I am reading a paper (Manuelli & Seshadri 2014) that use a neoclassical model to model technology diffusion. I cannot understand the arbitrage condition that are used to calculate the rent price on the machine service used in the production function. I reframe a simplified version of the model as below.
Assume that there are complete rental markets. And assume that the machine services can be provided by machines of different vintages $\gamma$. The maximization problem on production side is $$\max_{k_t, l_t} p_{ct}F(k_t,l_t) - \sum_{\tau=-\infty}^{t}\left[q_{k t}(\tau)+c_{k t}(\tau)\right] m_{k t}(\tau) - w_{t}l_{t}$$, where $k_{t}=\sum_{\tau=-\infty}^{t} m_{k t}(\tau) \left(1-\delta_{k \tau}\right)^{t-\tau}$, $m_{k t}(\tau)$ is the quantity of machines of vintage $\tau$, $q_{k t}(\tau)$ is the rental price of the machine, and $c_{k t}(\tau)$ is the operating cost of the machine.
Given no arbitrage, the price of the machine of vintage $\tau$ should be $q_{k t}(\tau)=p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}$, where $r$ is the interest rate. In addition the paper says that for a new machine at time $t$, no arbitrage arguments imply $$q_{k t}(t)=p_{k t}\left[1-R_{t}(1)\left(1-\delta_{k t}\right)\right]+\delta_{k t} C(t+1, T-1)$$, where $C(t+1, T-1) \equiv \sum_{j=0}^{T-1} R_{t+1}(j) c_{k t+1+j}$ and $R_{t}(j) \equiv \prod_{k=1}^{j}\left(1+r_{t+k}\right)^{-1}$. The authors explain that the first term in the RHS translates the price of a machine into its flow equivalent while the second term in the RHS captures "the increase in cost per unit of machine services associated with operating a one year old machine, relative to a new machine".
What I cannot understand is the intuition of this no arbitrage equation of new machine, mainly for the part of the present discounted value of the operating cost.
Update:
I find that I may misunderstand the notation. This no-arbitrage requirement is not for a new machine but a machine with vintage $t$, i.e. $q_{k t}(\tau = t)$. So this is actually the rental price at time $t$ of a machine produced at $t=0$. Also it seems that the $p_{k t}$ in the equation should be $p_{k t}(t)$, i.e. the price of the machine with vintage $\tau =t$ at time $t$. The reason for my argument is below.
Actually I find that in the end of the paper there is an appendix section that derive this equation (though with no mention in the main text). It says that the formula $q_{k t}(\tau)=p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}$ "has the drawback that it depends on the price of used machines". And "since there are no data available on used machine prices we now proceed to derive an expression for $q_{k t}(\tau)$ using simple arbitrage arguments". And the derived equation (in a full version) is $$q_{k t}(t)=p_{k t}(t)\left[1-R_{t}(1)\left(1-\delta_{k t}\right) \frac{\gamma_{c t}}{\gamma_{c t+1}}\right]+\left(1-\Delta_{t+1}\right) C(t+1, T-1)$$.
I can generally follow this derivation, which is quite complicated. But again, I don't understand the intuition of this "simple arbitrage arguments" as well as the intuition behind this derivation. It seems that the authors only have the new machine price at each $t$, i.e. $p_{kt}(0)$, and now from the equation they can estimate $q_{k t}(0)$. (Now I get even one more question: since there are also vintages in the market that provide machine services at each time, how can we derive their rental prices? Is new machine rental price enough to estimate the model?)
I copy the derivation in the appendix below for reference. This time I keep a full model version as I am not sure whether there will be some typos in the original derivation. The additional part is the machine quality $v(x_t)$ and machine cost $\gamma_{ct}$ so that $p_{k t}(t)=\frac{v\left(\mathbf{x}_{t}\right)}{\gamma_{c t}}$ and $\tilde{k}_{t}(\tau) \equiv v\left(\mathbf{x}_{\tau}\right)\left(1-\delta_{k \tau}\right)^{t-\tau}$. And now $k_{t}=\sum_{\tau=-\infty}^{t} m_{k t}(\tau) \tilde{k}_{t}(\tau)$.
