We show that unsynchronized wage adjustment, via the relative wage effect, can lower the required rate of increasing returns to uncertainty to an empirically plausible level and make it invariant to relative labor risk propensity. Keywords: increasing returns; Unsynchronized payroll reconciliation; Relative wages; Relative aversion to risk in childbirth; indeterminacy; Sunspot. One point of this paper is that unsynchronized wage adjustment, through the relative wage effect, can lower the required rate of increasing returns to uncertainty to an empirically plausible level and make it invariant to relative labor risk propensity.
A point made in this paper is that unsynchronized wage adjustment can both lower the degree of increasing returns to scale required for indeterminacy to an empirically plausible level and make it invariant to the relative risk aversion associated with childbirth. To assign a value to the hazard rate of wage adjustmentθ, we note that (1−θ) corresponds to the quarterly frequency of wage adjustments, and the average duration of newly set wages for individual workers is given by 1/(1−θ). The middle column of the table shows the results under synchronized wage regulation and the three.
The message from the table is therefore that unsynchronized wage adjustment can lower the required rate of increasing returns to uncertainty to an empirically plausible level. Unsynchronized wage adjustment makes the required increasing returns to uncertainty invariant to the relative risk propensity of the job.
SOME INTUITION
Putting the above results together, we conclude that unsynchronized wage adjustment not only lowers the required rate of increasing returns to indeterminacy to an empirically plausible level, it also makes it invariant to the relative risk aversion of work. As a consequence, an indeterminate equilibrium and thus sunspot-driven cyclical fluctuations can arise for empirically plausible increasing returns regardless of the value of the relative risk aversion in the work. Unsynchronized wage adjustment effectively dampens the costs of raising working hours and relieves the counterforce resulting from the relative risk aversion of work.
With unsynchronized wage adjustment, firms can optimally increase employment in response to an intended capital increase without increasing equilibrium wages as they would with synchronized wage adjustment. The fact that labor can be more elastic to capital if θ >0 can be seen from (5): a change on the right-hand side due to a large response of labor to capital may well be matched by a change on the left-hand side now. To understand why χmultiplied byǫ enters the left-hand side of (5), we note that with unsynchronized wage adjustment, even households that can adjust their wages will choose not to increase their wages much, since firms can substitute their more expensive labor services and against the cheaper ones provided by other households who cannot change their wages.
The strength of this deterrence depends on the ease of substitution across the differentiated labor skills, as measured by ǫ, and on how quickly the marginal disutility of labor changes as labor hours decrease, as determined by χ. This substitution across differentiated labor skills following a relative wage change results in a small aggregate wage increase and a large aggregate employment response to the intended capital increase. Since labor is more elastic to capital, the marginal product of capital does not fall as quickly or even rises with capital, and a self-fulfilling equilibrium is more likely.
The presence of χ on the left side of the equation via a positive θ essentially cancels its effect on the right side, making the required increasing returns to scale for indeterminacy invariant to χ. To clarify our intuition in a more rigorous way, we can use (6) and (8)-(11) to solve for the wage inflation on the left side of (5) in terms of labor and capital. We note that φso set is greater than ρ−1 and that the implied steady-state capital depreciation rate is strictly between 0 and 1 for any value of η in the above range; and asη varies over this range, φ varies from arbitrarily close to ρ−1 to arbitrarily close to infinity.
Since kw increases with η andρ <1, unsynchronized wage adjustment given existing returns makes it more efficient to generate an elastic response of labor to an intended capital increase, since a change on the right-hand side of (12) can be matched with a change on the left-hand side due to a large increase in labor and the response is no longer sensitive to χ, as its effects on both sides of the equation essentially balance out.
