<p>The profit maximisation problems give:
<span class="math-container">$$
\max_{L_c} p_c \sqrt{L_c} - w L_c,
$$</span>
and
<span class="math-container">$$
\max_{L_f} p_f \sqrt{L_f} - w L_f
$$</span>
This gives the following first order conditions:
<span class="math-container">$$
\frac{p_c}{2w} = \sqrt{L_c} \text{ and } \frac{p_f}{2w} = \sqrt{L_f}.
$$</span>
Substituting into the profit functions and adding up gives:
<span class="math-container">$$
\pi = \frac{(p_c)^2 + (p_f)^2}{2w} - w \underbrace{(L_f + L_w)}_{80}.
$$</span></p>
<p>Next, as the utilty function is Cobb-Douglas, we have:
<span class="math-container">$$
C = \frac{80w + M + \pi}{8 p_c} \text{ and } F = \frac{80w + M + \pi}{2 p_f}
$$</span>
Substituting out the profits <span class="math-container">$\pi$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;C = \frac{M + \frac{(p_c)^2 + (p_f)^2}{2w}}{8p_c} = \sqrt{L_c} = \frac{p_c}{2w},\\
&amp;F = \frac{M + \frac{(p_c)^2 + (p_f)^2}{2w}}{2p_f} = \sqrt{L_f} = \frac{p_f}{2w}
\end{align*}
$$</span>
Solving for <span class="math-container">$M$</span>:
<span class="math-container">$$
M = \frac{7 (p_c)^2 - (p_f)^2}{2w} = \frac{(p_f)^2 - (p_c)^2}{2w}.
$$</span>
So,
<span class="math-container">$$
(p_f)^2 = 4 (p_c)^2 \to p_f = 2 p_c.
$$</span>
Then squaring the first order conditions gives:
<span class="math-container">$$
L_f = \frac{(p_f)^2}{4 w^2} = \frac{4 (p_c)^2}{4 w^2} = 4 L_c.
$$</span>
Then:
<span class="math-container">$$
80 = L_f + L_c = 5 L_c \to L_c = 16 \text{ and } L_f = 64.
$$</span>
As such,
<span class="math-container">$$
C = \sqrt{L_c} = 4 \text{ and } F = \sqrt{L_f} = 8.
$$</span></p>

CC BY-SA 4.0
2023-09-18T12:09:56.467Z
2023-09-18T12:09:56.467Z
tdm
<p>This question was asked in a micro exam last semester and I just dont know the answer, have been thinking about it for weeks.
Please help.</p>
<p>In a Robinson Cruse Economy Robinson produces Coconuts (C) and Fish (F) with Labour (L).</p>
<p>The Production functions are:</p>
<p>C(Lc) = Lc^0.5</p>
<p>F(Lf) = Lf^0.5</p>
<p>his utility function is: u= C*F</p>
<p>A politician wants Robinson to introduce a tax on the consumption of coconuts of 300% (The consumer has to pay C<em>pc and in addition 3</em>pc*C to the government)</p>
<p>Robinson thinks that he will be worse off after the introduction of the tax. The politician tells him that wont be the case, because all the tax money will be paid back to the consumer as „coconut money&quot;.</p>
<p>Task: Use the Robinson Crusoe Modell to find out how much C and F Robinson consumes with tax. The coconut money he gets from the government doesnt depend on his consumption.</p>
<p>The optimum without tax is: F=C= 40^0.5</p>
<p>(when the consumer chooses C and F, to maximise C<em>F s.t. pc</em>C+3<em>pc</em>C+pf<em>F= w</em>80+profit+M (M=coconut money), he assumed that the profit and M dont depend on his consumption)</p>

CC BY-SA 4.0
2023-09-18T22:55:01.037Z
Note that Mr. Crusoe is absolutely correct that he will be worse off under the tax scheme. The resources they are taxing away must necessarily be going to be put toward something he values *less* than he does the coconuts.  Otherwise he would be contributing to this other thing without the necessity of a tax, simple advertising would do.
Perkins
<microeconomics><general-equilibrium>
Robinson Crusoe with tax
AJl
# Robinson Crusoe with tax

(Asked by: username_0 on 2023-09-18)

This question was asked in a micro exam last semester and I just dont know the answer, have been thinking about it for weeks.
Please help.

