Download Competitive Input Markets

Competitive
Input Markets
Chapter 16
Slides by Pamela L. Hall
Western Washington University
©2005, Southwestern
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
2
Figure 16.1 Factor market under
perfect competition
3
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
4
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
5
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)
6
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
7
Figure 16.2 Market supply curve
for labor
8
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
9
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
10
Figure 16.3 Economic rent
11
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
12
Figure 16.4 Low level of economic rent
associated with an elastic …
13
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
14
Figure 16.5 Economic rent associated
with an inelastic supply curve
15
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
16
Figure 16.6 Land rent
17
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
18
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
19
Table 16.1 Optimal Tax Rates by
Sector
20
Figure 16.7 Ricardian rent
21
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
22
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
23
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
24
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)
25
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 perunit 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
26
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
27
Figure 16.8 Isoprofit lines and
production functions
28
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
29
Figure 16.9 Flowchart illustrating the
different types of efficiencies for a firm
30
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
31
Figure 16.10 First-order condition
for profit maximization …
32
Figure 16.10 First-order condition
for profit maximization …
33
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
34
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)
35
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
36
Figure 16.11 Labor demand curve
with variable capital
37
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
38
Figure 16.12 Perfect competition in
both the factor and output market
39
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
40
Figure 16.13 Value of the marginal product
and marginal revenue product …
41
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
42
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
43