Download Some Simple Models of Labor Market Equilibrium

Some Simple Models of Labor Market Equilibrium
1. Monopsony and Minimum Wages.
Let’s consider an industry in which a single firm employs all the labor.
MFC = w(L) + L∙w′(L)
w(L) (labor supply)
a
b
wmin
w0
pF′(L) = VMP
L0 L1 L*
L
If w(L) is the labor supply curve facing the firm (and industry), this firm maximizes Π =
pF(L) – w(L)L. FOC are pF′(L) = w(L) + L∙w′(L), or VMP = MFC, where MFC
(marginal factor cost) is the derivative of total labor costs (w(L)∙L) wrt L. VMP (value of
marginal product) is the marginal revenue from another unit of L.
The monopsonist maximizes profits at point a, where VMP=MFC. It pays a wage of w0
and employs L0 units of labor. This is less than the socially efficient level, L*.
Imposing a binding minimum wage at wmin changes the MFC curve to the bold line. The
profit-maximizing firm again sets VMP=MFC, which now occurs at point b. The firm
now employs L1 units of labor, which is more than before. So both wages and
employment rise as a result of the minimum wage law. Note that, at least in the case
where L is the firm’s only input, output = F(L) will rise as well.
Finally, note that if this firm had some pricing power in product markets, the above
tendency to increase output should translate into a endogenous decline in product prices.
This prediction tends to be at odds with empirical evidence on the effects of minimum
wages on prices. One way out of this dilemma (that respects similar studies’ effects of
zero effects on employment) would be monopsonistic competition models, like Bhaskar
and To, Economic Journal 109.455 (Apr 1999): 190-203.
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2. Competitive Industry Labor Demand
The simplest way to move from the firm and household level to the market is to imagine a fixed
number of firms in an industry (endogenous entry decisions of firms can complicate matters), but
large enough in number so that each firm takes factor and product prices as given. In this case, it
is well known that we can derive an industry level factor demand curve by horizontally summing
(i.e. summing over quantities demanded) the demand curves of the individual firms. This yields a
market level demand curve of the form LD = LD(w) when we are thinking of a single factor, L in
isolation. Theory says it must be downward sloping. More generally, the industry is
characterized by a system of factor demand equations of the form:
x1 = D1(p, w1 , w2 , … wn)
x2 = D2(p, w1 , w2 , … wn)
. = ……..
(1)
…….
xn = Dn(p, w1 , w2 , … wn)
where the x’s are industry level input demands, the w’s are input prices, and p is the price of the
industry’s output. The derivatives of (1) wrt p and w must satisfy the properties derived for the
individual firm’s labor demand (e.g. negative definiteness and symmetry). In sum, the industrylevel factor demand relationship maps prices into quantities and has the same properties as
firm-level demand.
Definition (Hamermesh 1993): factors i and j are p-complements iff ∂xi/∂wj < 0 in (1).
Otherwise they are p-substitutes. In other words, xi and xj are p-complements if an increase in the
price of factor j leads firms to use less of factor i. Note two things about this: First, we could just
as well define p-complements as occurring when “an increase in the price of one input reduces
the demand for the other” because (recall our notes on single-firm factor demand) factor demand
responses are predicted to be symmetric. Second, since own factor demand effects must be
negative, p-complementarity means the quantities of xi and xj both fall when the price of either
one rises.
Now suppose there is also a system of factor supply equations of the form:
x1 = S1(Y, w1 , w2 , … wn)
x2 = S2(Y, w1 , w2 , … wn)
. = ……..
(2)
…….
xn = Sn(Y, w1 , w2 , … wn)
where Y is a shift variable, like household income. If certain regularity conditions hold, we can
solve (1) and (2) for the vectors of equilibrium factor prices and quantities, w and x, as a function
of the exogenous variables (p and Y).
If we ignore interactions between factors in this context, we can answer some important questions
about labor markets at the industry level using a simple diagrammatic approach. For example:
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3. Payroll Tax Incidence
S´
$1
w1
b
Firms’ share
a
w0
w1-$1
S
Workers’ share
D
L1
L0
L
Consider a tax of one dollar per unit of labor supplied, levied on workers. This shifts the
labor supply curve vertically (upwards) by one dollar, from S to S´. (Why? Because
supplying L units of labor at a wage of $w per hour is the same as supplying L units of
labor at $w+1 per hour, then paying a tax of one dollar per hour). So whatever amount of
labor was supplied at $w before, will now be supplied at $w+1.
The shift in the supply curve moves the equilibrium from point a to point b, so the
equilibrium wage rises from w0 to w1 and the amount of labor exchanged falls from L0 to
L1. Total tax revenues collected by the government will be $1 times the amount of labor
