Consider the following one-shot version of a labour market matching model. Let the labour force be normalized at 1, who, because there is only one period, all start out as unemployed. There is a very large number of firms who can enter the market and search for a worker. Firms who engage in search first have to pay a fixed cost, $k$. If a measure $v$ of firms enters the labour market, a constant returns to scale matching function $m(1,v)$ gives us the total measure of matches in the economy.
Within each match, the firm and the worker bargain for the wage, $w$, so that the workers get a constant proportion of $y$. Denote this proportion by $\beta$, which is interpreted as the bargaining power of the worker. Assume $\frac{k}{y} < 1 - \beta$ for the firm.
Define market tightness as $ b \equiv \frac{1}{v}$ and assume that the arrival rate for a firm is given by: $a_{F} = 1 - e^{-b}$.
Consider the firms can enter the labour market freely if they pay the entry cost, then what is the equilibrium value of $b$? Describe it graphically? Does it always exist? Is it unique?
My solution: The value of a vacancy: $$V = -k + a_{F}(b)(J-V),$$ and the value of a filled job: $$J = y-w.$$ If the firms enter the labour market freely then $V=0$. Then from these two equations I am left with this equation $$ 1 - e^{-b} = a_{F}(b) = \frac{k}{y (1-\beta)}.$$
Graphically, the function looks something like this:
From the graph, it looks like that $b^*$ is unique, but how do I know if it always exists or not?