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

Question

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.

Answer 1

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 $r,\pi$.

Answer 2

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.