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 ...
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 ...
5
Given that the utility maximization problem is:
\begin{eqnarray*} \max_{\{y_j \geq 0 : 1\leq j\leq J\}} && \sum_{j=1}^{J} u(y_j) \\ \text{s.t.} && \sum_{j=1}^{J} p_jy_j \leq m\end{eqnarray*}
where $p_j > 0$ for every $j$, and $m > 0$.
If $u$ is convex and strictly increasing, then in optimum consumer will always spend all his money on ...
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
$$y_1^* = \frac{\...
5
Partial answer:
for simplicity let $P_c =1$.
The budget constraint: $c= wn + \Pi$
Simplify (plug in $\Pi$): $c= F(n)- fc$
Goods clearing: $c = F(n)$
The household's budget constraint is inconsistent w/ goods market clearing.
The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy.
...
3
Why don't you just plug in some values for $x_1$ and $x_2$?
Start with something like
$x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$.
Then increase $x_1$ or $x_2$ and let
$x_1=2, x_2=1$ and find utility $u(2,1)=1+5*2=11$, and
$x_1=1, x_2=2$ and find utility $u(1,2)=1+5*2=11$.
The utilities of both bundles are identical, and the goods seem to be substitutable....
3
We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences:
$u_1(x_{11}, x_{12}) = x_{11}$
$u_2(x_{21}, x_{22}) = x_{22}$
Equilibrium price vector $(p_1, p_2=1)$ and allocation $((x_{11}, x_{12}), (x_{21}, x_{22}))$ satisfy the following:
Optimality Conditions (...
3
Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, price $p$ supports the allocation in which 1 consumes $\left(\frac{1+3p}{1+p}, \frac{1+3p}{1+p}\right)$ and 2 consumes $\left(\frac{3+p}{1+p}, \frac{3+p}{1+p}\...
2
An indirect proof. Suppose
$$
x(p,w) = w\cdot x(p,1)
$$
does not hold. This is equivalent with stating
$$
U(x(p,w)) \neq U(w\cdot x(p,1)).
$$
(To be precise: $x(p,w)$ and $x(p,1)$ may be set valued. In this case we are talking about two elements at least one of which is not included in both sets.)
Case 1.
$$
U(x(p,w)) > U(w\cdot x(p,1))
$$
As $U$ is ...
2
I am not sure whether the question asks about the function being convex, or simply the demand correspondence/function being a convex set. The latter is often asked and if that is the case here is an answer.
This is a general answer to your question which may be more useful to you and future readers than a specific one.
The following theorem holds in ...
2
In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed point is, in fact, a Walrasian equilibrium.
Depending on the exact construction of the proof, sometimes the fixed point will only give you the equilibrium ...
2
So WARP says that if bundle x' you choose under budget B' is affordable at budget B'' (and you don't choose x' - i.e x'$\not=$x''), then x'' cannot be affordable under budget B'.
The picture you showed gives the 2 bundles that the person chooses. Those are given in the picture. As you can see, x' is not affordable at budget B'' and bundle x'' is not ...
2
Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$.
In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$.
And because of the preferences in optimum also $x_1 = 4 y_1$ and $x_2=y_2$ holds.
You use that to get 2's demand as $x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$ and a similar bur slightly different formula for $4 y^*_1 = ...
2
It is almost true.
There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom.
However, if we ask that
$$v \cdot S(p,w) v <0 $$
whenever $v \not = \alpha p$ for any scalar $\alpha$ (i.e. $S$ is negative definite for all vectors except those proportional to price), then the Weak Axiom holds.
2
To solve for competitive equilibrium, we can first find the demand :
Demand for commodity $X$ by A is $x_A = \frac{5}{p_x}$ if $p_x < 1$, $x_A \in [0,5]$ if $p_x = 1$, $x_A = 0$ otherwise.
Demand for commodity $X$ by B is $x_B = \frac{(30p_x+5)}{2p_x}$.
Now we can equate demand and supply and solve for $p_x$.
$x_A + x_B = 30$ yields $p_x = \frac{1}{2}$.
2
Coalition formation is not the same as majority voting. Say, in a single good economy, individual 1 has the entire endowment, so $\mathbf e=(1,0,0,0,0,0)$. Then individual 1 forms a blocking coalition to any allocation $\mathbf x=(x_1,\dots,x_6)$ where $x_1<1$.
1
I prefer another approach. I use duality to find the cost function and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than generally, for simplicity.
Begin with the indirect utility function:
$$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}} = \left(\...
1
Under what constraints is $u_1+u_2$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to highlight that the equilibrium you found also maximizes the sum of utilities.
In general, the maximum of a sum of functions need not equal the maximum of any ...
1
For case 1, you can argue $z$ will be unique by contradiction:
Suppose ad absurdum there is another $z'$ that is feasible (i.e. $p^Tz' =w$) optimal and $z'\neq z$. Then you can consider a convex combination of $z$ and $z'$: $\bar z = \beta z + (1-\beta) z'$, for $\beta\in(0,1)$. Notice that $\bar z$ is still feasible (because it is a combination of two ...
1
for this problem you must conciser two possible branches of the utility function:
$$u(\text{x})=x_1+x_3\ \ \text{if} \ \ x_1<x_2$$
$$u(\text{x})=x_2+x_3\ \ \text{if} \ \ x_1>x_2$$
The demands of these then proceed how you would for any case of perfect substitutes. However you must list them for each case.
Hope this helps
1
Hint: Imagine that there are two coffee bars, $A$ and $B$. There is only one type of coffee in the world. My preferences are such that I always want 1 unit of sugar with 1 unit of coffee; if I consume units of coffee and sugar in the ratio $1:1$, additional units of only one of the two don't give me extra utility.
At coffee bar $A$ they sell coffee and ...
1
Walras' law describes market equilibria conditions, which states roughly speaking that if there exists within an exchange economy market equilibria for $n-1$ good markets, then the last good market $n$ is also in equilibrium. Thus, we have a market system with $n$ equations and $n-1$ independent variables, where the last market is linear dependent from the ...
1
You seem to be looking for the Second Welfare Theorem.
The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.
1
Excess demand is the function describing the amount of quantity demanded above quantity supplied at each price level. Mathematically, it is allowed to be negative (so then we have excess quantity supplied actually).
Remember that the Walrasian construct imagines an "auction" situation when transactions do not take place as long as the market doesn't clear....
1
If the prices $p_1$ and $p_2$ are positive than as you pointed out the equations
\begin{eqnarray*}
\frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\
\\
\frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20.
\end{eqnarray*}
hold. This is troublesome, because subtracting the first equation from the second yields
\...
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