<p>Under the stated assumptions, we have that (using $N_t = Y_t$ and $L_t = L$)</p>

<p>$$u_t - u_{t-1} = \frac {L-Y_t}{L} - \frac {L-Y_{t-1}}{L}  = \frac {Y_{t-1} - Y_t}{L}$$</p>

<p>The "growth rate of output" is defined as</p>

<p>$$g_{yt} \equiv \frac{Y_t-Y_{t-1}}{Y_{t-1}}$$</p>

<p>So we can manipulate the first equation as</p>

<p>$$u_t - u_{t-1}  = -\frac {Y_t-Y_{t-1}}{L} \frac{Y_{t-1}}{Y_{t-1}} = -g_{yt}\cdot (1-u_{t-1})$$</p>

<p>$$\implies u_t - u_{t-1} = -g_{yt} + g_{yt}\cdot u_{t-1}$$</p>

<p>So we do have an approximation. Such are fine for theoretical expositions (educational or not), but whether they should be acceptable in real-world studies depends on the actual magnitudes of the two rates involved. For example with a growth rate of $3\%$ and a previous-period unemployment rate of $5\%$ we would get an approximation error of $0.0015$.  The true drop in unemployment would be over-estimated ($3$ percentage points instead of $2.85$ percentage points), which appears acceptable.</p>

CC BY-SA 3.0
2015-08-30T22:57:49.817Z
Alecos, I'm not sure I understand you. How can the right-side be equal?
An old man in the sea.
CC BY-SA 3.0
2015-08-31T07:51:49.327Z
Thanks for the more elaborate exposition. p.s.: in these last few days I asked a few questions. If you find them interesting, could you help me? Either way, thanks for this answer. ;)
An old man in the sea.
CC BY-SA 3.0
2015-08-30T22:16:42.447Z
2015-08-30T23:32:44.637Z
2015-08-30T23:32:44.637Z
Alecos Papadopoulos
<p>I get the same equation as you do. So you are correct. But tough usually $u_t - u_{t-1} \neq -g_{y_{t}}$, for small values of $g_{y_{t}}$ 
$$
u_t - u_{t-1} \approx - g_{y_{t}}
$$
does hold. In macroeconomics these kind of (in my opinion annoying) approximations are used every now and then. The author perhaps neglected to mention that he was using an approximation. For example the <a href="https://en.wikipedia.org/wiki/Fisher_equation" rel="nofollow">Fisher equation</a> is <a href="https://en.wikipedia.org/wiki/Fisher_hypothesis" rel="nofollow">sometimes written</a> as $i = r + \pi$, even though it is actually $i \approx r + \pi$ and that only for small values of $r,\pi$.</p>

CC BY-SA 3.0
2015-08-30T12:37:58.807Z
Yes, I think you're right. These implicit approximations usually written as equalities bother me a lot! Thanks for the help  ;)
An old man in the sea.
CC BY-SA 3.0
2015-08-30T09:18:20.233Z
2015-08-30T09:18:20.233Z
Giskard
<p>I'm reading Blanchard's Macroeconomics, and on page 206 of the 5th edition, he writes: 
«Let $u_t$ denote the unemployment rate in year t, and $u_{t-1}$ the unemployment rate in year t-1, and $g_{yt}$ the growth rate of output from year t-1 to year t. Then under these two conditions ($Y_t=N_t$ i.e. Output=Employment, and labour force $L_t=L$ constant), the following relation would hold: $$ u_t-u_{t-1}=-g_{yt}$$</p>

<p>But what I get is $\frac{Y_{t-1}-Y_t}{L}=-\frac{Y_t-Y_{t-1}}{Y_{t-1}}$ which is not true... where did I go wrong, or is there a typo?</p>

<p>Any help would be appreciated.</p>

CC BY-SA 3.0
2015-08-29T22:30:29.327Z
It is hard to help because in its current form the question is not self contained, i.e. not all variables are defined nor are all equations given.
Giskard
CC BY-SA 3.0
2015-08-29T22:38:16.717Z
@denesp thanks for the help. Could you please tell me what's missing ?
An old man in the sea.
CC BY-SA 3.0
2015-08-29T22:49:39.480Z
@denesp or what I should assume to reach that conclusion?
An old man in the sea.
CC BY-SA 3.0
2015-08-30T08:27:57.057Z
For example, what is $N$?
FooBar
CC BY-SA 3.0
2015-08-30T09:00:07.960Z
@FooBar it's employment
An old man in the sea.
<macroeconomics>
A simple question on a relationship between output growth rate and unemployment rate
An old man in the sea.
# A simple question on a relationship between output growth rate and unemployment rate

