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Please kindly instruct me on solving the following in a general equilibrium framework with standard budget constraint, $$ u^{1}\left ( x \right )= max\left [ \frac{x_{1}}{10}, \frac{x_{2}}{10}\right ]+ min\left [ x_{1},x_{2} \right ] $$ , with $e^{1} = (10,10)$

  1. What would this consumer demand at prices $p = (\frac{3}{4}; \frac{1}{4})$?

Given the other two consumer utilities

$$u^{2}\left ( x \right )= x_{1}^{\frac{2}{5}}x_{2}^{\frac{3}{5}} $$

$$u^{3}\left ( x \right )= x_{1}^{\frac{3}{5}}x_{2}^{\frac{2}{5}}$$ , with $e^{2} = (4,6)$ , $e^{3} = (4,4)$

  1. Find one Walrasian equilibrium (and determine both equilibrium prices and the equilibrium allocation).

I have completely no idea how to start with it...

Thank you!

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closed as off-topic by Giskard, Bayesian, luchonacho, BKay, Adam Bailey Mar 6 '17 at 22:24

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, Bayesian, luchonacho, BKay, Adam Bailey

  • $\begingroup$ For (2) I suggest you view this playlist on youtube: youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx I am sure it will be helpful. $\endgroup$ – Amit Mar 3 '17 at 20:09
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    $\begingroup$ For (1) I can give you the hint: $u(x_1, x_2) = \frac{x_1}{10} + x_2$ when $x_1 > x_2$ and $u(x_1, x_2) = \frac{x_2}{10} + x_1$ otherwise. Now plot the indifference curves, and the budget line. $\endgroup$ – Amit Mar 3 '17 at 20:13
  • $\begingroup$ so is this the correct IC? $$\left\{\begin{matrix} x_{2}=-\frac{1}{10}x_{1}+\bar{U}, x_{1}> x_{2} \\ x_{2}=-10x_{1}+10\bar{U}, x_{2}> x_{1} \end{matrix}\right.$$ $\endgroup$ – K. Rob Mar 4 '17 at 14:18
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Let us name three consumers A, B and C, and two goods X and Y. Equilibrium price vector $(p_x, p_y=1)$ and allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the three consumers, i.e. it must lie on the demand functions)

  • $(x_A, y_A) = \begin{cases} \left(\frac{10p_x+10}{p_x}, 0\right) & \text{if } p_x \leq \frac{1}{10} \\ \left(10,10\right) & \text{if } \frac{1}{10} \leq p_x \leq 10 \\ \left(0, 10p_x+10\right) & \text{if } p_x \geq 10\end{cases} $
  • $(x_B, y_B) = \left(\frac{2(4p_x + 6)}{5p_x}, \frac{3(4p_x + 6)}{5}\right)$
  • $(x_C, y_C) = \left(\frac{3(4p_x + 4)}{5p_x}, \frac{2(4p_x + 4)}{5}\right)$

Feasibility Conditions

  • $x_A + x_B + x_C = 18$
  • $y_A + y_B + y_C = 20$

Solving the above gives price vector $(p_x, p_y) = (1.2, 1)$ that supports the allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C)) = ((10, 10), (3.6, 6.48), (4.4,3.52))$ in equilibrium.

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  • $\begingroup$ so does it mean that, for example, if the equilibrium price solved for consumer 2 and 3 is $p_{1}>10$ or $p_{1}<\frac{1}{10}$ with $p_{2}=1$, there will be no equilibrium at all? $\endgroup$ – K. Rob Mar 4 '17 at 14:40

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