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The story of my question is

enter image description here

I have multiple question.

(1) when John doesn’t work in the underground economy at all , t=0, how can I find the optimal value of $l$ and consumption bundle and his utility.

(2) And when john work in the underground economy, t>0, how can write his optimization problem? Can I say that $l=0$ for t>0 by using the equation (1) in the picture? And in this case, when he is not fined, which way I can find his optimal consumption bundle and his expected utility?

(3) finally, I would like to derive an inequality in order to describe for which parameter values John elects to work in the underground economy. And how can I decide whether John is willing to work in the underground economy or not if m and w increase by the same dollar amount ?

Sorry for not posting my answer, because I am not familiar with such a question, my answer is really trivial.

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  • Case 1 If John choose not to work in the underground economy, his utility maximization problem is

\begin{eqnarray*} \max_{x_1, x_2, l} && x_1x_2(1-l) \\ \text{s.t.} && p_1x_1 + p_2x_2 \leq wl \\ && 0 \leq l\leq 1 \end{eqnarray*}

Solution to this problem is \begin{eqnarray*}(x_1^*, x_2^*, l^*) = \left(\frac{w}{3p_1}, \frac{w}{3p_2}, \frac{2}{3}\right)\end{eqnarray*}

  • Case 2 If John choose to work in the underground economy then he'll not work in the legitimate sector since wage in the underground economy is higher, consequently his utility maximization problem is

\begin{eqnarray*} \max_{x_1, x_2, t} && (1-q)x_1x_2(1-t) \\ \text{s.t.} && p_1x_1 + p_2x_2 \leq mt \\ && 0 \leq t\leq 1 \end{eqnarray*}

Solution to this problem is \begin{eqnarray*}(x_1^*, x_2^*, t^*) = \left(\frac{m}{3p_1}, \frac{m}{3p_2}, \frac{2}{3}\right)\end{eqnarray*}

Comparing utilities in the two cases above give us the condition under which John will choose to work for the two sectors :

  • If $(1-q)m^2 \geq w^2$, he'll work for the underground economy.
  • If $(1-q)m^2 \leq w^2$, he'll work for the legitimate sector.
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