COMPUTING SUNSPOTS EQUILIBRIA
When the degree of scale changes is less than 1.1, we have e=s= 4, and the system has a unique equilibrium, where the expectation errors are uniquely determined by the fundamental shockεt,. When the degree of scaling changes is greater than 1.1, we have e = s = 3 and the system has a one-dimensional uncertainty, so there is room for sunspots to affect the forecast errorsηt and other endogenous variables. To put into perspective the dynamics of our model in light of those obtained by Benhabib and Wen (2004) and Wen (1998), we invoke the same values of parameters as in these papers when solving our model numerically, except for the two parameters relating to the labor market frictions, the elasticity of substitution of differentiated labor skills, ǫ, and the risk of wages.
To illustrate the effect of indeterminacy on equilibrium dynamics, we present the impulse responses of the model to the fundamental shocks under determinism, and to both the fundamental shocks and the sunspot shock under indeterminacy. Specifically, we examine two versions of our model, one with the degree of returns to scale equal to 1.1 (the case of determinism), and the other with the degree of returns to scale equal to 1.11 (the case of indeterminacy), as in Benhabib and Wen (2004) and Wen (1998). When considering the impulse responses to one type of shocks, we turn off the other type(s) of shocks by setting the corresponding variance(s) to zero.
In addition, when accounting for impulse responses to a consumption shock, we set the steady-state ratio of public consumption to output to zero; also, when considering impulse responses to a government spending shock, we set the steady-state ratio µ/c to zero; and when considering impulse responses to a sunspot shock at uncertainty, we set both the steady-state ratios g/y and µ/c to zero. This is the same procedure that Benhabib and Wen (2004) followed when they studied the impulse responses of their model to different shocks. The solid lines are the responses to a positive consumption shock of one standard deviation, and the dashed lines are the responses to a positive consumption shock of the country with one standard deviation.
The range of responses here is larger than in Benhabib and Wen (2004), especially for output, investment and hours worked, suggesting a diffusion mechanism. The solid lines are the responses to a one standard deviation positive consumption shock, the dashed lines are the responses to a one standard deviation positive government consumption shock, and the dashed dotted lines are the responses to a one standard deviation sunspot shock leading to an upward correction of forecast errors. 9 These responses are similar to those obtained by Benhabib and Wen (2004) under uncertainty: in addition to volume movements and relative volatility, after demand shocks, persistent cycles (except for the consumption response to the consumption shock), including i.i.d. The magnitudes of the sunspot shock responses are much larger here than in Benhabib and Wen (2004), suggesting that labor market frictions are key to the significance of the model's response to sunspot shocks.
In summary, while the model's impulse responses to demand shocks under uncertainty are as reasonable as those in Benhabib and Wen (2004) in matching the business cycle as shown by Wen (1998), sunspot shocks seem much more important here than in Benhabib and Wen (2004 ) due to the presence of labor market frictions.
CONCLUDING REMARK
We show that the stability property of the system (5)-(11) for the case of synchronized wage regulation can be obtained by analyzing a three-equation system. We show that the stability property of the system (5)-(11) for the case of unsynchronized wage regulation can be obtained by analyzing a six-equation system. Constant Returns and Small Markups in US International Finance Discussion Paper 483, Board of Governors of the Federal Reserve System.
The impact of affirmative action on labor demand: a test of some implications of the Le Chatelier principle. New evidence on labor supply: employment versus hour elasticities by gender and marital status. A simple scheme for estimating an intertemporal model of labor supply and consumption in the presence of taxes and uncertainty.
How well a small structural model with sticky prices and wages fits US postwar replacement over time: Another look at life-cycle labor supply. Dynamic General Equilibrium Models with Imperfectly Competitive Product Markets. ed.), Frontier of Business Cycle Research. Note: The χ value marked with ⋆ in the first column of the table is the baseline calibration for χ.
Minimal degree of increasing returns to scale that can generate indeterminacy as a function of the relative risk aversion in labor (all other parameters assume their calibrated values). Minimal degree of increasing returns to scale that indeterminacy can generate as a function of the relative risk aversion in labor (with three alternative hazard rates for unsynchronized wage adjustment while all other parameters assume their calibrated values).
