Chapter 16 Labor Markets
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.
Max U = U(c,h) St. 24w c wh = 0 Lagrangian expression Utility Maximization Model U( c, h) λ(24w c wh) FOCs U 1. λ 0 c c U 2. wλ 0 h h Dividing 2 by 1 gives U h MU h MRS U c MU c 3. 24w c wh 0 h for c w To maximize utility, the individual should work up to the point where the marginal rate of substitution of leisure (h) for consumption (c) is equal to the real wage rate (w), provided that the SOC are satisfied; MRS h for c is diminishing.
Consumption=c Income and Substitution Effects The real wage rate increases from w 0 to w 1 (the price of leisure increases). The constraint rotates to the right because the individual can consume more for the same amount of labor, ie., w 1 l > w 0 l. The individual is made better off by an increase in w because the individual is the supplier of labor. c = w 0 (24) c 1 c 0 B C A 0 h 1 h 0 Substitution Effect Income Effect c = w 1 (24 h) ); slope=-w 1 c = w 0 (24 h); slope=-w 0 h = 24, so c = 0. U 1 U 0 Leisure=h Consumption=c c 1 c = w 0 (24 h) c 0 B A C 0 h 0 h 1 Substitution Effect Income Effect 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 consumption is a normal good). c = w 1 (24 h) Leisure= h 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 h 0 to h 1 (l increases) as w increases. In the right graph, the income effect outweighs the substitution effect, so h increases from h 0 to h 1 (l decreases) as w increases. This example suggests that the supply of labor could be backward bending (negatively sloped) as in the right graph. U 0 U 1
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 U 0 = 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 = -l c = h c. 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: l c (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 l c (w,u) = l(w,n) = l[w,e(w,u)]. Partially differentiate both sides with respect to w to get, l c E Then realizing that E/ w = -lc and substituting n for E,. E c. E n h (leisure) is a normal good ( h/ n > 0), so l/ n is <0. U U 0 n. (+or-) = (+) + (+)(-) c Now let U U 0 and rearrange terms to get, The change in labor supplied as w changes equals the change in labor supplied from the substitution effect when U is held constant plus the change in labor supplied from the income effect resulting from an appropriate change in nonlabor income). Thus, labor supply could be backward bending if the income effect outweighs the substitution effect. If l is larger (eg., the person is working 18 hr/day), the negative income effect may be greater in absolute value than the positive substitution effect.
Market Supply Curve for Labor The market supply curve for labor is the horizontal sum of the amounts of labor supplied by all individuals in the labor market. A higher real wage rate might cause each person in the market to work more hours and it might induce more individuals to enter the labor market; thus, total labor might increase. Real wage w * 0 l * D S Quantity of labor D is demand for labor by firms and S is supply of labor by individuals. If w were above w*, involuntary unemployment would exist and individuals would bid down the real wage. If w were below w*, firms would want to hire more labor than was available, so firms would bid up the real wage.
Mandated Labor Benefits Such as health insurance and paid leave Suppose labor supply is l S = a + bw and labor demand is l D = c dw and that l S = l D. Solve for w to get, w* = (c a)/(b + d). Now suppose government mandates a labor benefit that costs firms t per unit of hired labor. Labor now costs firms w + t per unit. Suppose that the monetary value of the benefits to workers is k per unit of labor supplied. Workers now receive w + k per unit of labor supplied. l S = l D now requires a + b(w + k) = c d(w + t), and the net wage is w** = (c a)/(b + d) (bk + dt)/(b + d) = w* (bk + dt)/(b + d). If k = 0 (the benefit is 0) and t is the tax on employment faced by firms, the demand for labor decreases (shifts left). Supply does not shift, so the equilibrium amount of labor hired decreases. The employee s share of the tax is given by the ratio d/(b + d). As long as k < t, the equilibrium amount of labor hired will decrease because demand shifts left by more than supply shifts right. If k = t, the new equilibrium wage falls by t because w** = w* t and the equilibrium amount of labor hired does not change because demand and supply shifts offset each other.
Causes of Wage Variation Human capital. A firm s demand for labor depends on the worker s marginal productivity, so differences in productivity should lead to different wages. Education Experience On-the-job training Compensating Wage Differentials. Some jobs are more desirable than others (Assistant Professor at UH), so workers are willing to receive a lower real wage. Other jobs are less desirable (truck driver in Iraq), so workers must be compensated for the unpleasantness. Monopsony. The firm is not a price taker for the inputs it buys.
Monopsony in Labor Purchase Many employers are not perfect competitors in purchasing inputs. In fact, most large firms are not. They are not price takers for w. The supply curve faced by the firm is the market labor supply curve. S is the average cost of labor equal to w = wl/l. The Marginal Expense of labor (ME Labor Supply L ) is the marginal cost of l w ME l = marginal labor equal to wl/ l. cost of labor Thus, the ME curve for labor is above the supply curve for labor because the firm must pay all workers more (not just the last one hired) to get one more worker. Similar to wl l PQ Q S = average cost of labor l MR ME P P Q Q l l on w l Supply of labor is positively sloped. the Demand for the good is negatively sloped. output side. Using the product rule because w = f(l). If the firm were a perfectly competitive buyer of labor, / = 0 and ME l = w. This means the elasticity of supply for labor equals positive infinity (perfectly elastic supply of labor; horizontal labor supply curve).
We have already stated that profit maximization requires ME l = MRP l. Therefore, the monopsonist purchases l 1 labor at w 1 price. This equilibrium is determined at point A where ME l = MRP l. w w * w 1 B A ME l C S l average cost of labor curve = market supply of labor l 1 l * l MRP l = P(MP l ) The perfectly competitive purchaser would purchase l * at w * (determined by point C where w = MRP l. The relationship between l and w is determined by the MRP l curve. However, there is actually no such thing as the monopsonist s demand curve for labor, just as there was no monopolist s supply curve for output. MRP l is not a monopsonist s demand curve for labor, but it does help determine one optimum point of w 1 l 1. This point represents the firm s demand for labor given P and MPP l.
Bilateral Monopoly A seller of an input may act as a monopolist if it is the only potential supplier of the input. This may occur in the cases of labor unions or where a particular input is available from a single seller. The seller of the input acts just as our earlier monopolist acted. If buyers of the input are perfect competitors, the monopolist may choose to operate anywhere on its demand curve as shown below. Monopolist selling labor (eg., labor union) w w 2 w ** MC l of seller MR l faced by seller D l faced by seller = MRP l of buyers 0 l 2 l ** l
When a monopolistic seller faces a monopsonistic buyer, a Bilateral Monopoly exists. The Monopolist (seller) wants to operate at MR S =MC S at point E 1 (P 1, Q 1 ), while the Monopsonist (buyer) wants to operate at MRP B =ME B at point E 2 (P 2, Q 2 ). They must negotiate a price and quantity somewhere between P 1 and P 2 and between Q 1 and Q 2. B The seller s MC curve is not the seller s supply curve for labor, but it is the supply curve faced by the buyer (no S curve for monopoly). S B S = MRP B S The buyer s MRP curve is not the buyer s demand curve for labor, but it is the demand curve faced by the seller (no D curve for monopsony).