Bertsekas (1976) introduces a component replacement example in which the current state of a component $x\in [0,1]$ is determined at the beginning of each period, and the agent makes a decision whether or not to replace the component at cost $R>0$. The replacement resets $x$ into $0$, and $C(x)$ is the expected cost of having the component at state $x$ for a single period. Bertsekas show that in the infinite horizon, the optimal policy takes the threshold form of $$ Replace\;\;\qquad if \; x\geq \gamma^* \\ Don't\;replace \; if\;x<\gamma^* $$ where $\gamma^*$ is the smallest scalar for which $$ R+C(0)+\beta \int_{0}^{1}{J^*(z)dF(z|0)=C(\gamma^*)+\beta \int_{0}^{1}J^*(z)dF(z|\gamma ^*)}, $$ and $J^*(z)$ the value function is $$ J^*(x)=min[R+C(0)+\beta \int_{0}^{1}{J^*(z)dF(z|0)},C(x)+\beta \int_{0}^{1}J^*(z)dF(z|x)]. $$ Since replacement occurs only at $x\geq \gamma^*$, $J^*(z)=C(\gamma^*)+\beta \int_{0}^{1}J^*(z)dF(z|\gamma^*)$ for $x<\gamma^*.$ So far So good. Now, it is presented in Rust(1985) that when $F(z|x)=1-\exp(-\lambda(z-x))$, i.e. exponential transition p.d., $\gamma^*$ is solution to $$ R(1-\beta )=\int_{0}^{\gamma^*}\partial C(y)/\partial y [1-\beta \exp(-\lambda(1-\beta)y)]dy, $$ which came from the first equation, but without showing derivation. I tried to derive the last equation from the first, but could not make it as it is represented. Any comments or hints on the derivation would be appreciated, thanks in advance.
references:
- Bertsekas, Dimitri P. "Dynamic programming and stochastic control." (1976).
- Rust, John. "Stationary equilibrium in a market for durable assets." Econometrica: Journal of the Econometric Society (1985): 783-805.
