# How to show there is no a Walrasian equilibrium?

There are 3 agents, one seller and two buyers. There are two indivisible goods, apple(a) and banana(b), and one divisible good, money(m). The seller's endowment is $$W_s=(1,1,0)$$, and buyers's are $$W_{b1}=W_{b2}=(0,0,10)$$. The utility function of the seller is $$U_s(a,b,m)=m$$. For buyer 1, $$u_{b1}(1,1,m)=3+m$$, and $$u_{b1}(a,b,m)=m$$ if $$(a,b)\neq(1,1)$$. For buyer 2, $$u_{b2}(a,b,m)=2+m$$ if $$(a,b)\neq(0,0)$$ and $$u_{b2}(0,0,m)=m$$.

I do not know how to show this economy does not have a WE.

• Have you calculated how much each buyer buys at each price? Dec 5, 2023 at 9:28
• Yes, it is the same as what Amit answered. Dec 7, 2023 at 6:38

## 1 Answer

Given the exchange economy,

• Three Goods: X (Apple), Y (Banana), M (Money)
• Three Agents: Seller (S), Buyer 1 (A), Buyer 2 (B)
• Endowment of the seller: $$W_S=(1,1,0)$$; Endowment of the buyers: $$W_A=W_B=(0,0,10)$$. Seller has apple and banana, buyers have the money.
• Utility functions: For $$i\in\{S,A,B\}$$, define $$u_i:\{0,1\}\times\{0,1\}\times\mathbb{R}_+\rightarrow\mathbb{R}$$ as follows:

$$u_S(x_S,y_S, m_S)=m_S$$,

$$u_A(x_A,y_A, m_A)=3\min(x_A,y_A)+m_A$$,

$$u_B(x_B,y_B, m_B)=2\max(x_B,y_B)+m_B$$

We can quickly rule out the possibility of having $$p_M=0$$ because that would lead to unbounded demand for $$M$$.

So, we let $$p_M=1$$ be the numeraire. We'll now consider the following possibilities:

• $$p_X=0$$ and $$p_Y=0$$. Seller is indifferent between selling and buying apples and bananas, but given that both are free, A will demand 1 unit of apple and 1 unit of banana, and B will also demand either 1 unit of apple or 1 unit of banana. So, at least one of the two will be in excess demand.

• $$p_X=0$$ and $$p_Y>0$$. B will demand 1 unit of X. A will definitely demand X and Y in such a way that quantity demanded of X is at least as much as quantity demanded of Y. So the aggregate demand for X will be at least one unit plus the aggregate demand for Y. However, at these prices, the supply of X will be at most as much as the supply of Y. Therefore, markets will not clear.

• $$p_X>0$$ and $$p_Y=0$$. This also ruled out by symmetric argument as in the case before.

• $$p_X>0$$ and $$p_Y>0$$. Since $$p_X>0$$ and $$p_Y>0$$, observing the seller's utility function we know that seller would want to sell both the goods irrespective of the prices. So the supply of X is 1 and supply of Y is 1.

Case 1: $$p_X+p_Y>3$$, In this case, A's demand is $$x^d_A=y^d_A=0$$ and B will demand either just $$1$$ unit of X, or just $$1$$ unit of Y, or $$0$$ unit of both X and Y depending on the prices. No matter what happens both the markets never clear at the same time.

Case 2: $$p_X+p_Y<3$$, In this case, A's demand is $$x^d_A=y^d_A=1$$ and B will demand either just $$1$$ unit of X, or just $$1$$ unit of Y depending on whichever is cheaper. This is because $$p_X+p_Y\leq 3$$ implies $$\min(p_X,p_Y)<2$$. Again no matter what happens there will be excess demand for the cheaper of the two goods, or for one of them if they are equally priced.

Case 3: $$p_X+p_Y=3$$, In this case A can either demand 0 unit of both, or 1 unit of both. If A demand 0 unit of both then Case 1 reasoning applies. If A demand 1 unit of both the goods then Case 2 reasoning applies. Either ways, competitive equilibrium does not exist.

• This is correct Dec 6, 2023 at 14:41
• Many thanks, Amit. Dec 7, 2023 at 6:38