LABOR ECONOMICS
Lecture 1: Labor Demand, Market Equilibrium, and
Economic Efficiency
Prof. Saul Hoffman
Université de Paris 1 Panthéon-Sorbonne
March, 2013
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COMPETITIVE SHORT RUN LABOR DEMAND
“Marginal Productivity Theory” of wages; emphasis on determinants of demand
Model: Competitive input and output markets (p and w exogenous). Short run - K is fixed.
Only choice variable is…. ?
Derived from π-Max: π(Q) = pQ - C(Q), where Q=f(L,K)
Rewrite in terms of inputs to emphasize corresponding choice of inputs:
Choose L to Max π(L; p, K) = pf(L,K) - wL - rK.
Solution (FOC): ∂π/∂L = pfL(L*; K) - w = 0 → pfL(L*; K) = w.
This is famous rule of π-max input demand. LHS is MRP or VMP.
FOC identifies best L: L* = L(w; p, K) labor demand function → π-max L for any w, p, K.
Solution (SOC): For maximum, must have ∂2π(L*; K)/∂L2 < 0.
Here, ∂2π(L*; K)/∂L2 = pfLL. If prod function exhibits Dim Marg Returns, then fLL< 0 and SOC
for a max holds.
See Graph.
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COMPARATIVE STATICS OF LABOR DEMAND
Most important part of any theory. Refutable. In terms of observables.
What happens to L* when w changes and firm adjusts so as to max π at new wage?
Reveals shape of Labor Demand Curve (∂L*/∂w)
General: FOC must hold both before and after change in exogenous variable (w)
Method #1: Rewrite FOC as an identity and then differentiate both sides wrt variable
of interest.
• Subst labor demand function into FOC to get an identity → pfL(L*(w, p; K); K) ≡ w.
In words: MRP always equals the wage when L is chosen so that MRP = wage
• Take derivative: pfLL x ∂L*(w; p,K)/∂w ≡ ∂w/∂w (=1)
• Rearrange to get ∂L*(w; p,K)/∂w = 1/pfLL.
• This is < 0, b/c fLL < 0 from SOC. Almost always true that comparative statics
depends on SOC.
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COMPARATIVE STATICS (continued)
• Method #2: The Implicit Function Approach
If y=f(x) is explicit function linking x & y, then g(x,y) = y-f(x) = 0 is
corresponding implicit function. Note: this is standard form of FOC, e.g.,
pfL(L*; K) - w = 0
• Implicit Function Theorem: if g(x,y)= y-f(x)=0, then dy/dx=-gX/gY
• In Labor Demand Problem: g(L,w) = pfL(L*)–w=0
• By IFT: dL*/dw = -gw/gL = - (-1)/pfLL = 1/pfLL
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EXTENSIONS
Can compute comparative statics for any exogenous variable.
What happens to L* if p changes?
Use same method, take derivative with respect to p.
pfL(L*(w, p; K); K) ≡ w .
pfLL(∂L*/∂p) + fL(∂p/∂p) ≡ ∂w/∂p (=0)
∂L*/∂p = - fL/pfLL = - (pos)/neg = > 0.
Interpretation: p increases b/c of demand, not cost (w is constant). See graph
By IFT: g(L,w) = pfL(L*) –w =0
dL*/dp = -gp/gL = - fL/pfLL
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LONG RUN LABOR DEMAND
Same problem but K is now variable
Max π(L) = pf(L,K) - wL – rK
FOCs:
∂π/∂L = pfL(L*, K*) - w = 0 → pfL(L*,K*) = w
∂π/∂K = pfK(L*, K*) - r = 0 → pfK(L*,K*) = r
Ratio of two equations gives familiar expression from cost-minimization:
fL(L*,K*)/fK(L*,K*) = w/r or ratio of MP = ratio of factor prices
SOC: fLL< 0; fKK < 0; fLL fKK - fLk2 > 0
Comparative statics are more complicated, but result….
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Labor Market Equilibrium
• Find total market demand
• Labor Supply?
• Equilibrium and its implications
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ECONOMIC EFFICIENCY
• Efficiency issue #1: Allocate a fixed amount of L across two
production processes to maximize the value of output
(studying, campaigning, allocating resources generally):
• Max Z (L1,L2 ) = p1f(L1,K) + p2f(L2,K) + λ ( L1 + L2 - L)
• ∂Z/∂L1 = p1fL1(L1*) + λ = 0
• ∂Z/∂L2 = p2fL2(L2*) + λ = 0
• Solution is (L1*, L2*) such that p1MP1(L1*)=p2MP2(L2*) and L1* + L2*= L
• Think about this. Very useful, with lots of applications
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Numerical Example
• Let p1=p2=1
• Q1=f(L1,K)=10L1.5; Q2=f(L2,K)=40L2.5; L1 + L2 = 20
• Find L1* and L2* to maximize value of output
• Solution:
• p1MP1(L1*)=p2MP2(L2*)→
• L1* + L2*= L
• Evaluate and solve simultaneously to get: L1*= ? & L2*= ?
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ECONOMIC EFFICIENCY (cont.)
Problem # 2: Find Pareto Efficient (PE) allocation of inputs (L, K) in production of two
outputs: Choose L & K to Max Q1 for given value of Q2, L, K.
MAX Z =f1(L1,K1) + λ(f2(L2, K2) - Q2) + λL(L1 + L2 - L) + λK(K1 + K2 - K)
FOCs:
1) ∂Z/∂L1 = fL1+ λL = 0 & 2) ∂Z/∂K1 = fK1+ λK = 0 →
3) ∂Z/∂L2 =λfL2+ λL = 0 & 4) ∂Z/∂K2= λfK2 + λK = 0.
• Dividing (1) by (2) and (3) by (4), we have fL1/fK1= λL/λK & fL2/fK2= λL/λK
• LHS of these equations is ratio of Marginal Products = MRTS(L*,K*)
• Therefore, PE choice of L* and K* must satisfy equality of MRTS for 1 & 2
Pareto condition is naturally solved in competitive mkts where firms face same prices w and
r for their inputs
Firm 1 chooses L and K such that MRTS(L1*, K1*) = w/r. Firm 2 does same, choosing L and K
such that MRTS(L2*, K2*) = w/r.