Labor demand in the long run Lecture notes

Yesterday We wrote down the profit maximization problem for the firm in the short-run. And derived the short-run labor demand curve. Negatively sloped. Next, we wrote down the problem in the long-run, when the firm can adjust both K and E. We wanted to derive the long-run labor demand curve.

The two-stage problem For ease of interpretationm, we can split the problem for the firm into two stages: Choose the optimal combination of K, E, for a given q. Choose the optimal level of q. Then, we can think about what happens when Dw: A “scale effect” (on q). A “substitution effect” (on the combination of K,E).

The graphical analysis We depicted the two effects graphically. Concluding that a Dw will: Decrease employment (both scale and substitution effects). Have an ambiguous effect on K (conflicting scale and substitution effects).

A more formal analysis Stage 1: Cost minimization The minimization problem. Solution: The “conditional demands” for E and K. They depend only on q, and w/r. The “cost function” of the firm.

Porperties of the cost function C is increasing in each of its arguments (w, r and q). C is concave in w and r. C satisfies Shephard’s lemma.

Properties of the conditional factor demands How do E and K vary with w, r? Conditional labor demand. Conditional demand for capital. Symmetry of cross-price effects.

Variation in the level of output Effect of an exogenous change in q on the total cost. We can differentiate with respect to q ad take into account the optimality condition. Total cost rises with the level of output. At least one factor increases, but the other one doesn’t have to. Common to assume “normal inputs”.

Stage 2: Choosing the level of production The firm chooses the level of output (the isoquant) that maximizes profits. The maximization problem. The first order condition. Price equals marginal cost.

Imperfect competition Price equals marginal cost, times a “mark-up”. See Cahuc & Zylberberg.

The long-run unconditional demands

The “laws” of demand The unconditional demand for a factor decreases with its cost.