# Robinson Crusoe with tax

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 Cpc and in addition 3pc*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 CF s.t. pcC+3pcC+pfF= w80+profit+M (M=coconut money), he assumed that the profit and M dont depend on his consumption)

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• 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. Sep 18 at 22:55

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, $$C = \sqrt{L_c} = 4 \text{ and } F = \sqrt{L_f} = 8.$$