## 经济代写|宏观经济学代写Macroeconomics代考|Efficiency wages

The idea behind efficiency wages is that the productivity of labour depends on effort, and that effort depends on wages. Because of these links, firms prefer to pay a wage that is higher than the market equilibrium wage. But at this wage there is unemployment. The most basic version of this story – one that applies to very poor countries – is that a higher wage makes workers healthier as they can eat better. But there are many other ways to make the argument. For example, it is mentioned that Henry Ford paid his workers twice the running wage to get the best and reduce turnover, and, as we will see in the next section, sometimes firms pay a higher wage to elicit effort because they have difficulty in monitoring workers’ effort. $^9$
To see this, let us consider a general model in which the firm’s profits are
$$\pi=Y-w L,$$
where
$$Y=F(e L),$$

with $F^{\prime}>0$ and $F^{\prime \prime}<0$. We denote by $e$ the effort or effectiveness of the worker. The crucial assumption is that this effectiveness is increasing in the real wage: $$e=e(w),$$ with $e^{\prime}>0$. With all these assumption we can rewrite the firm problem as
$$\operatorname{Max}_{L, w} F(e(w) L)-w L,$$
which has first-order conditions
$$\frac{\partial \pi}{\partial L}=F^{\prime} e-w=0,$$
and
$$\frac{\partial \pi}{\partial w}=F^{\prime} L e^{\prime}(w)-L=0 .$$
Combining (16.34) and (16.35) we have
$$\frac{w e^{\prime}(w)}{e(w)}=1 .$$
The wage that satisfies this condition is called the efficiency wage. This condition means that the elasticity of effort with respect to wage is equal to one: a $1 \%$ increase in the wage translates into an equal increase in effective labour.

Why does this create unemployment? Notice that (16.34) and (16.35) is a system that defines both the wage and employment. If the optimal solution is $w^$ and $L^$, total labour demand is $N L^*$ where $N$ indicates the number of firms. If the supply of labour exceeds this number, there is unemployment because firms will not want to reduce their wages to clear the market. ${ }^{10} \mathrm{We}$ can also extend this model to include the idea that effort depends on the wage the firm pays relative to what other firms pay, or existing labour market conditions. Summers and Heston (1988) do this and the insights are essentially the same.

The model provides an intuitive explanation for a permanent disequilibrium in labour markets. What explains the relation between wages and effort? To dig a bit deeper we need a framework that can generate this relationship. Our next model does exactly that.

## 经济代写|宏观经济学代写Macroeconomics代考|Wages and effort: The Shapiro-Stiglitz model

Whén you’ré àt work, your bóss oobviously cannnót perfectly monitor your effort, right? This méñs you have a moral hazard problem: the firm would like to pay you based on your effort, but it can only observe your production. It turns out that the solution to this moral hazard problem leads to a form of efficiency wages.

Following Shapiro and Stiglitz (1984), consider a model in continuous time with identical workers who have an instantaneous discount rate of $\rho$ and a utility at any given instant that is given by
$$u(t)=\left{\begin{array}{cl} w(t)-e(t) & , \text { if employed } \ 0, & \text {, otherwise } \end{array}\right.$$
where again $w$ is wage and $e$ is effort. For simplicity, we assume that effort can take two values, $e \in{0, \bar{e}}$. At any given point in time, the worker can be in one of three states: $E$ means that she is employed and exerting effort ( $e=\bar{e}), S$ means that she is employed and shirking $(e=0)$, and $U$ denotes unemployment. We assume that there is an exogenous instantaneous probability that the worker will lose her job at any instant, which is given by $b$. In addition, there is an instantaneous probability $q$ that a shirking worker will be caught by the firm, capturing the monitoring technology. Finally, the rate at which unemployed workers find jobs is given by $a$. This is taken to be exogenous by individual agents, but will be determined in equilibrium for the economy as a whole. Firms in this model will simply maximise profits, as given by
$$\pi(t)=F(\bar{e} E(t))-w(t)[E(t)+S(t)],$$
where $F(\cdot)$ is as before, and $E(t)$ and $S(t)$ denote the number of employees who are exerting effort and shirking, respectively.

In order to analyse the choices of workers, we need to compare their utility in each of the states, $E$, $S$, and $U$. Let us denote $V_i$ the intertemporal utility of being in state $i$; it follows that
$$\rho V_E=(w-\bar{e})+b\left(V_U-V_E\right) .$$
How do we know that? Again, we use our standard asset pricing intuition that we found in the first section of this chapter. The asset here is being employed and exerting effort, which pays a dividend of $w-\bar{e}$ per unit of time. The capital gain (or loss) is the possibility of losing the job, which happens with probability $b$ and yields a gain of $V_U-V_E$. The rate of return that an agent requires to hold a unit of this asset is given by $\rho$. Using the intuition that that the total required return be equal to dividends plus capital gain, we reach (16.38). A similar reasoning gives us
$$\rho V_S=w+(b+q)\left(V_U-V_S\right),$$
because the probability of losing your job when shirking is $b+q$. Finally, unemployment pays zero dividends (no unemployment insurance), which yields ${ }^{11}$
$$\rho V_U=a\left(V_E-V_U\right) .$$

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Efficiency wages

$$\pi=Y-w L,$$

$$Y=F(e L),$$

$$\operatorname{Max}_{L, w} F(e(w) L)-w L,$$

$$\frac{\partial \pi}{\partial L}=F^{\prime} e-w=0,$$

$$\frac{\partial \pi}{\partial w}=F^{\prime} L e^{\prime}(w)-L=0 .$$

$$\frac{w e^{\prime}(w)}{e(w)}=1$$

$u(t)=\backslash l e f t{$ 给出的任何给定时刻的实用程序
$w(t)-e(t)$, if employed 0, , otherwise
〈正确的。
whereagain $\$ w \$$iswageand \$$ e $\$$iseffort. Forsimplicity, weassumethateffortcantaketwovalues \backslash pi (t)=F(\backslash \operatorname{bar}{\mathrm{e}} \mathrm{E}(\mathrm{t}))-\mathrm{w}(\mathrm{t})[\mathrm{E}(\mathrm{t})+\mathrm{S}(\mathrm{t})] \ \$$ 其中$F(\cdot)$和以前一样，并且$E(t)$和$S(t)$分别表示付出努力和偷濑的员工人数。 为了分析工人的选择，我们需要比较他们在每个州的效用，$E, S$，和$U$. 让我们表示$V_i$处于状态的跨期 效用$i$；它遵循 $$\rho V_E=(w-\bar{e})+b\left(V_U-V_E\right) .$$ 我们怎么知道? 同样，我们使用我们在本章第一节中找到的标准资产定价直觉。这里的资产正在被使用 并付出努力，它支付的红利为$w-\bar{e}$每单位时间。资本收益（或损失) 是失去工作的可能性，这种可能 性发生$b$并产生增益$V_U-V_E$. 代理人持有该资产单位所需的回报率由下式给出$\rho$. 使用总要求回报等于股 息加上资本收益的直觉，我们达到 (16.38)。类似的推理给了我们 $$\rho V_S=w+(b+q)\left(V_U-V_S\right),$$ 因为偷濑时丢掉工作的概率是$b+q$. 最后，失业支付的红利为霝 (没有失业保险)，这会产生${ }^{11}\$
$$\rho V_U=a\left(V_E-V_U\right) .$$

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