exchanged in the new equilibrium, which equals the combined area of the two rectangles
labeled “Firms’ share” and “Workers’ share”. How much of this total tax burden is
‘borne’ by firms? Firms pay no taxes, but now pay w1 instead of w0 for labor.
Multiplying this difference by the amount of labor employed yields the rectangle labeled
“firms’ share”. Thus, even though firms are not ‘physically’ paying any taxes to the
government, they do share in the tax burden. Workers receive a higher wage, w1 than
they did before the tax was imposed (w0), but they also pay $1 in tax per unit of labor
sold. The difference between their old and new situation is therefore w0 – (w1-$1).
Multiplying this by the amount of labor employed yields the rectangle labeled “workers’
share”, which is less than the total amount workers are physically paying to the
government ($1∙L0). Thus, workers bear only part of the tax they are ‘physically’ paying;
the rest is shifted to firms via a higher equilibrium wage.
How much of the tax is shifted? The share of the tax shifted to firms rises with the
elasticity of labor supply, and falls with the elasticity of labor demand. In general, as in
any market, the inelastic side of the market bears the burden of the tax.
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Exercises:
1. Re-draw the above figure for the case of a very elastic labor supply curve and a very
inelastic labor demand curve, and show that in this case almost all of a tax on workers is
shifted from workers to firms.
2. How would the diagram change if the tax was a percentage of wages (e.g. like U.S.
Social Security taxes) rather than a dollar amount per unit of labor supplied?
3. Re-draw the above figure for the case of a one dollar tax per unit of labor, levied on
firms instead of workers. Show that:
(a) the new equilibrium level of L is exactly the same as if workers paid the tax, but this
time the wage falls instead of rising.
(b) the incidence of the tax is exactly the same as if it were levied on workers. In other
words, the distribution of the tax burden is completely unaffected by which party
‘physically’ pays the tax.
4. Some Simple Economics of Mandated Benefits (Summers, AER May 1989)
S0
S1
$α
w1+$1
w0
w1+$α
firms’ share
workers’ share
a
b
w1
c
$1
D0
D1
L
Consider a law that compels firms to give workers a benefit that costs one dollar to
provide for every unit of labor hired. If this benefit is not valued by workers at all, this is
just like a one-dollar unit tax levied on firms: it shifts the labor demand curve down by
$1, and moves the equilibrium from point a to point b. Wages fall, meaning that part of
the tax burden is shifted from firms to workers, and the amount of labor exchanged falls
too. So mandated benefits are indeed ‘job killers’.
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But suppose that the benefit firms are forced to give to workers is valued by workers at
$α per unit, where 0 < α < 1. This means that the mandated benefit shifts the labor
supply curve down by $α, to S1. Why? (be sure work it out). This puts the new
equilibrium at point c, at an even lower wage, but a higher level of employment. At the
new equilibrium, firms pay a wage of w1 (much lower than before) but must provide
benefits costing $1, so the difference in unit labor costs between the new and old
equilibrium is w1 + $1 - w0. Multiplying this by L yields the rectangle labeled “firms’
share”. Likewise, at the new equilibrium, workers receive a wage of w1 (much lower
than before) but receive a benefit that is worth $α to them. So the difference between the
old and new equilibrium to them is w0 - (w1+$α). Multiplying this by L yields the
rectangle labeled “workers’ share”. Note that the total burden of the mandated benefit is
the sum of these rectangles, or L(1- α), which is smaller than the burden of a pure tax. As
before, the division of this burden depends on the relative elasticities of demand and
supply, but its total amount now depends on how much workers value the benefit.
Exercise: Show, diagramatically, that (as the above formula for the total burden
suggests) a mandated benefit has zero allocative or distributional effect on labor markets
when α=1, i.e. when workers fully value the benefit that is mandated. Show that
equilibrium wages will, however, fall by the full cost of supplying the benefit, i.e. by $1
in the above example. Finally, consider the case where workers value the mandated
benefit by more than one dollar. This might apply to benefits (like insurance) that run into
adverse selection problems when voluntarily provided by firms or private providers;
problems which are avoided when provision is mandated. Who benefits, and who loses?
5. The one-good general-equilibrium model.
Now imagine an entire closed economy (not trading with other economies) where a fixed
but large number, N, of identical firms all produce the same good, y using a vector of
inputs, x. Each of these firms is small enough that it takes the economy’s vector of factor
