# Find the competitive equilibrium of the following economy

The following question was given as a part of a task in microeconomic theory course. It is not from some textbook and since I still haven't figured a way to solve it I will leave it here. Thank you in advance.

Let $$f:\mathbb{R}_{+}^2\to \mathbb{R}$$ be a single output production function such that

$$f(z_1, z_2) = (z_1^\delta+z_2^\delta)^{\frac{1}{\delta}},\quad \text{with 0<\delta<1}$$

Let $$X = \mathbb{R}^3$$ be the consumption set. The vector of commodities $$(z_1, z_2, q)$$ is a typical element of $$X$$. There are $$N=\{1, 2, \dots, n\}$$ individuals (with $$n\geq 2$$) where the typical individual is denoted by $$i$$. Each $$i$$ has a preference relation $$\succsim_i$$ defined over $$X$$ and the utility representation of $$\succsim_i$$ is given by $$u_i(z_1, z_2, q) = i(z_1+z_2) +q$$. Individual $$1$$ owns the technology, but has an endowment $$\mathcal{E}_1 =(0, 0, 0)$$. For every $$i\in N-\{1\}$$ and some pair of non-negative real numbers $$a_1$$, $$a_2$$, $$\mathcal{E}_i =(\frac{a_1}{n-1}, \frac{a_2}{n-1}, 0)$$.

Which is the competitive (Arrow-Debreu) equilibrium for this economy?

I believe it will be easier if we set $$n=2$$ to solve the problem.

• Should it be $X=\mathbb{R}^3_+$, or is negative consumption really allowed here? Commented Jan 8 at 22:37
• As it is given, it is $X = \mathbb{R}^3$ Commented Jan 8 at 22:39

There will be no equilibrium. Both technology and endowment play no role in the argument.

Assume there would be an equilibrium with price system $$(p_1,p_2,p_3)$$.

Notice that all prices must be strictly positive since all goods are desirable.

The optimality condition of agent $$1$$ implies that $$p_1=p_3$$. If, say, $$p_1, agent $$1$$ could always improve by consuming one unit less of good $$3$$ and consuming $$p_3/p_1$$ units more of good $$1$$. Since negative consumption is allowed, this would always be feasible and an improvement.

Similarly, we must have $$p_1=2 p_3$$. Otherwise, agent $$2$$ could improve.

All in all, we get $$0, which is clearly absurd.

• Sorry.... I don't follow...you write price system $(p_1, p_2, p_2)$ and you say optimality condition of agent $1$ is $p_1=p_3$.... and what do you mean by optimality condition? Please, could you explain a bit more? Commented Jan 8 at 23:20
• If a bundle is preferred by an agent to what they consume, it must cost more. That is all I am using here. If it helps you, you can also argue that equality of marginal rates of substitution is necessary here since there are boundary constraints. Commented Jan 8 at 23:24
• Yes, that would change things a lot. That's why I asked. Commented Jan 8 at 23:38
• The firm maximizes profit. Who owns it is irrelevant at that stage. This is just tedious undergraduate microeconomics. Commented Jan 8 at 23:55
• Profit maximization has nothing to do with ownership structure. The firm maximizes profit. Agent 1 consumes. Their income is the firm's profit, but the decisions are made separately. And yes, it should be $p_3$. Commented Jan 9 at 0:04

Assuming that the economy has two agents, $$n=2$$ and since individual $$1$$ owns the production, this means that the firm belongs to agent $$1$$ who has no endowment.

Suppose that the firm produces the commodity good $$q$$ by using as inputs the commodities $$z_1$$ and $$z_2$$. Then the problem of agent $$1$$ is

$$\text{max}_{(z_1,z_2,q)}\{z_1+z_2+q\},\quad\text{s.t. p_1z_1+p_2z_2+p_3q\leq \pi_f}$$

Where $$\pi_f = p_3f(z_1,z_2)-(p_1z_2+p_2z_2)$$ is the profit function and $$f(z_1,z_2)=q$$

Also the problem of agent $$2$$ is

$$\text{max}_{(z_1,z_2,q)}\{2(z_1+z_2)+q\},\quad\text{s.t. p_1z_1+p_2z_2+p_3q\leq p_1\alpha_1+p_2\alpha_2}$$

By solving the lagrangian for agent $$1$$ and agent $$2$$, we have that $$p_1=p_2=p_3$$ and $$p_1=p_2=2p_3$$ respectively. The only vector of prices that satisfies both of the latter conditions about the prices is $$(p_1, p_2, p_3) = (0,0,0)$$

Then (either) there is not an equilibrium for this economy, because prices must be non-negative and at least some of them strictly positive.

Or else the only equilibrium that satisfy the given economy is the one that implies a zero profit condition (which I am not sure if this is just equivalent to the fact that $$(p_1, p_2, p_3) = (0,0,0)$$)

• Using the same variables for both agents' consumptions and the firm's inputs is a bit confusing. Commented Jan 21 at 12:34
• Yes, but it saves time to the paper in order to solve the problem :P Do you want me to change the notation? Commented Jan 21 at 12:35
• It does ignore the firm's problem. If you want to try the case in which consumptions have to be nonnegative too, this needs to be done. Commented Jan 21 at 12:37
• yes, the firms problem does not play much of a role. Do I miss a case? Commented Jan 21 at 12:40
• The demand of the consumers will look different if nonnegativity constraints bind and then the actual amount of profit can matter. Commented Jan 21 at 12:53