Labour Demand
- Sources for labour demand theory:
Cahuc, Carcillo and Zylberberg, Ch. 4
Hamermesh Labor Demand Ch.2 p.18-33,44-55 (theory), Ch. 3 (see: empirical estimates)
- Interest in labour demand?
- Employment and wages determined by labour demand’s interaction with supply.
- So labour demand provides a partial explanation of employment-wage outcomes:
- establishes link between: productivity and wages
- partial explanation of wage structure, inequality, income distribution
- identifies factors behind employment distribution between sectors.
- Extreme cases:
- flat labour supply: demand determines only employment
(sensible perhaps for narrowly defined labour markets)
- vertical labour supply: demand determines only wages.
(perhaps most sensible for aggregate labour markets)
- Analysis of policies that affect labour costs depends on labour demand elasticities:
- effects of minimum wages, labour standards laws, payroll taxes, pay equity
policies, etc.
- implications of international trade and trade policy for wages and employment
depend of how labour demand is determined.
- In uncompetitive labour markets labour demand parameters affect outcomes.
e.g. in bargaining and union sector models: wage-elasticity of labour demand can be
a key determinant of bargaining power.
Theory of Labour Demand
- A branch of the theory of the firm: factor demand theory.
- Roots in the 1870s-1890s
- A theory of income distribution: "marginal productivity" theory.
- Marshall (1890s) and Hicks (1930s):
- Post-WWII: part of production theory.
- Following provides an overview of key results from theory.
Labour the Only (Variable) Input: the Marginal Productivity Theory of Labour Demand
- Firm maximizes profits.
- not an appropriate assumption for all cases (Hamermesh has a brief section on non-profit
employers).
- Production function:
Y = f(L) Y = output
L = quantity of labour hired
f', f'' are first and second derivatives of f(L)
f'>0 labour is productive (f' ≡marginal product of labour)
f''<0 diminishing returns.
(note: there may be other "fixed" inputs implicit in f(L) )
- Profit:
p f(L) - wL p = output price
w = wage rate
- Assume competitive labour and output markets:
- the firm is a price and wage taker: p and w fixed.
i.e. firm is too small to affect prices.
- so: dp/dL = dw/dL = 0 (Cahuc, Carcillo and Zylberberg allow for
possibility that dp/dL may not be zero in
their base case -- see below)
- Profit maximization:
max p f(L) - w L
<L>
f.o.c: d(Profit)/dL = p f'(L) - w = 0
so: p f'(L) = w
(value of marginal product = wage ; this holds at L* in the diagram)
or: f'(L) = w/p
(marginal product = real (product) wage)
- second order condition (for maximum) requires:
pf''(L) <0 (need diminishing returns!)
- The first order condition suggests:
Ld = Ld(w,p; form of f)
- labour demand curve for the firm.
- downward sloping:
p f'(L) - w = 0
p f''(L) dL - dw = 0
dL/dw = 1/pf''(L) < 0
e.g., Y = ALa , 0<a<1
f.o.c. p a A La-1 - w = 0
L = (paA/w)1/(1-a)
- Determinants of a firm’s labour demand:
- Wage rate
- Determinants of (value of) productivity:
- demand depends on output market conditions (here via output price: p)
- technology: form or parameters of f, (i.e. a, A in the example)
- A : could also represent the effect of a fixed input on Y. In which case quantity of
the fixed input will also determine L.
- Market level labour demand:
- sum of labour demand across all firms hiring the type of labour.
Lmarket = Σi Li(w,p ; parameters of f) sum taken over all firms (i – denotes a firm)
then: dLmarket/dw = Σi dLi/dw < 0
- A possible complication at the market level?
- feedback effects via the output market are possible:
all firms change hiring decisions, supply of output (Y) changes, this affects
output price, this in turn affects labour demand.
dLmarket/dw = Σi dLi/dw + Σi [ (dLi/dp) x (dp/dY ) x (dY/dw)]
(-) (+) (-) (-)
|______|
(+)
- feedback through the output market dampens effect of the wage on labour demand.
( see discussion of Hick-Marshall laws of derived demand)
- The simple labour demand model has some important implications:
- Wages: labour is paid in line with the value of its marginal product (p f'(L) ).
- leads to demand-side explanations of wages and wage patterns.
- wage levels are rooted in determinants of (marginal) productivity
- technology
- quantities (and qualities) of other inputs
- output market conditions
examples:
Industrial Revolution and wage growth:
- new technologies, more physical capital combined with L.
- MP theory: wage will rise with productivity.
