# A simple question on a relationship between output growth rate and unemployment rate

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.

• 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 Aug 29 '15 at 22:30
• @denesp thanks for the help. Could you please tell me what's missing ? – An old man in the sea. Aug 29 '15 at 22:38
• @denesp or what I should assume to reach that conclusion? – An old man in the sea. Aug 29 '15 at 22:49
• For example, what is $N$? – FooBar Aug 30 '15 at 8:27
• @FooBar it's employment – An old man in the sea. Aug 30 '15 at 9:00

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 is sometimes written as $i = r + \pi$, even though it is actually $i \approx r + \pi$ and that only for small values of $r,\pi$.

• 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. Aug 30 '15 at 12:37

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.

• Alecos, I'm not sure I understand you. How can the right-side be equal? – An old man in the sea. Aug 30 '15 at 22:57
• 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. Aug 31 '15 at 7:51