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Consider an economy with four goods, two individuals and two firms. Firm 1 produces good $x$, firm 2 produces good $y$. Consumers' utilities are $u_1(x,y,z,w)=\min\{x,2y\}$ and $u_2(x,y,z,w)=\min\{2x,y\}$. Production function of firm 1 is $x=(1+a)z$, production function of firm 2 is $y=w$, where $a$ is a small positive number. Consumer 1(resp. 2) owns firm 1(resp. 2). Initially, consumer 1(resp. 2) has $1$ unit of good $z$ (resp. $w$).

The question asks for an equilibrium. My approach is first stipulating a price, then try to deduce the equilibrium, but I fails to do so, since there are 4 goods here and they are not symmetric. Any help is appreciated.

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  • $\begingroup$ Shouldn't price be endogenous. Given that this is homework (at least seems like), you can perhaps show your attempt. For example, writing objective functions for the two individuals. Or (hint:) consider boundary cases $\endgroup$
    – Dayne
    Dec 2 '20 at 15:33
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First, note that both firms have constant returns to scale. This implies that firms must make zero profit in equilibrium. If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist. So who owns which firm does not really matter. We know from the first welfare theorem that every equilibrium must be Pareto efficient. With the given utility functions, this means that both firms have to produce a positive amount. But this pins down the relative prices of $x$ and $z$, and of $y$ and $w$, respectively. So this reduces the problem to one in which you have to find two prices. You can set one of the remaining prices to be $1$, and then calculate the excess demand for one good. FInd the price that makes this excess demand zero. As a consequence of Walras' law, this price will clear all markets.

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  • $\begingroup$ "If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist" - but OP has mentioned that each of the individual has limited input/endowment ($z,w=1$). So how can the production be scaled beyond that? We can say however that since $z,w$ do not appear in utility function, they will exhaust their endowment to produce $x,y$ and then exchange. $\endgroup$
    – Dayne
    Dec 3 '20 at 10:45
  • $\begingroup$ One condition of equilibrium is that a firm chooses a profit-maximizing production plan while taking prices as given. That the plan is actually feasible is part of the equilibrium conditions, not part of the firm's maximization problem. $\endgroup$ Dec 3 '20 at 10:50

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