Wage inequality in US:
- key role of demand shifts: - trade (via output market effects)
- technological change and productivity of
skilled and unskilled workers.
(Katz and Murphy (1992) an example)
- Distribution of employment: product market conditions and technology are important
determinants of employment structure.
- Labour (factor) allocation models:
- marginal productivity theory of labour demand is at their core.
Firms hire until:
pi fi'(Li) = wi for a representative firm in sector i
- Fixed amount of labour to be allocated: L* (so fixed total labour supply).
- Add assumption that labour moves to where rewards are highest.
Migrate from sector i to sector j if:
wi < wj (ignoring mobility costs)
- Equilibrium?
wi = wj
so: pi fi'(Li) = pj fj'(Lj)
and: Li + Lj = L* (supply = demand for labour)
- Result:
- labour is allocated between sectors according to where its value is highest;
- value of labour in each sector reflects: relative output prices (pi vs. pj) and relative
marginal products (f’ in each sector).
- Multiple sector models:
Shifts from agriculture ? manufacturing ? services
- Migration models:
Alberta vs. Maritime provinces
- Labour allocation rooted in demand determinants between uses of labour.
Extending the Basic Model of Labour:
Some extensions:
- Relaxing the price taker, wage taker assumptions.
- Multiple variable inputs (will focus on 2 factor case)
- Hours-Employment Choice for Employers.
Firm not a Price Taker
- Let: p=p(Y) p'(Y)<0 (p' is dp/dY)
i.e., firm faces a downward sloping demand for its output.
- so feedback of hiring decision to output market is present even at the firm level.
- Now choice of L affects output price, so:
max p[f(L)] f(L) – wL ( p(Y)=p[f(L)] since Y=f(L) )
<L>
f.o.c.
[p + Y dp/dY] f'(L) - w = 0
- marginal revenue = p + Y×dp/dY = p(1+1/eD) eD = elasticity of output demand
(dY/dp)(p/Y)
- "marginal revenue" product = wage
( Cahuc, Carcillo and Zylberberg write this as: f'(L) = v w/P
where v is defined as the markup v = 1/(1+1/eD ) in my notation 1/eD is the
same as their nYP
Markup? usually means markup of price over cost; here: w/f' = marginal cost of
producing output so condition is P = v w/f' )
- implied demand function:
Ld = Ld(w ; output demand parameters, technology)
- This extension does not change results of the model much.
- now demand parameters replace the output price as a determinant of labour demand.
Two Factors of Production: Labour Demand with Substitution
- Production function:
Y = F(K,L) K = quantity of physical capital
L = labour
assume: FK>0, FL>0 marginal products positive (FL= ∂Y∂L , FK= ∂Y∂K )
FKK<0, FLL<0 diminishing returns to K and L (strict concavity of f)
FLK>0 more of one input boosts productivity of the other input.
(where FLL= ∂FL∂L , Fkk= ∂FK∂K , FLK= ∂FL∂K = ∂FK∂L )
- Profit:
max p F(K,L) -wL - rK r = rental price of K (per period price)
<K,L p = output price, w= wage
Competition: price taker so w, r and p are fixed.
foc's:
K: p FK - r = 0
L: p FL - w = 0
- Hire L and K until factor prices equal value marginal products.
i.e. like the simple one-input version of labour demand.
- Combining foc’s gives:
FL/FK = w/r
marginal rate of technical substitution = w/r
- diagrams?
slope of isoquant = slope of iscost
- isoquant: combinations of K and L with same output level.
Y = F(K,L)
dY= FK dK + FL dL = 0
dK/dL = -FL/FK tradeoff of K for L that keeps Y constant.
- isocost: K and L combinations with the same cost level
Cost = wL + rK
d(Cost) = w dL + r dK = 0
dK/dL = - w/r tradeoff of K for L that keeps costs constant.
- Geometry: output is being produced in the least cost way.
(see diagram below)
- Usual Labour demand function with two inputs?
Ld = Ld(w,r,p; technology) note: now depends on both input prices.
- Constant Output (Conditional) Labour Demand Function (Ldc):
- Given output (Y*) the firm minimizes costs to maximize profits:
min wL + rK s.t. Y*=F(K,L)
<K,L
i.e. minimize the Lagrangianfor this problem: wL + rK + l [Y*-F(K,L)]
f.o.c's ( λ = Lagrange multiplier – here it is interpreted as “marginal cost”)
K: w - λ FL = 0
L: r - λ FK = 0
so again:
FL/FK = w/r (see diagram next page)
can solve f.o.c. and constraint for:
Ldc = Ldc(w,r,Y*; technology)
- constant output or conditional demand function.
