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
w(L) is the labor supply curve facing the firm (and industry)
MFC (marginal factor cost) is the derivative of total labor costs (w(L)∙L) wrt L.
VMP is the marginal revenue from another unit of L.
The monoponist 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.
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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 and 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
labelled “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 labelled
“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 labelled “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.
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4. Some Simple Economics of Mandated Benefits (Summers, AER May 1989)
S0
$α
w1+$1
w0
w1+$α
firms’ share
workers’ share
S1
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’.
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 labelled “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 labelled “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.
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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 a 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)
. = ……..
(3)
…….
wn = Fn(x1 , x2 , … xn)
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 much simpler than the effects of prices on quantities at the
firm level, which requires us to solve a maximization problem. 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:
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Exercise: Show that, for a Cobb-Douglas production function, all factors are psubstitutes, and q-complements.
Some additional notes on the one-sector general-equilibrium model:
1. This model is often used as an interpretive framework for studying the effects of
changes in factor endowments (including immigration, cohort size, etc) on (the
distribution of) wages.
2. Note that, if factors move together, the predicted effects of factor flows can be
unexpected. For example, while an inflow of any one factor alone is predicted to lower
its equilibrium return (this is just the ‘law’ – or if you prefer, assumption—of diminishing
returns to any one factor), a ‘balanced’ inflow of all factors will have zero effect on factor
prices if the production function exhibits constant returns to scale. So, if immigrants
bring capital with them when they immigrate…. More deeply, this discussion the
question of whether economies have any fixed factors, and if so, what are they? (land,
climate, political stability, business culture….).
3. 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 with the same production function
will equalize their factor prices, even when no factor flows are possible between the
countries. Thus, international factor movements have zero effects on equilibrium factor
prices in this model as well.