7
votes
Under what condition is a cost function strictly concave in prices?
No function that is homogeneous of degree one, is at the same time strictly concave in its arguments. If the function is differentiable (or non-differentiable at a finite number of points), then the ...
- 2,692
6
votes
Is Varian's definition of continuity of preference equivalent to standard definitions?
Here is how one can show that Definition 1 implies Definition 2. We do the contrapositive, we show that if Definition 2 fails then Definition 1 will fail too.
Suppose that $x\succ y$, but for every $\...
- 9,814
6
votes
Accepted
Is Varian's definition of continuity of preference equivalent to standard definitions?
What Varian (Microeconomic Analysis, p 95) says is that:
If $x$ is strictly preferred to $y$ and if $z$ is a bundle that is close enough to $x$ then $z$ must be strictly preferred to $y$.
This is a ...
- 8,642
5
votes
Prove that if a production function is such that f'>0 and f''<0, then f'<Average Product
Assuming $f''<0$ implies strict concavity and hence
$$tf(y) + (1-t)f(x)<f(ty+(1-t)x) \Leftrightarrow f(y) - f(x)<\frac{f(ty+(1-t)x)-f(x)}{t}$$
$$f(y) - f(x)<\frac{f(x + t(y-x))-f(x)}{t(y-x)...
- 313
5
votes
Accepted
Weak preferences and negative transitivity
Probably it can be done easier if you do both steps separately ($\implies$ and $\impliedby$), but here is a proof that does both at the same time:
\begin{align*}
&x\succ y \vee x\sim y\\
\iff\;&...
- 454
5
votes
Accepted
Under what condition is a cost function strictly concave in prices?
As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems.
So let $f:\mathbb{R}^n_+\to\mathbb{R}...
- 9,814
5
votes
Accepted
Prove all cost functions are concave in input prices and demand for inputs is downward
Let $x(w, q)$ denote the solution to the cost minimization problem :
\begin{eqnarray*} \min_{x} & \ w\cdot x \\ \text{s.t.} & \ \ f(x) \geq q \end{eqnarray*}
where $f$ is the production ...
- 5,267
4
votes
Basic Solow Growth Model: Stability Proof
For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is
$\dot{k} = s f(k) - \delta k = \phi(k)$
Then we have
$\frac{\partial \...
- 141
4
votes
Accepted
Prove that the profits of the firm weakly decreases with input prices
From FOC, we know that:
\begin{align}
\nabla_x\pi(\mathbf{x},\mathbf{w})=p\nabla f(\mathbf{x})-\mathbf{w}=\mathbf{0} \tag{1}
\end{align}
This will be true at equilibrium, i.e. for any given $\mathbf{w}...
- 1,630
4
votes
Accepted
Can I assume utility functions strictly increasing?
You have to show that something does not hold universally true. To show this, you just have to show that there is at least one exception- a counterexample. For this counterexample, you can make any ...
- 9,814
4
votes
Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function
Take a strictly increasing mapping $f:\mathbb{N} \to [0,1)$, such as
$$
f(y) = 1 - \frac{1}{y+1}.
$$
Then
$$
U(x,y) = x + f(y)
$$
represents the Lexicographic preference in the $\mathbb{N}^2$ choice ...
- 26.7k
3
votes
Accepted
Does x ≽ y imply x > y or x ~ y in preferences?
The usual definition of $x \succeq y$ is that either $x \sim y$ or $x \succ y$ hold. Your proof and the assumption of completeness are not necessary.
One could also start from $\succeq$ and $\sim$ ...
- 26.7k
3
votes
Accepted
How do you establish uniqueness of a rational preference relation?
Assume that, towards a contradiction, that both $\succeq$ and $\succeq^\ast$ rationalise the choice function and that they are different.
The fact that $\succeq$ and $\succeq^\ast$ are different means ...
- 8,642
3
votes
Accepted
Proof of Expected utility theorem with three outcomes
In order to do that, you need to define $u(·)$ as a utility function on "sure things" rather than on lotteries. In your example, you need to think in terms of the set of possible prizes to the ...
- 711
3
votes
Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?
I don't think it is true in a standard pure exchange economy the question is referring to. Consider the following counterexample:
Suppose
$I = \{1,2\}$ and $u_1(x_1, m_1) = \sqrt{x_1} + m_1$ and $...
