Let $X$ ~ Exp($\lambda$)
Consider testing $H_0 : \lambda \geq \lambda_0$ vs. $H_1 : \lambda < \lambda_0$
I'm trying to find the power function of the test that rejects the null hypothesis iff $X$ >= some constant c.
$\pi(\lambda | \delta_c$) = $1 - (1-\exp(-\lambda c))$ = $\exp(-\lambda c)$
Is that correct?
Additionally, I'm looking for a specific $c$ that gives our test size $\alpha$ on the interval $0 < \alpha<1$ .
This looks good. The decay rate $\lambda$ is the true, unknown parameter. Your alternative hypothesis is that the true value of $\lambda$ is smaller than some postulated value $\lambda_{0}$, which means that $X$ will decay more slowly . So, your rejection region should have the form $[c,\infty]$ - you reject $H_0$ when $X$ is large (compared to the expected value under the null).
So, it's fine, as far as it goes. But normally, you'd expect $c$ to depend on $\lambda_{0}$. If you want to perform that test at the $\alpha$-level, then the critical region will be the upper $\alpha$-tail of $X$ under the null hypothesis. Thus $\alpha=\textrm{e}^{-\lambda_{0}c}$. Solving this for $c$ gives: $$c=\frac{-\ln(\alpha)}{\lambda_0}.$$
The power is therefore $$\pi(\lambda)=\textrm{e}^{-\lambda \frac{-\ln(\alpha)}{\lambda_{0}}}=\alpha^{\frac{\lambda}{\lambda_{0}}}.$$