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Consider a production economy with 2 consumers and 2 firms. There are 3 goods: leisure $l$ and two goods $x,y$. The utility function of both consumers are $u_i(l_i,x_i,y_i)=l_i(x_i)^2(y_i)^2$ and the production functions of the 2 firms are $x=2\sqrt{l}$ and $y=2x$. Each consumer is only endowed with 6 units of leisure. Consumer 1 is the sole owner of the first firm (the one that produces $x$) and consumer 2 is the sole owner of the second firm (the one that makes $y$). Suppose the wage rate is 1.

How do I find a Walrasian equilibrium for this economy?

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  • $\begingroup$ You solve for the demand of the consumers, the profit-maximizing plan of from 1, and use the fact that if firm 2 is active (necessary, since equilibria are Pareto efficient), constant returns to scale pins down the ratio of the prices of the two consumption goods. So you just have to find one price ratio under which supply is equal to demand. $\endgroup$ Oct 29, 2023 at 22:07

1 Answer 1

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Solving the profit maximisation problem of firm producing $x$,

$\max_{x\geq 0, l_X\geq 0} \ p_Xx - l_X$ subject to $x = 2\sqrt{l_X}$

gives the demand for labor by firm producing $x$ and supply of $x$ as: $l_X(p_X)=p_X^2$, $x^s(p_X)=2p_X$. Profits earned by this firm is $\pi_X(p_X)=p_X^2$.

Solving the profit maximisation problem of firm producing $y$,

$\max_{x\geq 0, y\geq 0} \ p_Yy - p_Xx$ subject to $y = 2x$

yields the condition for positive production of $y$ as: $p_Y = \dfrac{p_X}{2}$. Profits earned by this firm is $\pi_Y(p_X, p_Y)=0$.

Now we can solve the utility maximisation problem of consumer $i$ \begin{eqnarray*} \max_{l_i\geq 0,x_i\geq 0,y_i\geq 0} & l_ix_i^2y_i^2 \\ \text{s.t. } & p_Xx_i+p_Yy_i+l_i = m_i \end{eqnarray*} Here $m_1=6+\pi_X = 6+p_X^2$ and $m_2 = 6+\pi_Y = 6$.

Solving the utility maximisation problem, we get the demand for leisure as $l_1 = \dfrac{6+p_X^2}{5}$, $l_2 = \dfrac{6}{5}$. Using demand = supply condition in the labor-market, we get

$l_1+l_2+l_X=12$ i.e. $\dfrac{6+p_X^2}{5}+\dfrac{6}{5}+p_X^2 = 12$

which gives $p_X=2\sqrt{2}$. Using $p_Y = \dfrac{p_X}{2}$, we get $p_Y = \sqrt{2}$. Corresponding demand for labor by firm producing $x$ equals $l_X=8$, demands for leisures by the two individuals are respectively $l_1=2.8$ and $l_2=1.2$.

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