prices, w, as fixed. The supply of factors to the entire economy, however, is fixed.
Denote the economy’s factor endowment vector by X. Thus, for the economy as a whole,
X is exogenous, while the vector of input prices, w, is endogenously determined. (In
principle the price of the single output good is endogenous too, but we need a numeraire
commodity and it is the natural one. So we set its price equal to one; effectively this
means we measure wages and other factor prices in terms of units of output –GDP if you
prefer—that are paid to each factor owner.)
Referring back to the first-order conditions for profit-maximization by a single,
representative firm and setting p=1, we note that each firm’s use of inputs must satisfy:
w1 = F1(x1 , x2 , … xn)
w2 = F2(x1 , x2 , … xn)
. = ……..
…….
wn = Fn(x1 , x2 , … xn)
(3)
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where F is the representative firm’s production function and the x’s are the amounts of
factors used by a representative firm. Finally, note that since all the firms are identical,
we must have xi = Xi/ N for all i. Thus (3) gives us a system of equations that gives the
economy’s equilibrium factor prices, w, as a function of the economy’s factor
endowment vector, X. Note that the predicted effect of an increase in the economy’s
endowment of a factor i on factor j’s equilibrium price is simply given by the appropriate
term of the Hessian matrix of the production function. So, in some sense, this
relationship is mathematically more ‘immediate’ than the effects of prices on quantities at
the firm level summarized in (1): in order to derive those factor demand functions, we
had to invert the matrix of FOCs represented by equation (3). Finally, note that concavity
of the production function (Fii < 0) means that the ‘own’ effects of increases in factor
endowments on factor prices should be negative.
Definition (Hamermesh 1993): factors i and j are q-complements iff ∂wi/∂Xj > 0.
Otherwise they are q-substitutes. In other words, xi and xj are q-complements if an
increase in the economy’s endowment of factor j leads an increase in the equilibrium
price of factor i. Thus, another term for q-complements in the trade literature is
“friends”. Note three things about this: First, every factor is its own enemy, since Fii < 0
for all i. Second, as for p-complementarity, q-complementarity is symmetric. So we can
just as well say that two factors are q-complements iff “an increase in the endowed
quantity of one raises the equilibrium price of the other”. Third, whether a pair of factors
are p-complements has little or nothing to do with whether they are q-complements. To
illustrate this, do the following exercise:
Exercise: Show that, for a Cobb-Douglas production function, all factors are psubstitutes, and q-complements.
One-good general-equilibrium models like the above have frequently been used as an
interpretive framework for estimating the effects of changes in factor endowments (such
as immigration, cohort size, and expansions in higher education) on (the distribution of)
wages. Because estimating these effects usually requires functional form assumptions,
we’ll engage in one last digression before discussing empirical approaches:
understanding the CES production function.
6. The CES production function.
a) The elasticity of substitution
To set the stage, we first define the “elasticity of substitution” between two factors of
production. Imagine a firm that uses two inputs, say L and K. Suppose it produces a
fixed amount of output, Q , at the minimum possible total cost, TC = wL +rK. As is
well known, the cost-minimizing point is where the isoquant for Q (whose slope equals
the ratio of marginal products, MPL/ MPK) equals the slope of a tangent isocost curve,
whose slope equals w/r (both slopes are in absolute values), as shown in the figure below.
Next, consider how this firm would respond to an increase in the relative price of labor,
w/r. As shown in the figure below, its input ratio, K/L, will rise from (K/L)0 to (K/L)1.
This is the familiar pure substitution effect of a factor price increase, now framed in terms
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of input ratios (i.e. capital intensity in this particular example) instead of the levels of the
individual inputs.
K
(K/L)1
Slope =
(-w/r)1
(K/L)0
Isoquant for Q
Slope = (-w/r)0
L
Now, notice that the rate of increase in the input ratio, K/L, when w/r rises is a natural
indicator of how easy it is for the firm to substitute K for L in response to factor price
changes: For example, if the isoquants are straight lines (the polar case where K and L
are perfect substitutes), even small increases in w/r can lead the firm to switch from using
no capital (K/L =0) to using only capital (K/L= ∞). At the opposite extreme of zero
substitutability between inputs (i.e. a Leontief production function, whose isoquants are
rectangular), the cost-minimizing input mix does not respond to factor price changes at
all. For intermediate cases, it therefore seems natural to define the elasticity of
substitution:

