Equilibrium value of $b$ (market tightness) in a search model
From

$$ 1 - e^{-b} = a_{F}(b) = \frac{k}{y (1-\beta)}$$

we get

$$b = -\ln\left(1-\frac{k}{y (1-\beta)}\right) \tag{1}$$

For this to exist (be a real number) it must be the case that

$$1-\frac{k}{y (1-\beta)} &gt; 0 \implies \frac k y &lt; (1-\beta) \tag{2}$$

which is already assumed.  

Then, uniqueness of the solution of $(1)$ is guaranteed because all arguments/parameters in the right-hand side enter only once. So for any vector $(y,k,\beta)$ satisfying what it needs to satisfy, we get a unique $b$.