Time Allocation • Income is fixed when allocating income among goods that provide utility. • Time is fixed when allocating time between leisure and working to earn income for consumption of goods, and time must be used as it passes. • Time can be spent – Working (earning income for consumption), – Consuming goods, – Maintaining ones self, (not earning income for consumption) – Sleeping. • Understanding this time allocation decision helps understand labor supply.

Two-Good Model • • • • • • • • •

Individuals spend time in leisure or working at a real wage rate of w, which allows consumption of goods, and they receive utility from both leisure and consumption. Utility = U(c,h), where c is consumption and h is hours of leisure (two composite goods). Constraint 1 is l + h = 24, where l is hours worked. Constraint 2 is c = wl, or consumption equals income. Combining Constraints 1 and 2 gives c = w(24 – h), or c + wh = 24w; where 24w equals maximum possible real consumption of goods per day. 24w is a person’s full income. A person’s full income (24w) can be spent by working for consumption of goods (c = wl) or not working (wh) and enjoying leisure. The opportunity cost of leisure is w per hour, or earnings forgone by not working an hour. The real wage rate (w) is the price of leisure. The price of $1 of consumption is $1.

Utility Maximization Model Max U = U(c,h) St. 24w – c – wh = 0 Lagrangian expression

U (c, h) λ(24w c wh)

FOCs U 1. λ 0 3. 24w c wh 0 c c U To maximize utility, the individual 2. wλ 0 should work up to the point where h h the marginal rate of substitution of Dividing 2 by 1 gives leisure (h) for consumption (c) is U h MU h MRS h for c w U c MU c

equal to the real wage rate (w), provided that the SOC are satisfied; MRSh for c is diminishing.

Income and Substitution Effects The real wage rate increases from w0 to w1 (the price of leisure increases). The constraint rotates to the right because the individual can consume more for the same amount of labor, ie., w1l > w0l. The individual is made better off by an increase in w because the individual is the supplier of labor. Consumption=c

This example assumes the individual will not reduce consumption when income increases in going from point B to point C (although consumption could increase because C consumption is a normal good).

Consumption=c

c1

C

c = w0(24)

B A

c0 0

h1 h0 Substitution Effect Income Effect

c1 c = w (24 – h) 0 U1 c = w1(24 – h) ); slope=-w1 c0 c = w0(24 – h); slope=-w0 U0 0 Leisure=h

h = 24, so c = 0.

B A

h0 h1

U1 c = w1(24 – h) U0 Leisure= h

Substitution Effect Income Effect

The substitution effect is always negative; w h in going from point A to point B. The income effect will be positive because leisure is a normal good; w h in going from point B to point C. The income and substitution effects work in opposite directions because h is a normal good. In the left graph, the substitution effect from an increase in the real wage rate outweighs the income effect, so h declines from h0 to h1 (l increases) as w increases. In the right graph, the income effect outweighs the substitution effect, so h increases from h0 to h1 (l decreases) as w increases. This example suggests that the supply of labor could be backward bending (negatively sloped) as in the right graph.

Mathematical Analysis of Labor Supply Change the constraint to add nonlabor income (n). c = wl + n n shifts the budget constraint out in a parallel manner, and if leisure is a normal good (∂h/∂n > 0), so ∂l/∂n < 0. An increase in n will increase the demand for leisure and reduce the supply of l. Thus, we have the individual’s labor supply function as l(w,n); the number of hours of labor supplied is a function of the real wage rate and nonlabor income. n plays the role of nominal income in the typical utility maximization problem. Look at the dual problem, which is Min E = c – wl Choose values of c and h that minimize additional St U0 = U(c,h) expenditures (n=c-wl) to achieve a given level of utility. Solve the dual for minimum E and apply the envelop theorem to get, ∂E/∂w = -lc = hc. This shows that a labor supply function can be calculated from the expenditure function by partial differentiation, but this is a compensated labor supply function because utility is held constant: lc(w, U). The compensated labor supply function is different from the uncompensated labor supply function: Max U=U(c,h); S.T. 24w-c-wh=0; Solve for c and h; h(w,n)=-l(w, n) is the uncompensated labor supply function.

Slutsky Equation for Labor Supply Explore the substitution and income effects of a change in w. First, expenditures minimized in the Dual [E(w, U)] are like nonlabor income (n) in the Primal. At the optimal point lc(w,U) = l(w,n) = l[w,E(w,U)]. Partially differentiate both sides with respect to w to get,

l c l l E Then realizing that ∂E/∂w = -lc and substituting n for E, . w w E w l c l l l l l c l l l . Now let U U0 and rearrange terms to get, w w E w n w w h (leisure) is a normal good (∂h/∂n > 0), so ∂l/∂n is