 log( K / L)
along an isoquant (i.e. holding output constant).
 log( MPL / MPK )
as a measure of how substitutable any two inputs are for one another. (Note that I have
expressed σ as a function of the ratio of marginal products, rather than w/r –which would
have been equivalent— to emphasize the fact that σ is a property of the production
function only. This property of course has consequences for how a firm responds to price
changes, but it is purely a feature of the ‘shape’ of the production function).
b) The CES production function.
Notice that σ, as defined above, is a local property of a production function; this ratio of
derivatives will in general change as we move along an isoquant, and in general will vary
with the level of output, Q, as well.1 If we want just a single, overall indicator of how
substitutable inputs are for a particular firm (or other productive unit), it is helpful to have
1
If the slope of the tangent isoquant does not change along a ray from the origin (i.e. as we expand output
holding K/L fixed), the production function is said to be homothetic.
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a production function where σ is the same at all points in the diagram. That way we can
refer to “the” elasticity of substitution for that firm.
About 50 years ago, economists proved that there exists only one functional form for the
production function with this property; this is the CES (“constant elasticity of
substitution) production function. In the case of two inputs, L and K, it has the formula:

Q   L L   K K 

1/ 
(4)
This production function has constant returns to scale iff αL + αK =1, which is often
assumed. The parameter ρ can take on any value from minus infinity to plus one, and is
related to the elasticity of substitution in production, σ, via the simple formula:
σ = 1/(1- ρ).
(5)
The relationship between ρ, σ and the type of production function for some important
special cases is summarized in the table below:
Ρ
1
0
-∞
σ
∞
1
0
Production function
Perfect substitutes: Q= αLL+ αKK
Cobb Douglas
Leontief: Q= min(L, K)
Isoquants
Straight lines
“Typical”, convex
Rectangular
Suppose now that a firm with a CES production function was behaving optimally, by
minimizing its cost for any given output. Can we work out what its input ratio would be?
To do this, recall that at an optimum, (MPL/ MPK) = w/r, and calculate the ratio of
marginal products:
MPL  (1 /  )  L L   K K 




MPK  (1 /  )  L L   K K 
[(1/  ) 1]
[(1/  ) 1]
  L L 1
(6)
  K K  1
(7)
Taking the ratio of these and using the optimality condition yields:
w MPL  L  L 


 
r MPK  K  K 
 1
(8)
Solving this for L/K as a function of w/r yields a simple expression for the firm’s optimal
input mix as a function of the input price ratio it faces:
1
1

