No arbitrage arguments on production machine in a neoclassical model

Question

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.

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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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Answer 1

The no-arbitrage condition simply states that the (rental) price of the machine should equal it's net cost. So, there are no profits, i.e. I am indifferent between renting or not. If the price is below or above the cost then there is still arbitrage (money on the table).

Due to the definition of $q$, we want the rental cost per period (i.e. in each year). So, the LHS gives you the price to rent in each period. The RHS gives you the net cost to rent. The net cost is made up of two parts.

Let's start with the first RHS term. For this, we need to figure out how much would buying a machine cost us if we were paying it off every year (that is its yearly cost, i.e. rental cost). Think of it like a mortgage or buying a car. You could buy it in cash for the total price, or convert that value into monthly/yearly payments. Here we want to look at yearly payments. So we do that conversion in the first term of the RHS.

So, first step is take the price of buying the machine and figure out, what is the equivalent cost per period (e.g. year) if I were renting it to myself (without a profit).

So let's say a machine costs 100 USD and it lives for 10 years (i.e. it fully depreciates afterwards, at which point its value is set to 0). Then, one could think that the equivalent rent could be 10 USD a year. Well, not quite. First, costs in the future need to be discounted with interest $r$ (because money in the future is worth less than money today). Similarly, years closer to the present have to be discounted less times. So your $R$ term is taking care of that for each year. Then, the machine also gets old due to depreciation $\delta$. So your 9 year old machine is worth less than your 2 year old machine, because it is not as good anymore. So the rental rate of that older machine would need to be adjusted too. Hence, each year, the machine is worth a little less than it used to (by $\delta$ %).

So your first RHS term is just translating the price of buying the machine into yearly costs. To do that, you need to take care of discounting ($R$) all future flows and for each year, you take out one yearly depreciation ($\delta$). Here, the machine lives forever, but eventually has very low value, due to $\delta$ that keeps getting cut off in each period. If we set $\delta$ = 1 after $t=10$ then we would have my example of a machine living 10 years.

Now the second part of RHS. There is also opportunity cost from not buying the machine, hence why I referred to "net" cost above. We need to account for this. If I buy and use a new machine I can save $x$ cost per year compared to my current one. We need to "net" out this benefit for the cost, because the new machine is partially paying for itself by being better than the old one, so its real net cost needs to reflect that. So, since we are comparing costs, we will just flip this benefit into a cost. Instead of looking at the benefit (cost saving) of using a new machine vs. the old one, the authors look at the cost (foregone benefit) of keeping on using the old machine vs. the new one. Mathematically it's the same, just with a minus sign in front.

Note, we could also have written the arbitrage condition as the total price of buying the machine (having it forever) vs. the total costs. However, economists tend to use values per period, e.g. yearly in our analysis. You also have such a setting. You need this, since you're first equation looks at (writes out) a problem in each period. That is why it makes sense to translate the total price of the machine into it's yearly equivalent cost which is called by economists as the "rental" cost, even if you're not actually renting. That is also why there is this calculation in the first RHS term.

Answer 2

Given no arbitrage, the price of the machine of vintage $\tau$ should be $q_{k t}(\tau)=p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}$...

"No arbitrage" is a bit of a misnomer here. In this sentence, it really means "zero profit". The rental price must be equal to the price of owning a machine for one period. For example, if $$ q_{k t}(\tau) > p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}, $$ then one can buy the machine at $p_{k t}(\tau)$, rent it out for one period at $q_{k t}(\tau)$, then sell at $p_{k t+1}(\tau)$ for strictly positive profit, which is not possible if the rental market is competitive and allows free entry.

In addition the paper says that for a new machine at time $t$, no arbitrage arguments imply... $$ q_{k t}(t)=p_{k t}(t) \left[1-R_{t}(1)\left(1-\delta_{k t}\right)\right]+\delta_{k t} C(t+1, T-1) $$

The RHS can be re-written as $$ p_{k t}(t) - \frac{1}{1+r_{t+1}} (1 - \delta_{kt} )p_{k t}(t) - \frac{1}{1+r_{t+1}} \delta_{kt} C(t + 1, T-1). $$ In other words (letting $\tau = t$ in the first equation), $$ p_{k t+1}(t) = \frac{1}{1+r_{t+1}} (1 - \delta_{kt} )p_{k t}(t) - \frac{1}{1+r_{t+1}} \delta_{kt} C(t + 1, T-1), $$ which is a necessary condition for equilibrium. In equilibrium, in all periods, $$ \mbox{price of a machine} = \mbox{marginal product} - \mbox{marginal cost}. $$ A machine that is one year older at $t+1$has marginal product diminished by $1-\delta_{kt}$, which is the first term. It also accrues additional marginal cost, which is the second term. ("No arbitrage" is again a bit of misnomer.)