Following Barro's book about the Ramsey Model, when we work with variables expressed in units per effective labor we end up with the following system of 2 differential equations \begin{equation} \frac{\dot{\hat{k}}(t)}{\hat{k}(t)} = \frac{f(\hat{k}(t))}{\hat{k}(t)} - \frac{\hat{c}}{\hat{k}(t)} - (x+n+\delta) \\ \frac{\dot{\hat{c}}(t)}{\hat{c}(t)} = \theta^{-1}[f'(\hat{k}(t)) -\delta -\rho-\theta x] \end{equation} and the transversality condition \begin{equation} \lim_{t \to \infty} \left\{ \hat{k}(t) \exp\left( - \int_{0}^{\infty} [f'(\hat{k}(s)) - \delta - x - n] \, ds \right) \right\} = 0 \end{equation}
where $\hat{k}(t) \equiv \frac{K(t) }{A(t) L(t) }$, with $K(t)$ denoting physical capital, $A(t)$ denoting the stock of technology at time $t$ which evolves according to $A(t)=e^{xt}$, and $L=e^{nt}$ is the population size. Then, $\delta$ is the depreciation rate of capital, $\rho$ is the subjective discounting rate of the household, and $\theta^{-1}$ is the intertemporal elasticity of substitution.
Then, with $(\gamma_{\hat{k}})^{\ast}$ and $(\gamma_{\hat{c}})^{\ast} $ referring to the steady state growth rate of capital and consumption per effective worker, respectively, the book says that at the steady state the differential equation for consumption implies \begin{equation} \hat{c}=f(\hat{k})-(x+n+\delta)\hat{k}-\hat{k}(\gamma_{\hat{k}})^{\ast} \end{equation} Then, differentiating the last expression for time \begin{equation} \dot{\hat{c}}=\dot{\hat{k}}[f'(\hat{k})-(x+n+\delta) - (\gamma_{\hat{k}})^{\ast}] \end{equation} must hold in the steady state. Then the book claims that the expression in the square brackets is positive from the transversality condition and so $(\gamma_{\hat{k}})^{\ast}$ and $(\gamma_{\hat{c}})^{\ast}$ must have the same sign.
It is clear that to satisfy the transversality condition it must be: $f'(\hat{k}(t)) - \delta - x - n >0$. What I do not understand is how can I infer from the transversality condition that $f'(\hat{k}(t)) - \delta - x - n - (\gamma_{\hat{k}})^{\ast} >0$ and why $(\gamma_{\hat{k}})^{\ast}$ and $(\gamma_{\hat{c}})^{\ast}$ have the same sign?
