I am trying to determine the conditions under which the following term is an increasing, concave function of $x_t$:
$$\sum_{t=0}^\infty \delta^{t+1} \frac{x_{t+1}(x_{t})}{h_{t+1}(x_t)}A$$ where $\delta <1$; the elasticity of $x_{t+1}$ to $x_t$ is $0< \alpha <1$ (as $\ln{x_{t+1}} = u + \alpha \ln{x_t}$); and the elasticity of $h_{t+1}$ to $x_t$ is $0< \beta <1$.
I suspect this is wrong: simplify the term as $\frac{A}{1-\delta} \sum_{t=0}^\infty\frac{x_{t+1}(x_{t})}{h_{t+1}(x_t)}$. This has an elasticity (wrt $x_t$) of $\frac{\alpha}{1-\alpha} - \frac{\beta}{1-\alpha}$, which implies that the term is concave as long as $2\alpha - \beta < 1$.
Would be grateful to receive any pointers.