11.12.2024,11:07 Question 1 Not yet answered Marked out of 30.00 Consider the following pure exchange economy with two consumers and two goods, I and y Consumer's utility functions are given by u_(1)(x,y)=min 2x,y and u_(2)(x,y)=(x+y-1)^2+sqrt (10) Suppose that the initial endowments of Consumer 1 and Consumer 2 satisfy omega _(1)+omega _(2)=(2,2) Instructions: If you answer is not an integer,round your answer to 2 decimal places (for example, 1/3 is 0.33,1/2 is 0.5 and 2/3 is 0.67). __ First suppose that the initial endowments of Consumer 1 and Consumer 2 are omega _(1)=(1,1) and omega _(2)=(1,1) Normalize prices in this economy such that p_(x)=pgeqslant 0 and p_(y)=1 For each p below.calculate demands of both consumers for both goods and identify whether the resulting profile constitutes a Walrasian equilibrium. for p=1/2 x_(1): y_(1):square x_(2):square y_(2): square Is it a WE equilibrium? Yes No for p=2 x_(1): y_(1):square x_(2):square y_(2): square Is it a WE equilibrium? Yes No Pure Exchange Economy

To determine the demands of both consumers for both goods and identify whether the resulting profile constitutes a Walrasian equilibrium, we need to solve the consumer optimization problems for each consumer given the price vector p.<br /><br />For Consumer 1:<br />$u_{1}(x,y)=min\{ 2x,y\} $<br />The consumer aims to maximize their utility subject to their budget constraint. The budget constraint is given by:<br />$p_{x}x+p_{y}y=\omega_{1}p_{x}+\omega_{1}p_{y}$<br />Substituting the given values, we have:<br />$\frac{1}{2}x+y=2\cdot \frac{1}{2}+1\cdot 1$<br />Simplifying, we get:<br />$x+2y=3$<br />The consumer's optimization problem is:<br />$max_{x,y} min\{ 2x,y\} $<br />subject to:<br />$x+2y=3$<br />The consumer will choose the bundle $(x,y)$ such that $2x=y$. Substituting this into the budget constraint, we get:<br />$2x+2x=3$<br />Solving for x, we get:<br />$x=\frac{3}{4}$<br />Substituting this back into $2x=y$, we get:<br />$y=\frac{3}{2}$<br />Therefore, Consumer 1's demand for good x is $\frac{3}{4}$ and for good y is $\frac{3}{2}$.<br /><br />For Consumer 2:<br />$u_{2}(x,y)=(x+y-1)^{2}+\sqrt {10}$<br />The consumer aims to maximize their utility subject to their budget constraint. The budget constraint is given by:<br />$p_{x}x+p_{y}y=\omega_{2}p_{x}+\omega_{2}p_{y}$<br />Substituting the given values, we have:<br />$\frac{1}{2}x+y=2\cdot \frac{1}{2}+1\cdot 1$<br />Simplifying, we get:<br />$x+2y=3$<br />The consumer's optimization problem is:<br />$max_{x,y} (x+y-1)^{2}+\sqrt {10}$<br />subject to:<br />$x+2y=3$<br />The consumer will choose the bundle $(x,y)$ such that $x=2y-3$. Substituting this into the budget constraint, we get:<br />$2y-3+2y=3$<br />Solving for y, we get:<br />$y=\frac{3}{2}$<br />Substituting this back into $x=2y-3$, we get:<br />$x=0$<br />Therefore, Consumer 2's demand for good x is 0 and for good y is $\frac{3}{2}$.<br /><br />Now, let's check if the resulting profile constitutes a Walrasian equilibrium. A Walrasian equilibrium occurs when the demands of both consumers for both goods are such that the market clears, meaning that the total demand for each good equals the total supply.<br /><br />In this case, the total demand for good x is $\frac{3}{4}+0=\frac{3}{4}$ and the total supply is 2. The total demand for good y is $\frac{3}{2}+\frac{3}{2}=3$ and the total supply is 2. Since the total demand for both goods does not equal the total supply, the resulting profile does not constitute a Walrasian equilibrium.<br /><br />Therefore, the answer is:<br />for $p=1/2$ $x_{1}:$ $\frac{3}{4}$ $y_{1}:\frac{3}{2}$ $x_{2}:0$ $y_{2}:\frac{3}{2}$<br />Is it a WE equilibrium?<br />No