CLASS #12: LABOR DEMAND AND LABOR SUPPLY

• Labor Demand
• How the demand for labor changes with:
  • productivity shocks (oil shocks)
  • the aggregate level of capital
• Labor supply
• cross country differences in real wages

THE DEMAND FOR LABOR

in order to understand labor demand we have to study the problem of the firm (we assume that the labor market is competitive)

$$max_{K,N} F(K,N) + (1-\delta)K -(1+r)K -wN$$

The FOCs for the firm are:

$\begin{array}{l} (F_1(K,N) + (1-\delta)) = (1+r) \\ F_2(K,N) = w \end{array}$

The second condition, for given level of aggregate capital, gives the demand for labor. In particular, if the production function is Cobb-Douglas, we obtain:

$$(1-\alpha) A K^{\alpha} N^{-\alpha} = w$$

Firms demand labor until the marginal productivity of labor is equal to the wage.

• The demand for labor is a decreasing function of the wage (why?)
• The demand for labor shifts with:
  • Productivity shocks (why?)
  • Changes the aggregate level of capital (why?)

Note that w is the real wage: amount of "cookies" per hour of work Let us assume that labor supply is fixed (N=1), then we obtain that cross country differences in the wage depend on:

• differences in productivity
• differences in the aggregate level of capital (both physical and human capital)

Notice that the real wage can be rewritten as:

$$(1-\alpha) A (K/N)^{\alpha} = w$$

The "living standards" in a country (real wage) depend on the amount of capital per worker, as well as on productivity.

THE SUPPLY OF LABOR

To study the supply of labor we use a static model:

• The amount of capital K is inherited from the previous period.
• The individual has 1 unit of time available, which she can spend working(N) or enjoying leisure(L): N=1-L.
• The agent has utility over both consumption and leisure.

$$max_{\{C,N\}} U(C,L)$$

subject to:

C=(1+r)K+w(1-L)

Substitute the constraint in the objective function and obtain:

max_L U((1+r)K+w(1-L),L)

The first order condition is:

U_C(C,L)w=U_L(C,L)

Individuals will work until the marginal productivity of consumption times the real wage is equal to the marginal productivity of labor.
Let us use the following functional form for the utility function:

$$U(C,L)= \gamma log(C)+ (1-\gamma) log(L)$$

Then the first order conditions become:

$$1-N = \frac{C}{w} \frac{1-\gamma}{\gamma}$$

• An increase in wages will increase labor supply, for given C.
• An increase in consumption will decrease labor supply, for given wage.

For a more general form of the utility function an increase in wages may not increase labor supply.

Graphical analysis

• Substitution effect: as wages increase leisure is more expensive $\rightarrow$ work more
• Wealth effect: as wages increase the agent has more wealth. If leisure is a normal good $\rightarrow$ work less
• for low levels of wage (and hence of consumption) the marginal utility of consumption is high an increase in wages induces people to work more
• for high levels of levels of wage (and hence of consumption), the marginal utility of consumption is low, and the marginal utility of leisure high, an increase in wages may induce people to work less