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Suppose we have the following problem:

$\max \int_0^\infty \exp(-\rho t) u(c(t))dt$

where $c(t)$ is consumption at time $t$. Subject to:

$\dot{k}(t)= f(k(t))- c(t) - \delta k(t)$.

where $k$ is capital, $f(.)$ production function and $\delta$ depreciation.

This problem is not hard to solve. However, suppose that we now add a twist. Suppose that agent can choose between two production functions:

$f_A(k(t))>f_B(k(t))$

but if agent chooses $f_A(k(t))$ there is a downside that depreciation is larger $\delta_A >\delta_B$.

I am not sure how to solve this problem. My idea was as follows:

  1. I will set up two separate optimization problems; problem A where I only use parameters for A production function and problem B where I only use B production function,
  2. Calculate the optimal $c_A$ and $c_B$ and corresponding utilities,
  3. Compare $U(c_A)$ to $U(c_B)$.

My reasoning is that if the person has choice between two production technologies they would simply pick which one gives highest utility. Is this valid reasoning?

However, what if the person can switch back and forth? Can we still apply this recipe (if my recipe is valid)?

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    $\begingroup$ Incorrect comment see below: $$\textrm{}$$ Looks like you would update your law of motion to: $$k(t+1)-k(t)=\max\{f_A(k(t))-\delta_A k(t), f_B(k(t))-\delta_B k(t)\}-c(t)$$ There would be no switching if both $f_A(k(t))$ and $f_B(k(t))$ are concave. $\endgroup$
    – EconJohn
    Commented Dec 24, 2023 at 16:47
  • $\begingroup$ @EconJohn is that really it? If you want to post it as an answer I can accept it $\endgroup$
    – WilliamT
    Commented Dec 25, 2023 at 21:39
  • $\begingroup$ EconJohn: That was great but could you or WilliamT explain where the expression for $\dot{k}(t)$ comes from ? Thanks. $\endgroup$
    – mark leeds
    Commented Dec 26, 2023 at 5:20
  • $\begingroup$ @WilliamT, I have made a mistake. There can be switching even with concave functions. The law of motion is the same. to see switching with concave functions: desmos.com/calculator/vtumhnfhfd In case the link dies: imgur.com/a/kMNAaDF $\endgroup$
    – EconJohn
    Commented Dec 26, 2023 at 5:20
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    $\begingroup$ I am not sure if an ex ante choice of the production is relevant (or interesting). However, it would be interesting to write a problem where an individual (or a social planner) chooses optimally the technology. $\endgroup$ Commented Dec 26, 2023 at 11:14

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Your reasoning is valid, although I am not sure you can do this efficiently with continuous time...

Using discrete time to convey the same idea (with continuous time, no, I don't know how to do this, since then you need to linearize the Euler equations, but a linear approximation is not a good idea if you have to compare back and forth the value functions), something like:

$W(k) = \max \{V_a(k), V_b(k)\} + \beta W(k')$

Where

$V_i(k) = \max_{0\leq k'\leq z_i k^{\alpha} + (1-\delta_i)k} \{U(z_i k^{\alpha} + (1-\delta_i) k - k') \} + \beta V_i(k')$

And $z_a > z_b$, $\delta_a > \delta_b$

The control variable (namely, the choice variable, what you optimize) of the problem above is $k'$.

So, here I've chosen productivity as a fixed constant, but obviously you can make this way more messy and realistic...

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