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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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  • $\begingroup$ 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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