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Question:

Hypothetically, Robinson Crusoe is stuck on an island and can choose between working on gathering coconuts or leisure. The utility function is:

$U(C,L)=C^{2/5}L^{3/5}$

where C is the num of coconuts found, L is fraction of day leisurely, H is the fraction of day spent looking for coconuts

and the production function is: $f(H) = 8H^{1/2}$

To solve this I manipulated the production function and set up the Lagrangian to solve as a utility maximization problem

$\mathcal{L}$ = $C^{2/5}L^{3/5}+𝜆(1-\frac{C^{2}}{64}-L)$

...

$L=c\frac{\sqrt{3}}{8}$

...

$L = \frac{-9 + 3\sqrt{13}}{2}$

(not totally sure this is correct)

Robinson Crusoe also creates a firm that posts fixed prices for labor and coconuts who him as an agent also buys from. And the firm experiences market clearing. Agent Crusoe sells labor and buys coconuts with his wage income and his share dividends. Crusoe, Inc. buys labor and produces coconuts using Crusoe’s technology production. I'm pretty uncertain what I should do to find the price or quantity of coconuts produced.

PS this is my first time using latex, hope I did well :)

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    $\begingroup$ Small correction in the first part: $\frac{L}{C^2} = \frac{3}{64}$. $\endgroup$ Apr 26, 2023 at 2:35
  • $\begingroup$ I think a lot is missing from the problem. If Crusoe sells his products in the market, what's the market demand? The usual Robinson Crusoe economy is a production-exchange economy where the initial endowments are given; if that's the case, what are the initial endowments, and do I assume the existence of other goods? If otherwise Crusoe produces in his own firm, works as the labourer and eats from there (assuming an unlimited supply of raw coconuts), then budget constraint or optimal choices do not change. $\endgroup$ Apr 26, 2023 at 2:51
  • $\begingroup$ I have posted the answer. For details about how to approach and solve such problems, you can watch this playlist: youtube.com/playlist?list=PLUJGfL_499TLCq4g1sW64WiiojYkVRRIl $\endgroup$
    – Amit
    Apr 26, 2023 at 3:49

1 Answer 1

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If you want to find Pareto efficient allocation in this economy, then you can determine that by solving the following system for $(C, L, H)$:

  • $L+H=1$
  • $C = 8\sqrt{H}$
  • $\text{MRS} = \dfrac{3C}{2L} = \dfrac{4}{\sqrt{H}} = \text{MRT}$

and we get $(C, L, H) = \left(4,\frac{3}{4},\frac{1}{4}\right)$

Here is the picture: enter image description here

To determine the competitive equilibrium, we need to find price of labor $w^*$ and an allocation $(C^*, L^*, H^*)$ such that the following holds:

  1. Given $w^*$, $(C^*,H^*)$ solves the profit maximisation problem of the firm: \begin{eqnarray*} \max_{C\geq 0, H\geq 0} & C - w^*H \\ \text{s.t. } & C \leq 8\sqrt{H} \end{eqnarray*} Let $\pi^*$ denotes the optimal profits i.e. $\pi^* = C^* - w^*H^*$.
  2. Given $w^*$ and $\pi^*$, $(C^*,L^*)$ solves the utility maximisation problem of Robinson Crusoe: \begin{eqnarray*} \max_{C\geq 0, 0\leq L \leq 1} & C^{\frac{2}{5}}L^{\frac{3}{5}}\\ \text{s.t. } & C \leq w^*(1-L) + \pi^* \end{eqnarray*}
  3. $L^*+H^* = 1$

Solving the system, we get $w^*=8$ and $(C^*, L^*, H^*) = \left(4,\frac{3}{4},\frac{1}{4}\right)$ as the competitive equilibrium. Here is the picture: enter image description here

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    $\begingroup$ Great answer, but the OP also asked for the price of coconuts, so you should maybe add that coconuts are a numeraire good here and Agent Crusoe's income is $w^*H^*+\pi^*=4$. $\endgroup$
    – VARulle
    Apr 26, 2023 at 8:16
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    $\begingroup$ @VARulle Thanks. Actually, that is true without loss of generality. Here $w$ can be considered as a real wage which is equal to $\frac{W}{P}$, where $W>0$ is the nominal wage, and $P>0$ is the price of the coconuts. $\endgroup$
    – Amit
    Apr 26, 2023 at 10:16

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