6

You don't need to use any fancy theorem, the trick is to disentangle the definitions. Everything follows directly from the definitions. $x^0=x^*(p^0,w)$ means that $p^0x^0\leq w$ and that $u(x)\leq u(x^0)$ for all $x$ such that $p^0x\leq w$. In words $x_0$ is affordable and at least as good as every other affordable bundle (in many cases, better and not ...


6

That was tricky. The idea is as follows: First, under standard assumptions, demand is continuous. If you change prices a little bit, demand will not change a lot. In particular, if your excess demand for the first good was initially strictly positive, it will still be strictly positive for small changes in price. Now, the new bundle after the price was ...


5

You have found the value function $V(p,I)$ which in your case is given as $$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$ where $\alpha_2 = 1- \alpha_1$ and $I$ is income. The Hicks demand is a function of the utility level denoted $u$. An agent facing prices $p$ and income $I$ gets utility $V(p,I) = u$. Therefore ...


5

To solve $$\max_{x \geq 0} \ (x_1+1)^\alpha(x_2 + 1)^\beta$$ $$s.t. \ \ I \geq p_1x_1 + p_2x_2,$$ I would define $y_1 = x_1+1$ and $y_2 = x_2 + 1$ to get the problem $$\max_{y \geq 0} \ y_1^\alpha y_2^\beta$$ $$s.t. \ \ \bar I \geq p_1y_1 + p_2y_2,$$ where $\bar I := I + p_1 + p_2$. For $\alpha + \beta = 1$ the solution is well known to be $$y_1^* = \frac{\...


5

Partial answer: for simplicity let $P_c =1$. The budget constraint: $c= wn + \Pi$ Simplify (plug in $\Pi$): $c= F(n)- fc$ Goods clearing: $c = F(n)$ The household's budget constraint is inconsistent w/ goods market clearing. The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy. ...


3

Why don't you just plug in some values for $x_1$ and $x_2$? Start with something like $x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$. Then increase $x_1$ or $x_2$ and let $x_1=2, x_2=1$ and find utility $u(2,1)=1+5*2=11$, and $x_1=1, x_2=2$ and find utility $u(1,2)=1+5*2=11$. The utilities of both bundles are identical, and the goods seem to be substitutable....


2

To solve for competitive equilibrium, we can first find the demand : Demand for commodity $X$ by A is $x_A = \frac{5}{p_x}$ if $p_x < 1$, $x_A \in [0,5]$ if $p_x = 1$, $x_A = 0$ otherwise. Demand for commodity $X$ by B is $x_B = \frac{(30p_x+5)}{2p_x}$. Now we can equate demand and supply and solve for $p_x$. $x_A + x_B = 30$ yields $p_x = \frac{1}{2}$.


2

Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$. In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$. And because of the preferences in optimum also $x_1 = 4 y_1$ and $x_2=y_2$ holds. You use that to get 2's demand as $x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$ and a similar bur slightly different formula for $4 y^*_1 = ...


1

I prefer another approach. I use duality to find the cost function and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than generally, for simplicity. Begin with the indirect utility function: $$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}} = \left(\...


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