(Asked by: username_0 on 2015-08-29)

I'm reading Blanchard's Macroeconomics, and on page 206 of the 5th edition, he writes:
«Let $u\_t$ denote the unemployment rate in year t, and $u\_{t-1}$ the unemployment rate in year t-1, and $g\_{yt}$ the growth rate of output from year t-1 to year t. Then under these two conditions ($Y\_t=N\_t$ i.e. Output=Employment, and labour force $L\_t=L$ constant), the following relation would hold: $$ u\_t-u\_{t-1}=-g\_{yt}$$

But what I get is $\frac{Y\_{t-1}-Y\_t}{L}=-\frac{Y\_t-Y\_{t-1}}{Y\_{t-1}}$ which is not true... where did I go wrong, or is there a typo?

Any help would be appreciated.

**Tags:** macroeconomics
> **Comments:**
> **username_1**: It is hard to help because in its current form the question is not self contained, i.e. not all variables are defined nor are all equations given.  
> **username_0**: @denesp thanks for the help. Could you please tell me what's missing ?  
> **username_0**: @denesp or what I should assume to reach that conclusion?  
> **username_2**: For example, what is $N$?  
> **username_0**: @FooBar it's employment  

---

## 2 answers

### Answer by username_1 on 2015-08-30. Score: 3 | Accepted answer

I get the same equation as you do. So you are correct. But tough usually $u\_t - u\_{t-1} \neq -g\_{y\_{t}}$, for small values of $g\_{y\_{t}}$
$$
u\_t - u\_{t-1} \approx - g\_{y\_{t}}
$$
does hold. In macroeconomics these kind of (in my opinion annoying) approximations are used every now and then. The author perhaps neglected to mention that he was using an approximation. For example the [Fisher equation](https://en.wikipedia.org/wiki/Fisher_equation) is [sometimes written](https://en.wikipedia.org/wiki/Fisher_hypothesis) as $i = r + \pi$, even though it is actually $i \approx r + \pi$ and that only for small values of $r,\pi$.

> **Comments:**
> **username_0**: Yes, I think you're right. These implicit approximations usually written as equalities bother me a lot! Thanks for the help  ;)  

---

### Answer by username_3 on 2015-08-30. Score: 0

Under the stated assumptions, we have that (using $N\_t = Y\_t$ and $L\_t = L$)

$$u\_t - u\_{t-1} = \frac {L-Y\_t}{L} - \frac {L-Y\_{t-1}}{L} = \frac {Y\_{t-1} - Y\_t}{L}$$

The "growth rate of output" is defined as

$$g\_{yt} \equiv \frac{Y\_t-Y\_{t-1}}{Y\_{t-1}}$$

So we can manipulate the first equation as

$$u\_t - u\_{t-1} = -\frac {Y\_t-Y\_{t-1}}{L} \frac{Y\_{t-1}}{Y\_{t-1}} = -g\_{yt}\cdot (1-u\_{t-1})$$

$$\implies u\_t - u\_{t-1} = -g\_{yt} + g\_{yt}\cdot u\_{t-1}$$

So we do have an approximation. Such are fine for theoretical expositions (educational or not), but whether they should be acceptable in real-world studies depends on the actual magnitudes of the two rates involved. For example with a growth rate of $3\%$ and a previous-period unemployment rate of $5\%$ we would get an approximation error of $0.0015$. The true drop in unemployment would be over-estimated ($3$ percentage points instead of $2.85$ percentage points), which appears acceptable.

> **Comments:**
> **username_0**: Alecos, I'm not sure I understand you. How can the right-side be equal?  
> **username_0**: Thanks for the more elaborate exposition. p.s.: in these last few days I asked a few questions. If you find them interesting, could you help me? Either way, thanks for this answer. ;)  

---