  1   w   1
   w
L
  L   
  L   
K
 K   r 
 K   r 

(9)
Taking logs,
log( L / K )   log(  L /  K )   log( w / r )
(10)
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Differentiating (10) reveals that
 log( L / K )
  , as required by the definition of σ.
log( w / r )
An implication of (10) is that if you had (say) time-series data on a firm facing varying
input prices, you could estimate σ by running the linear regression suggested by (10).
One advantage of estimating σ this way (rather than calibration) is that the regression
doesn’t impose any restrictions on the sign of the estimate (which theory says must be
positive). Thus the data can reject the model.
c) Application to one-good general equilibrium models and estimation.
Consider now a country (or city) producing a single aggregate output Q from two inputs,
LH and LC, via the CES production function:

Q   H ( LH )    C ( LC ) 

1/ 
(11)
where LH are the economy’s exogenous endowments of “high-school-equivalent” labor,
and LC is “college-equivalent” labor. If the firms in this economy all minimize costs, the
equilibrium ratio of input prices (wC / wR ) must again equal the ratio of marginal
products; i.e. writing (8) for the inputs LH and LC:
wH  H

wC  C
L
  H
 LC



 1
(12)
Taking logs,
w
log  H
 wC

 
L
  log  H   (   1) log  H

 C 
 LC



(13)
Now imagine we had data on a bunch of economies (they will be U.S. cities empirically)
that had CES production functions and were identical in all respects except for their
exogenous relative endowments of LH and LC . If, in this sample of economies, we
regressed the log of the relative prices of these two factors, wH/wC , on the log of their
relative factor endowments, LH/LC, the resulting coefficient is an estimate of ρ-1 = -1/σ
(where the last equality follows from equation 5).
In words, suppose we looked at a cross-section of cities. If our framework is correct, we
expect to find that cities where high-school-equivalent labor, LH, is relatively abundant
(for example because of high levels of unskilled immigration) have a lower relative wage
of high-school-equivalent workers, i.e. a lower wH/wC.. Further, the magnitude of this
effect is an estimate of 1/σ: Large effects indicate that it is difficult for firms to substitute
LH for LC, while small effects indicate that substitution is easy. In the latter case,
economies can easily absorb large inflows of a particular type of labor without leading to
large changes in the relative wages of different labor types, because it is easy to substitute
one type of labor for the other.
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d) Extensions
Mathematically, it is straightforward to generalize the CES production function to more
than two inputs. When there are n inputs, x1 , x2 , … xn, the CES production function is
simply:




 1/ 
Q  1 x1   2 x2  ...   n xn
1/ 
 n

   i xi 
 i 1

(12)
which emphasizes the fact that the CES production function is nothing more than a
(weighted) geometric mean of the input levels. An advantage of the multifactor CES is
that a firm’s relative demand for any two factors (xi /xj) depends only on those two
factors’ relative prices, (wi /wj). A disadvantage of the multifactor CES is that it forces
the elasticity of substitution to be the same for all pairs of factors. The function has only
one σ that applies to all input pairs.
To address this limitation, many applied researchers have used the nested CES. For
example, suppose there are three inputs: capital (K), skilled labor (S) and unskilled labor
(U). To allow for “capital-skill complementarity” one could specify the following:




