competitive input markets

©2005, Southwestern

Slides by Pamela L. Hall

Western Washington University

Competitive Input Markets

Chapter 16

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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

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Figure 16.1 Factor market under perfect competition

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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

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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

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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)

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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 ℓS

1 = ℓ1(w) and ℓS2 = ℓ2(w)

• Total labor market supply is sum of amounts supplied by the two workers XS = ℓS

1 + ℓS2

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Figure 16.2 Market supply curve for labor

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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

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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

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Figure 16.3 Economic rent

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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

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

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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

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Figure 16.5 Economic rent associated with an inelastic supply curve

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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

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Figure 16.6 Land rent

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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

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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

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Table 16.1 Optimal Tax Rates by Sector

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Figure 16.7 Ricardian rent

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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

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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

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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

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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)

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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

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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

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Figure 16.8 Isoprofit lines and production functions

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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

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

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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

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Figure 16.10 First-order condition for profit maximization …

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Figure 16.10 First-order condition for profit maximization …

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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

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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 x j

Solving these F.O.C.s simultaneously for k inputs yields input demand functions, x j = 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)

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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

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Figure 16.11 Labor demand curve with variable capital

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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

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Figure 16.12 Perfect competition in both the factor and output market

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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

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

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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

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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

competitive input markets

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