# Question on the conditions for the existence of a Walrasian equilibrium

I have a production Economy with two consumers and one producer.

Consumers have a consumption set in $$R^2_+$$

Y is production possibility set and $$Y= \{y | max (2y_1+ y_2, y_1+2y_2)\le 0\}$$.

Consumers have an equal share of the firm and endowments $$e_1=e_2=(2,1)$$

Consumer 1 have the utility function $$u_1(x_{11},x_{12})= x_{11}-x_{12}$$

Consumer 2 have the utility function $$u_1(x_{21},x_{22})= x_{21}-x_{22}$$

Does this economy satisfies the conditions for the existence of a Walrasian equilibrium ? If it exist, identify it. Or if not, why?

What I think is that

There is no Walrasian equilibrium because the production set is empty. In order for there to be a Walrasian equilibrium, there must be a non-empty production set. This economy does not satisfy the conditions for existence of a Walrasian equilibrium because the production set is empty. A Walrasian equilibrium requires that there be a non-empty production set from which to choose an equilibrium allocation. In this economy, there are no production possibilities, so there can be no Walrasian equilibrium. In order for there to be a Walrasian equilibrium, the following conditions must be satisfied:

1. There must be a non-empty production set.

2. There must be a set of prices such that the market clears.

3. There must be an allocation of resources that is efficient. In this scenario, the production set is non- empty, the market clears at the equilibrium prices, and the allocation of resources is efficient. Therefore, there is a Walrasian equilibrium. In a Walrasian equilibrium, the market clears at the equilibrium prices and the allocation of resources is efficient.

What do you think about my answer and the question? Thanks a lot

• Which is your definition of 'production possibility set'? If $y_1$ and $y_2$ are allowed to be negative, the production possibility set is not empty. See for example Varian, Microeconomic Analysis: $y_j$ is negative if the $j^{th}$ good serves as a net input, positive if the good serves as a net output. Commented Sep 7, 2022 at 12:06
• The question on my lecture note didn’t state the the production possibility set. But what you said is confused my mind. I look at the Varian’s book. I see, you are right. Well, how can solve this question when y1 and 62 are allowed to be negative? Because the production possibility is given the function of maximum and y1 and Y2 are negative. I didn’t deal with such a question before. Can you suggest a way to deal with this question? Commented Sep 7, 2022 at 13:53
• To show that the production possibility set is not empty is sufficient an example. Try to find a pair $(y_1,y_2)$ for which the max above is negative. Commented Sep 7, 2022 at 14:23
• Don't be so negative, $y_1=y_2 = 0$ also works fine. Commented Sep 7, 2022 at 14:27
• @Yes, of course, it is the same. I said negative because studentp seems not at ease with negative numbers here, (0,0) is easier. Commented Sep 7, 2022 at 14:30

When confronted with situations like this its always nice to remember the key statement of the Sonnenschein Matel Debreu Theorem which is:

Any function $$z(p)$$ which satisfies homogeneity, continuity and walras law can be an excess demand function for an economy.

We note that the preferences for each of the consumers $$i\in\{1,2\}$$ generate the following demand structure:

$$x_{i1}^*=\frac{p_1e_{i1}+p_2e_{i2}}{p_1},\ \ \ \ \ \ \ \ \ \ \ x_{i2}^*=0$$ since undesirable by both consumers (given the fact it provides negative utility) there is no value assigned to the endowment and in turn a price of $$p_2=0$$. This modifies our demand structure to be:

$$x_{i1}^*=e_{i1},\ \ \ \ \ \ \ \ \ \ \ x_{i2}^*=0$$ Knowing this (ignoring the production aspect of our economy in whatever role it may play) we note that these demands are homogenous and continuous with constant values which in turn defines our excess demand function $$z(p)\equiv \sum_i (x_{i}-e_{i})$$ to be both continuous and homogenous (note in this case $$x_i$$ and $$e_i$$ are vectors).

Choosing some arbitrary price vector we can see that for any price walras law is satisfied (that is $$pz(p)\equiv 0$$. Using these facts we can construct a fixed point mapping with $$z(p)$$ and prove equilibrium existence.

NOTE: we could have worked this from first defining a fixed point mapping and made the proof that way but its more instructive to see how each of the pieces of this problem fit together. Production in this case would be irrelevant as we haven't defined how the firms allocation decision is tied to the consumers' problem.

I hope this helps.