In this section we describe a version of Alexopoulos’ (1998) efficiency wage model, modified to allow for distortionary income taxes. The basic structure of the model is similar to a standard RBC model with the exception of the labor market. In contrast to RBC models, we assume that a worker’s effort is imperfectly observable by firms. This is the greatest problem when you realize it doesn’t matter how hard you work you efforts are not observable by your administration. and your wage depends on the fact what kind of work you carry out. the majority of people have to take speedy cash payday loans speedy cash payday loans to survive not to lead a wretched existence.
Competitive firms offer contracts that induce workers not to shirk on the job. These contracts specify a real wage, an effort level, and a specification that a worker will be dismissed and paid only a fraction of the wage if he is caught shirking on the job. Given a no bonding constraint, the supply for labor will in general exceed the demand for labor, resulting in unemployment. Whether the ex-post utility of employed workers exceeds the utility of unemployed individuals depends on the nature of risk sharing among members of the household. In the version of Alexopoulos’ model discussed below, risk sharing is imperfect (by assumption) and unemployed workers are worse off, ex-post, than employed workers.
4.1. The Government
The government faces the flow budget constraint
where Gt is real government purchases, rt is the marginal tax rate, rt is the rental rate of capital, 0 < 5 < 1 is the depreciation rate, Wt is the real wage rate, nt is employment, and is lump-sum taxes. By assumption h, hours worked per worker, is constant so that hours and employment move in proportion to one another. The fiscal policy rule is of the form given by the last two rows of (3.1).
The representative household owns the stock of capital, makes all capital related decisions, and pays both capital income taxes and lump-sum taxes. The household consists of a unit measure continuum of individuals. If individuals earn labor income, they must pay taxes on it. Employed members of the household partly insure the income of unemployed members of the household.
The household accumulates capital according to
where Kt is the beginning of period t capital stock and It is time t investment. The household rents capital to firms at the competitively determined rate rt, and rental income, net of depreciation, is taxed at the margin. The household uses its rental income net of these taxes and any lump-sum taxes that it pays to buy new capital. It distributes any remaining funds equally among the individual members of the household. We denote this common income as
Members of the household derive their remaining income from selling labor services to firms or from partial unemployment insurance provided by the household. They are assumed to take both the terms of labor contracts and firms’ demand for labor parametrically. In addition, from the perspective of firms, all individuals look alike. So we can think of the employment outcome for any individual as being determined completely randomly. Some individuals will be employed, while others will be unemployed. Under our assumptions, no individual would choose to be unemployed, because the ex-post utility of such an individual will be less than or equal to that of an employed individual. If you lose your working place, you have no money, the consequences may be different you may speedy cash payday loans, loan with the help of which you may solve all the problem and take them back when you will stand firmly on the ground.
Employed workers will either work and exert the level of effort required by the labor contract, denoted et, or they will shirk. The labor contract stipulates that if a worker is caught shirking, they will be fired and receive only a fraction s of their wages. The technology for detecting shirkers is imperfect, so that a shirker is only caught with probability d.
The household only observes the initial employment status of its members, not whether they shirked or were fired. Each employed member of the household transfers ‘Irt units of income to a pool which is distributed equally among the unemployed members of the household. By assumption, the household chooses the level of the transfer so that unemployed members of the household will be at least as well off as any shirker caught by the firm would be. Finally, we assume that labor income is taxed. Members of the household who pay the insurance transfer receive no tax credit for it, while recipients of transfers do not pay taxes on that type of income.
Our assumptions imply that the consumption of an employed individual who does not shirk is constrained by
An employed individual who shirks but does not get caught faces the same constraint. An employed individual who shirks and is caught only receives the fraction s of his contractual wages. Hence, his consumption, Cts, is constrained by
Suppose that nt members of the household are employed while 1—nt are unemployed. This implies that the transfer received by each member of the unemployed is given by nt^t/(1—nt). Hence, the consumption of an unemployed individual, C“, is constrained by
The instantaneous utility of an individual with consumption level C, and a positive level of effort e, is given by
while the instantaneous utility of an individual with consumption level C who exerts no work effort is given by
where 77 > 0, T is the time endowment, and £ is the fixed cost of exerting nonzero effort.
Thus, an employed worker who does not shirk has utility
where et is determined by the contract offered by the firm.
An employed worker who shirks but is not caught has utility
Let nst be the number of shirkers and let d be the probability of a shirker being caught. Since there is a continuum of individuals, this implies that dnl is the number of shirkers caught and (1 — d) nst the number of shirkers not caught.
Notice that the effective leisure time of caught shirkers and unemployed individuals is the same. If the family sets the transfer so that their consumption and utility levels are the same this will imply that
The household takes the effort level and wage rate as given in the contracts offered by the firm. The household also takes firms’ labor demand as given. The only decisions the household makes are those regarding capital and the level of common income, in order to maximize the expected utility of an individual household member:
subject to (4.2), (4.3), (4.4), (4.5) and (4.6).
4.3. The Firm
A perfectly competitive firm produces output using the technology
where nt is the number of workers it hires. It maximizes its profits
According to (4.7) the expected utility of an employee who does not shirk is at least as great as the expected utility of an employee who shirks. Here we assume that all employed workers are monitored and the exogenous probability of being caught shirking is d. In equilibrium there is no shirking. Given a wage rate, Wt, we can think of (4.7) as indicating a maximal level of effort the firm will be able to extract from workers. Rearranging the constraint we see that
The firm takes the level of the intra-household transfer parametrically. This is what allows us to write the expression on the right-hand side as a function, from the firm’s point of view, only of its choice regarding Wt.
Alexopoulos (1998) shows that the first-order conditions for the firm, along with the expression for e(Wt) imply that CtjCl is a constant given by x where x satisfies
This is a nonlinear equation in x that can be solved numerically.
The level of employment, nt, which characterizes the solution to the firm’s problem will not in general coincide with the number of workers who wish to work at the contract characterized by (wt, e(wt)). As long as the demand for workers is less than the supply of workers, (4.7) will hold with equality and there will be equilibrium unemployment. We confine ourselves to calibrated versions of the model in which this is the case and in which all of the inequality constraints above hold with equality.
We use the log linearization procedure described by Christiano (1998) to solve for the competitive equilibrium of this economy.