PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 5

Bils (1987) observes that in many industries, a higher wage is paid for overtime hours (i.e., , hours in excess of 40 hours per week). He thus proposes to quantify the extent to which the marginal wage rises as firms ask their employees to work longer hours, by measuring the extent to which the average number of overtime hours per employee, V, rises with increases in the total number of hours worked per employee H, and then assuming that W(H) = +pV(H)], where w0 is the straight-time wage and p is the overtime premium (0.5 according to the U.S. statutory requirement).
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For example, he finds that when average hours per employee rise from 40 hours per week to 41 hours, the average number of overtime hours worked per employee rises by nearly 0.4 hours, while when they rise from 41 to 42 hours per week, overtime hours rise by another 0.5 hours. This increase in the fraction of hours that are overtime hours as average hours increase means not only that the marginal wage exceeds the average wage, but that the ratio of the marginal wage to the average wage rises as hours increase.

Assuming p = .5, Bils finds that an increase in average hours from 40 to 41 hours increases the average wage by about 0.5%, but increases the marginal wage by 4.6%. On average, he finds that the factor uj in (2.11) has an elasticity of 1.4 with respect to variations in average hours.24 Thus a log-linear approximation to (2.11) is again of the form (2.9), where in Bils’ work H refers to fluctuations in average hours per worker,25 and 6 = -1.4.
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PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 4

One reason might be monopsony power in the labor market. Suppose that each firm faces an upward-sloping firm-specific labor supply curve, and takes this into account in its hiring and production decisions. (The wage that the firm must pay may also depend upon other variables such as the overall level of employment in the economy, but these factors are taken as given by the individual firm, and can simply be treated as time-variation in the location of the firm-specific labor supply curve.)
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If w(H) is the wage that the firm must pay if it hires H hours of work, then W(H) = Hw(H), and ш = 1 + 6#^, where is the elasticity of the firm-specific labor supply curve. This might be either increasing or decreasing with increases in hours hired by the firm. The most plausible assumption, however, would probably be that the elasticity of labor supply decreases as the hours hired by the firm increase (it is hard to induce people to work more than a certain number of hours, even at very high wages, while on the other hand the opportunity cost of their time tends not to fall below a certain level even when the number of hours worked is small). Under this assumption, the factor oj is an increasing function of Я, and (2.9) again holds, with b
Alternatively, one might imagine that firms first hire a certain number of employees, and that they initially contract with them about a wage schedule which determines the wage as a function of hours worked. Subsequently, perhaps after receiving additional information about current demand conditions, the firms determine the hours of work. If all employees are asked to work the same number of hours at this stage, we may interpret W{H) in (2.11) as the wage schedule negotiated with each employee.

Now if the number of employees is chosen ex ante so as to minimize the cost of the number of hours that the firm expects to use, then ex ante expected hours per worker will be the level H* that minimizes the average wage W(H)/H22 At this point, the marginal wage should equal the average wage, and (assuming a unique minimum) in the case of small fluctuations in H around the value H*, to should be increasing in H. Again this would imply markups more countercyclical than would be suggested by (2.5).

Most observed wage contracts do not involve wages that increase continuously with the number of hours that the employee is asked to work. On the other hand, if one supposes that contractual wages are not the true shadow price of additional labor to a firm, because of the presence of implicit contracts of the kind assumed, for example, by Hall (1980), then one might suppose that the true cost to the firm rises in proportion to the employee’s disutility of working, even if the wages that are paid in the current period do not. This would be a reason to expect the effective wage schedule W(H) to be convex, so that the above analysis would apply.

PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 3

Indeed, Hall (1988) finds (using stock market returns to construct a user cost for capital) that pure profits in U.S. industry are close to zero. It furthermore makes sense that profits should be zero in the steady state, due to entry, which one should expect to eliminate persistent profits in the long run, even if entry does not respond quickly enough to eliminate cyclical fluctuations in profits. If we assume this, we can impose p = /i, so that there is only a single parameter to calibrate, that describes both the degree of returns to scale and the degree of market power. With fi = 1.25 and a labor share of .7, the parameters b is then ~A. Table 2 shows that, even letting a equal zero, such a value of b leads to markups that are strongly countercyclical though the correlations with lagged output remain higher in absolute value.
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The significance that one attaches to such findings obviously depends upon the size of the average markup (or degree of returns to scale) that one is willing to assume. Here it is worth remarking that a value of /i equal to 1.6 need not mean that any individual firm marks up its costs by 60%. The reason for this is that firms do not just mark up their labor costs but also their materials cost. To see what this implies about the markup, suppose that, as in Rotemberg and Woodford (1995), materials are a fixed proportion sM of aggregate output while value added constitutes only a fraction (1 — sm) of total costs. The marginal cost of producing one unit of gross output is then
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PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 2

A quantitative estimate of the elasticity a requires values for eKH and for the average value tjh the value of this elasticity in the case of the “steady-state” factor ratio around which one considers perturbations). Using (2.5), the latter parameter may be calibrated from the average labor share, given an estimate of the average markup, resulting in a = (1 — “ !)• (In this last expression, all symbols refer to the average or steady-state values of the variables.)
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With a markup \x near one, a labor share of .7 and an elasticity of substitution Ckh of .5, this formula gives a value of a equal to -.4. Table 2 reports the resulting correlations between ft and predicted declines in GDP for markups based on both the nonfinancial corporate and the private labor shares. Not surprisingly, markups are now much more countercyclical. However, the contemporaneous correlation of the markup with detrended output is still smaller in absolute value than the correlation with lagged output. Also, as in the case where we do not adjust the labor share, the correlations with leads of output are greater than the contemporaneous correlation. These are actually positive for output led more than 3 quarters.
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