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One can show that the optimal choice of $m_{k t}(\tau)$ requires that $$p_{c t} F_{k}(t) \tilde{k}_{t}(\tau)=p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}+c_{k t}(\tau)$$.
Iterating forward and denoting by $T(\tau, t)$ the number of periods of useful life that a $t-\tau$ old tractor has left, we get that $$p_{k t}(\tau)=\sum_{j=0}^{T(\tau, t)} R_{t}(j)\left\{p_{c t+j} F_{k}(t+j) \tilde{k}_{t+j}(\tau)-c_{k t+j}(\tau)\right\}$$.
Using the special structure of $k_{t}(\tau)$ it follows that $$\begin{aligned} p_{k t+1}(\tau)=& \gamma_{k \tau} v\left(\mathbf{x}_{\tau}\right)\left(1-\delta_{k \tau}\right)^{t+1-\tau} \sum_{j=0}^{T(\tau, t+1)} R_{t+1}(j) p_{c t+1+j} F_{k}(t+1+j)\left(1-\delta_{k \tau}\right)^{j} \\ &-\sum_{j=0}^{T(\tau, t+1)} R_{t+1}(j) c_{k t+1+j}(\tau) \end{aligned}$$ and $$\begin{aligned} p_{k t+1}(t)=& \frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)} p_{k t+1}(t+1)-\left[1-\frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)}\right] \sum_{j=0}^{T(t, t+1)} R_{t+1}(j) c_{k t+1+j} \\ &+\frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)}\left\{\sum_{j=0}^{T(t, t+1)} R_{t+1}(j) p_{c t+1+j} F_{k}(t+1+j)\left(1-\delta_{k t}\right)^{j}\left(1-(1+\varphi)^{j}\right)\right.\\ &\left.-\sum_{j=T(t, t+1)}^{T(t+1, t+1)} R_{t+1}(j)\left[p_{c t+1+j} F_{k}(t+1+j)\left(1-\delta_{k t+1}\right)^{j}-c_{k t+1+j}\right]\right\} \end{aligned}$$ where $\delta_{k t+1}=\delta_{k t}-\varphi\left(1-\delta_{k t}\right)$.
To simplify the presentation we assume that two consecutive vintages of tractors have the same depreciation and economic lifetime $(\varphi=0$ and $T(t+1, t+1)=T(t, t+1)+1)$. Moreover, we assume that operating costs vary over time, but are not a function of the vintage, i.e. $c_{k t+j}(\tau)=c_{k t+j},$ for all $\tau .$ In this case, using the previous formula for $\tau=t$ (one period old tractors) and $\tau=t+1$ (new tractors) it follows that $$\begin{aligned} p_{k t+1}(t)=& \frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)} p_{k t+1}(t+1)-\left[1-\frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)}\right] \sum_{j=0}^{T-1} R_{t+1}(j) c_{k t+1+j}(t) \\ &+\frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)} R_{t+1}(T)\left[p_{c t+1+T} F_{k}(t+1+T) v\left(\mathbf{x}_{t+1}\right)\left(1-\delta_{k t}\right)^{T}-c_{k t+1+T}\right] \end{aligned}$$.
However, the last term in square brackets must be zero, since a tractor of vintage $t+1$ is optimally scrapped when the marginal product of its remaining tractor services, $p_{c t+1+T} F_{k}(t+1+T) v\left(\mathbf{x}_{t+1}\right)\left(1-\delta_{k t}\right)^{T},$ equals the marginal cost of operating it, $c_{k t+1+T}$
Let the 'effective' depreciation of a vintage $t$ tractor between $t$ and $t+1$ be $\Delta_{t+1} \equiv$ $\frac{v\left(\mathbf{x}_{t}\right)\left(1-\delta_{k t}\right)}{v\left(\mathbf{x}_{t+1}\right)} .$ It follows that, $q_{k t}(t)=p_{k t}(t)\left[1-R_{t}(1)\left(1-\delta_{k t}\right) \frac{\gamma_{c t}}{\gamma_{c t+1}}\right]+\left(1-\Delta_{t+1}\right) C(t+1, T-1)$.
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