LABOR ECONOMICS Lecture 1 Labor Demand Market Equilibrium

LABOR ECONOMICS Lecture 1: Labor Demand, Market Equilibrium, and Economic Efficiency Prof. Saul Hoffman Université de Paris 1 Panthéon-Sorbonne March, 2013

2 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) = p. Q - 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) - w. L - r. K. Solution (FOC): ∂π/∂L = pf. L(L*; K) - w = 0 → pf. L(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)/∂L 2 < 0. Here, ∂2π(L*; K)/∂L 2 = pf. LL. If prod function exhibits Dim Marg Returns, then f. LL< 0 and SOC for a max holds. See Graph.

3 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 → pf. L(L*(w, p; K) ≡ w. In words: MRP always equals the wage when L is chosen so that MRP = wage • Take derivative: pf. LL x ∂L*(w; p, K)/∂w ≡ ∂w/∂w (=1) • Rearrange to get ∂L*(w; p, K)/∂w = 1/pf. LL. • This is < 0, b/c f. LL < 0 from SOC. Almost always true that comparative statics depends on SOC.

4 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. , pf. L(L*; K) - w = 0 • Implicit Function Theorem: if g(x, y)= y-f(x)=0, then dy/dx=-g. X/g. Y • In Labor Demand Problem: g(L, w) = pf. L(L*)–w=0 • By IFT: d. L*/dw = -gw/g. L = - (-1)/pf. LL = 1/pf. LL

5 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. pf. L(L*(w, p; K) ≡ w. pf. LL(∂L*/∂p) + f. L(∂p/∂p) ≡ ∂w/∂p (=0) ∂L*/∂p = - f. L/pf. LL = - (pos)/neg = > 0. Interpretation: p increases b/c of demand, not cost (w is constant). See graph By IFT: g(L, w) = pf. L(L*) –w =0 d. L*/dp = -gp/g. L = - f. L/pf. LL

6 LONG RUN LABOR DEMAND Same problem but K is now variable Max π(L) = pf(L, K) - w. L – r. K FOCs: ∂π/∂L = pf. L(L*, K*) - w = 0 → pf. L(L*, K*) = w ∂π/∂K = pf. K(L*, K*) - r = 0 → pf. K(L*, K*) = r Ratio of two equations gives familiar expression from cost-minimization: f. L(L*, K*)/f. K(L*, K*) = w/r or ratio of MP = ratio of factor prices SOC: f. LL< 0; f. KK < 0; f. LL f. KK - f. Lk 2 > 0 Comparative statics are more complicated, but result….

7 Labor Market Equilibrium • Find total market demand • Labor Supply? • Equilibrium and its implications

8 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 (L 1, L 2 ) = p 1 f(L 1, K) + p 2 f(L 2, K) + λ ( L 1 + L 2 - L) • ∂Z/∂L 1 = p 1 f. L 1(L 1*) + λ = 0 • ∂Z/∂L 2 = p 2 f. L 2(L 2*) + λ = 0 • Solution is (L 1*, L 2*) such that p 1 MP 1(L 1*)=p 2 MP 2(L 2*) and L 1* + L 2*= L • Think about this. Very useful, with lots of applications

9 Numerical Example • Let p 1=p 2=1 • Q 1=f(L 1, K)=10 L 1. 5; Q 2=f(L 2, K)=40 L 2. 5; L 1 + L 2 = 20 • Find L 1* and L 2* to maximize value of output • Solution: • p 1 MP 1(L 1*)=p 2 MP 2(L 2*)→ • L 1* + L 2*= L • Evaluate and solve simultaneously to get: L 1*= ? & L 2*= ?

10 ECONOMIC EFFICIENCY (cont. ) Problem # 2: Find Pareto Efficient (PE) allocation of inputs (L, K) in production of two outputs: Choose L & K to Max Q 1 for given value of Q 2, L, K. MAX Z =f 1(L 1, K 1) + λ(f 2(L 2, K 2) - Q 2) + λL(L 1 + L 2 - L) + λK(K 1 + K 2 - K) FOCs: 1) ∂Z/∂L 1 = f. L 1+ λL = 0 & 2) ∂Z/∂K 1 = f. K 1+ λK = 0 → 3) ∂Z/∂L 2 =λf. L 2+ λL = 0 & 4) ∂Z/∂K 2= λf. K 2 + λK = 0. • Dividing (1) by (2) and (3) by (4), we have f. L 1/f. K 1= λL/λK & f. L 2/f. K 2= 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(L 1*, K 1*) = w/r. Firm 2 does same, choosing L and K such that MRTS(L 2*, K 2*) = w/r.