## 经济代写|宏观经济学代写Macroeconomics代考|Introducing labour turnover

The model can be easily modified to introduce labour turnover. If the worker can lose his job, we need to introduce in the equation for the value of accepting an offer the possibility that the worker may be laid off and go back to the pool of the unemployed. We will assume this happens with probability $\lambda$ :
$$r W(w)=w+\lambda[U-W(w)] .$$
The equation for the value of being unemployed remains $(16.10)$, and still $W\left(w_R\right)=U$. Because $r W\left(w_R\right)=w_R$ we know that $r U=w_R \cdot(16.13)$ implies that $W(w)=\frac{w+\lambda I I}{(r+\lambda)}$, which replacing in (16.10) gives
$$r U=b+\frac{\alpha}{r+\lambda} \int_{w_{\mathbb{R}}}^{\infty}\left[w-w_R\right] d F(w),$$
or
$$r W\left(w_R\right)=w_R=b+\frac{\alpha}{r+\lambda} \int_{w_{\mathrm{R}}}^{\infty}\left[w-w_R\right] d F(w) .$$
The reservation wage falls the higher the turnover; as the job is not expected to last forever, the searcher becomes less picky.

This basic framework constitutes the basic model of functioning of the labour market. It’s implications will be used in the remainder of the chapter.

## 经济代写|宏观经济学代写Macroeconomics代考|Diamond-Mortensen-Pissarides model

We will put our job search value functions to work right away in one very influential way of analysing unemployment: thinking the labour market as a matching problem in which sellers (job-seeking workers) and buyers (employee-seeking firms) have to search for each other in order to find a match. If jobs and workers are heterogeneous, the process of finding the right match will be costly and take time, and unemployment will be the result of that protracted process. ${ }^7$

Let us consider a simple version of the search model of unemployment. The economy consists of workers and jobs. The number of employed workers is $E$ and that of unemployed workers is $U$ $(E+U=\bar{L}$ ); the number of vacant jobs is $V$ and that of filled jobs is $F$. (We will assume that one worker can fill one and only one job, so that $F=E$, but it is still useful to keep the notation separate.) Job opportunities can be created or eliminated freely, but there is a fixed cost $C$ (per unit of time) of maintaining a job. An employed worker produces $A$ units of output per unit of time $(A>C)$, and earns a wage $w$, which is determined in equilibrium. We leave aside the costs of job search, so the worker’s utility is $w$ if employed or zero if unemployed; the firm’s profit from a filled job is $A-w-C$, and $-\mathrm{C}$ from a vacant job.

The key assumption is that the matching between vacant jobs and unemployed workers is not instantaneous. We capture the flow of new jobs being created with a matching function
$$M=M(U, V)=K U^\beta V^\gamma,$$
with $\beta, \gamma \in[0,1]$. This can be interpreted as follows: the more unemployed workers looking for jobs, and the more vacant jobs available, the easier it will be to find a match. As such, it subsumes the searching decisions of firms and workers without considering them explicitly. Note that we can parameterise the extent of the thick market externalities: if $\beta+\gamma>1$, doubling the number of unemployed workers and vacant jobs more than doubles the rate of matching; if $\beta+\gamma<1$ the search process faces decreasing returns (crowding).

We also assume an exogenous rate of job destruction, which we again denote as $b$. This means that the number of employed workers evolves according to
$$\dot{E}=M(U, V)-b E .$$
We denote $a$ as the rate at which unemployed workers find new jobs and $\alpha$ as the rate at which vacant jobs are filled. It follows from these definitions that we will have
\begin{aligned} &a=\frac{M(U, V)}{U}, \ &\alpha=\frac{M(U, V)}{V} . \end{aligned}

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Introducing labour turnover

$$r W(w)=w+\lambda[U-W(w)]$$

$$r U=b+\frac{\alpha}{r+\lambda} \int_{w_{\mathbb{R}}}^{\infty}\left[w-w_R\right] d F(w),$$

$$r W\left(w_R\right)=w_R=b+\frac{\alpha}{r+\lambda} \int_{w_{\mathrm{R}}}^{\infty}\left[w-w_R\right] d F(w)$$

## 经济代写|宏观经济学代写Macroeconomics代考|Diamond-Mortensen-Pissarides model

$$M=M(U, V)=K U^\beta V^\gamma,$$

$$\dot{E}=M(U, V)-b E .$$

$$a=\frac{M(U, V)}{U}, \quad \alpha=\frac{M(U, V)}{V} .$$

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