In a Robinson Cruse Economy Robinson produces Coconuts (C) and Fish (F) with Labour (L).

The Production functions are:

C(Lc) = Lc^0.5

F(Lf) = Lf^0.5

his utility function is: u= C\*F

A politician wants Robinson to introduce a tax on the consumption of coconuts of 300% (The consumer has to pay C*pc and in addition 3*pc\*C to the government)

Robinson thinks that he will be worse off after the introduction of the tax. The politician tells him that wont be the case, because all the tax money will be paid back to the consumer as „coconut money".

Task: Use the Robinson Crusoe Modell to find out how much C and F Robinson consumes with tax. The coconut money he gets from the government doesnt depend on his consumption.

The optimum without tax is: F=C= 40^0.5

(when the consumer chooses C and F, to maximise C*F s.t. pc*C+3*pc*C+pf*F= w*80+profit+M (M=coconut money), he assumed that the profit and M dont depend on his consumption)

**Tags:** microeconomics, general-equilibrium
> **Comments:**
> **username_1**: Note that Mr. Crusoe is absolutely correct that he will be worse off under the tax scheme. The resources they are taxing away must necessarily be going to be put toward something he values *less* than he does the coconuts.  Otherwise he would be contributing to this other thing without the necessity of a tax, simple advertising would do.  

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## 1 answers

### Answer by username_2 on 2023-09-18. Score: 4 | Accepted answer

The profit maximisation problems give:
$$
\max\_{L\_c} p\_c \sqrt{L\_c} - w L\_c,
$$
and
$$
\max\_{L\_f} p\_f \sqrt{L\_f} - w L\_f
$$
This gives the following first order conditions:
$$
\frac{p\_c}{2w} = \sqrt{L\_c} \text{ and } \frac{p\_f}{2w} = \sqrt{L\_f}.
$$
Substituting into the profit functions and adding up gives:
$$
\pi = \frac{(p\_c)^2 + (p\_f)^2}{2w} - w \underbrace{(L\_f + L\_w)}\_{80}.
$$

Next, as the utilty function is Cobb-Douglas, we have:
$$
C = \frac{80w + M + \pi}{8 p\_c} \text{ and } F = \frac{80w + M + \pi}{2 p\_f}
$$
Substituting out the profits $\pi$ gives:
$$
\begin{align\*}
&C = \frac{M + \frac{(p\_c)^2 + (p\_f)^2}{2w}}{8p\_c} = \sqrt{L\_c} = \frac{p\_c}{2w},\\
&F = \frac{M + \frac{(p\_c)^2 + (p\_f)^2}{2w}}{2p\_f} = \sqrt{L\_f} = \frac{p\_f}{2w}
\end{align\*}
$$
Solving for $M$:
$$
M = \frac{7 (p\_c)^2 - (p\_f)^2}{2w} = \frac{(p\_f)^2 - (p\_c)^2}{2w}.
$$
So,
$$
(p\_f)^2 = 4 (p\_c)^2 \to p\_f = 2 p\_c.
$$
Then squaring the first order conditions gives:
$$
L\_f = \frac{(p\_f)^2}{4 w^2} = \frac{4 (p\_c)^2}{4 w^2} = 4 L\_c.
$$
Then:
$$
80 = L\_f + L\_c = 5 L\_c \to L\_c = 16 \text{ and } L\_f = 64.
$$
As such,
$$
C = \sqrt{L\_c} = 4 \text{ and } F = \sqrt{L\_f} = 8.
$$


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