- at the profit maximum conditional and unconditional demands are equal.
(Ld = Ldc)
(NOTE: this is also a theory of demand for K )
- Effects of a change in wage on labour demand:
∂Ld∂w= ∂Ldc∂w + ∂Ldc∂Y∂Y*∂w
first term: substitution effect – how labour demand changes with output constant.
second term: output of scale effect – changing w, changes costs of producing
output, this induces a change in output that affects labour demand.
- Notice that the substitution effect and scale/output effects are in the same direction.
i.e. so a rise in w lowers labour demand, fall in w raises labour demand.
- Examples of substitution:
- Substitution across time: robotics in the auto industry ; evolution of pulp and paper,
mining.
- Substitution at a point in time: production in same industry in high and low wage
countries.
(construction in Shanghai vs. Vancouver)
- History: is substitution the reason UK had the first industrial revolution?
Robert Allen: high wages, low energy prices and why UK has 1st
industrial revolution (K intensive production profitable).
- What factors determine the size of the wage effect on aggregate labour demand?
- Aggregate? summed across all employers of the above type of labour.
- An old question: Alfred Marshall 1890s, John Hicks 1930s.
“Hicks-Marshall Laws of Derived Demand”
- An answer is useful in thinking about the effects of policies that affect labour costs.
Hicks-Marshall Laws of Derived Demand:
- A set of rules for thinking about when labour demand is most sensitive to wage changes.
- Focus is on the size of the wage-elasticity of labour demand:
h = ∂Ld∂wwLD= ∂ln(Ld)∂ln(w)
- The “laws” highlight the roles of: - price elasticity of output demand (eD)
- elasticity of substitution between labour and capital (s)
- share of production costs due to labour (vL).
- the elasticity of supply of substitute inputs (hSK) – extended
version!
- Assuming competitive markets and constant returns to scale in production it can be shown that:
h = vLεD-(1-vL)σ
where: h = ∂Ld∂wwLD
eD =∂YD∂ppYD=dlnYDdlnp (p=output price, YD output demand)
σ = dKLdwr wrKL= dlnKLdlnwr (evaluated at given Y i.e. along isoquant)
vL= L wp Y and 1-vL = K rp Y
- this is how the results is most commonly stated result.
(a derivation based on Allen (1938) Mathematical Analysis for Economists pp. 317-
318, 340-343 and 369-373 is given below)
- Interpreting the Hicks-Marshall result:
h = vLεD-(1-vL)σ
- first term: vLεD
- reflects output effects :
- rising wages, raise the cost of producing output.
- in a competitive output market these higher costs translate into higher
output prices (aggregate effect: all firms responding)
- higher output prices lead to less output demand, less production and so less
labour demand.
- this output effect is captured by: vLeD (<0)
|h| is larger the larger is |eD| -- sensible: effect of a rise in p, caused by a rise
in w is large if consumers are responsive to price changes.
|h| is larger the larger is vL -- sensible: rise in w causes a larger rise in p if vL
(labour share of costs) is large.
- second term: -(1-vL)σ
- This is the substitution effect of the change in wages.
- Elasticity of substitution between K and L (s) defined as:
σ = dKLdwr wrKL= dlnKLdlnwr >0 holds output constant (on an isoquant)
note with cost minimization w/r=FL/FK so the expression could be written in terms
of FL/FK.
σ = dKLdFLFK FLFKKL= dlnKLdlnfLfK
- elasticity of substitution measures how K/L changes when w/r changes.
Higher s means greater change in K/L for a given change in w/r
i.e. larger degree of substitutability.
- value of s directly related to the shape of the isoquants:
(extremes: s=0 isoquants are right angles, no substitution
s=∞ isoquants are straight lines, infinite elasticity)
- larger is s the larger is the substitution effect of a change in w.
- so |h| is larger the larger is s.
- note that vL also affects the size of the substitution effect.
- larger vL the smaller the substitution effect (in absolute value).
- why? “the larger the quantity of labour is, the smaller the variations in the
quantity of labour expressed in percentage terms are.”
(CCZ p. 89)
- Hicks-Marshall laws of derived demand:
- Labour demand is more elastic (|h| is higher):
- the greater the elasticity of substitution (through σ and the substitution effect);
- the more elastic is output demand (via output effect);
- the higher share of costs due to labour (via the output effect) -- but this may be
countered by a smaller substitution effect.