cost curve for labor (AICL') • It is total input cost of labor (TICL) divided by labor • AICL is average cost per worker, which is worker’s wage rate • Associated with this AICL is marginal input cost of labor, MICL • Defined as addition to total input cost from hiring an additional unit of labor • Note that when AICL is neither rising nor falling, MICL is equal to it • The consequence of general relationship between average and marginal units • If average unit is neither rising nor falling, marginal unit will be equal to it • Same relationship holds for AIC and MIC as for average and marginal cost