T  S S  K K 
1/ 
(13)
and
Q   T T   U U 
1/
(14)
where T is an aggregate “technology” input. To get the firm’s ‘overall’ production
function, just substitute (13) into (14). Because ρ and ν are distinct, the elasticity of
substitution between S and K can differ from the elasticity of substitution between the
technology aggregate, T, and unskilled labor U.
7. Estimating the effects of factor supply shocks on wages
Empirically, the basic idea is to estimate the effects of immigration-induced
changes in the supply of different types of labor on the equilibrium prices of all labor
types. Modeling choices include:
The production function framework that is assumed: no framework; log-linear
system of factor price or share equations (consistent with translog production
function); CES (usually nested).
How labor is disaggregated: natives versus migrants; migrants by years since
arrival; education groups; experience groups).
The unit of analysis: city versus national (This is important because cities can
adjust to immigration shocks in some ways that are harder for nations: flows of
native workers, capital and goods trade).
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Instrumenting for factor supplies? Especially at the city level, unobserved
labor demand shocks (like a technology boom) can both attract immigrants and
raise wages; thus immigrant inflows may be endogenous.
Some key studies:
Grossman, J. B. (1982). The substitutability of natives and immigrants in production.
Review of Economics and Statistics, 64(4), 596-603.
Factor shares/translog approach; labor disaggregated by migrant and education
status; cities. .
Lalonde, R. J., and R. Topel, "Labor Market Adjustments to Increased Immigration", in J.
Abowd and R. Freeman, eds., Immigration, Trade, and the Labor Market,
Chicago, University of Chicago Press, 1991.
Log-linear system of wage equations; cities in two Census years; labor
disaggregated by natives and (migrants x years since arrival).
Card, D. "The Impact of the Mariel Boatlift on the Miami Labor Market", Industrial and
Labor Relations Review 43 (January 1990): 245-58.
Nonstructural approach, labor disaggregated by skill and migrant status; Miami
versus ‘control’ cities.
Borjas, G. "The Labor Demand Curve Is Downward Sloping: Re-examining the Impact
of Immigration on the Labor Market". Quarterly Journal of Economics 118(4)
(November 2003): 1335-1374.
Nonstructural and three-level CES approach; labor disaggregated by
experience/education cells; national level.
Card, David. “Immigration and Inequality” American Economic Review 99(2) (May
2009): 1-21.
CES approach; three education groups; panel-of-cities context. Introduces the
lagged stock of immigrants by origin country in a city, interacted with current
national inflows by origin country, to generate an instrument for immigrant
inflows.
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Limitations to the basic one-sector general-equilibrium model as an interpretive
lens:
1. It is commonplace to think of immigrant inflows as flows of labor only. But if
immigrants bring some other factor with them (capital, entrepreneurship), or if
immigrant inflows induce other factor flows (e.g. endogenous capital inflows in a
model with free capital mobility), immigration can have no effect on wages. For
example, with a CRS production function, a balanced inflow of factors has no effect on
any factor prices. Agglomeration effects or externalities can also create positive own
wage effects of immigration. See Beaudry, Green and Sand (2014) for a model where
migrants bring entrepreneurship.
2. Predictions can also change dramatically if we allow for multiple goods and trade.
For example, in the ‘classic’ trade model, Samuelson’s factor price equalization
theorem holds, which argues that free trade between countries (or cities) with the same
production function will equalize their factor prices, even when no factor flows are
possible between the countries. Thus, immigration can have zero effects on equilibrium
factor prices in this model as well (See Kuhn and Wooton for an early statement of this
result in the immigration context.)
Other effects of immigration:
On prices:
Cortes, Patricia. “The Effect of Low-Skilled Immigration on U.S. Prices: Evidence from
CPI Data”. Journal of Political Economy 116(3) (June 2008): 381-422.
On natives’ geographical migration:
Card, David. “Immigrant inflows, native outflows, and the local labor market impacts of
higher immigration”. Journal of Labor Economics 19: 22-64. (2001).
On natives’ occupational/industrial migration:
Peri, Giovanni; and Chad Sparber. “Task Specialization, Immigration, and Wages”
American Economic Journal: Applied Economics. Vol. 1 (3). p 135-69. July 2009.
On trade:
Peri, Giovanni and Francisco Requena. “The Trade Creation Effect of Immigrants:
Evidence from the Remarkable Case of Spain” NBER working paper no. 15625
(December 2009)
On automation:
Lewis, Ethan. “Immigration, Skill Mix, and Capital Skill Complementarity” The
Quarterly Journal of Economics (2011) 126(2): 1029-1069 (May)