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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.

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    $\begingroup$ Have you calculated how much each buyer buys at each price? $\endgroup$ Dec 5, 2023 at 9:28
  • $\begingroup$ Yes, it is the same as what Amit answered. $\endgroup$
    – user45481
    Dec 7, 2023 at 6:38

1 Answer 1

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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.

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  • $\begingroup$ This is correct $\endgroup$
    – High GPA
    Dec 6, 2023 at 14:41
  • $\begingroup$ Many thanks, Amit. $\endgroup$
    – user45481
    Dec 7, 2023 at 6:38

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