- 5,267
3
votes
Accepted
Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?
Edit: Edge cases suck; see comments. See also MWG Chapter 10 section C, D.
Suppose $(\vec x^*, \vec m^*)$ solves
$$\max \sum^I_{i=1} m_i + \phi_i(x_i)$$
but is not Pareto optimal.
$$\begin{align}
\...
- 6,407
3
votes
Accepted
Basic Solow Growth Model: Stability Proof
For stability, we want
$$\frac{\partial k_{t+1}}{\partial k_t}\Big|_{\bar k} <1 \implies \frac{(1-\delta) + \sigma A_0 f'(\bar k)}{1+n} <1$$
$$ \implies f'(\bar k) < \frac {\delta+n}{\...
- 31.9k
2
votes
Accepted
How to prove convexity + quasilinear preferences imply concave utility?
Consider any $x_2'$ and $x_2''$ in $\mathbb{R}_+$. Without loss of generality, let $x_2'' > x_2'$. We can choose $x_1'=f(x_2'')-f(x_2') > 0$ so that $U(0,x_2'')=U(x_1',x_2')$. Let $\lambda(x_1',...
- 5,267
2
votes
Accepted
Equivalence of Definitions of Continuity of Preferences
We want to show that for $\succcurlyeq$ on $X$, Def 1 $\iff$ Def 2
$\boxed \Longrightarrow$
Assume that $\succcurlyeq$ is continuous by Def 1.
Let us say $x \succ y$. Denote our open-balls as $B(x, ...
- 6,407
2
votes
Accepted
Expected Utility with expected value and variance
While it is true that a function has the expected utility form if and only if it is linear (in probabilities), it is not the case that any linear function can represent a preference that satisfies the ...
- 14.6k
2
votes
Expected Utility with expected value and variance
It would suffice to show that $U$ is linear. But is $U$ necessarily linear if it satisfies the vNM axioms?
Hint: No.
- 26.7k
2
votes
Prove that the profits of the firm weakly decreases with input prices
(Without using differentiation) When $w \leq w'$ it follows that $pf(x) − w · x \geq pf(x) − w' · x$ and so $\pi(p,w) \geq \pi(p,w')$.
EDIT 1. The last inequality (first left as an exercise) can be ...
- 2,692
2
votes
Accepted
Can one prove Pareto efficiency in an n-party system by showing all Pareto improvements between any two given parties are made?
@HerrK. got it right in his comment (he should have deleted the somewhat confusing "yes" from the beginning and then posted it as an answer) It is possible that no pairwise improvements are possible ...
- 26.7k
1
vote
Formal proof that IR_L IC_H are binding constraints
If $IR_L$ is not binding, you can increase both $T_L$ and $T_H$ by the same amount: Such increases do not violate either of the following:
$IR_L$ is not violated as it's initially not binding;
...
- 14.6k
1
vote
Accepted
Prove that $u$ is a utility function for $\succsim$
You're asked to prove that $u(x)\ge u(y)\;\Leftrightarrow\;x\succsim y$ for any $x,y\in X$, where $u(x)=|\{z\in X:z\prec x\}|$, i.e. the utility of $x$ is measured by the number of other alternatives ...
- 14.6k
1
vote
Solution to maximization not Pareto efficient
An example with two agents and two goods: let
$$
U_1(x) = 0, \hskip 20pt U_2(x) = x_1+x_2, \hskip 20pt w = (1,1).
$$
In this case allocating all the goods, so (1,1) to the first consumer solves the ...
- 26.7k
1
vote
Help with Monopolistic Competition Proof, Prove Love for Variety
You have
$$ \begin{align} u(\lambda a + (1 - \lambda) b, \lambda b + (1 - \lambda)a) &= \sqrt{\lambda a + (1 - \lambda)b} + \sqrt{\lambda b + (1 - \lambda)a} \\
&\geq \lambda\sqrt{a} + (1 - ...
- 197
1
vote
Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?
I believe you are referring to the following result: Any PE allocation maximizes $\sum_{i=1}^{I}\phi_{i}(x_{i})$, but it is hard to know precisely since you are not specific about feasibility.
Let me ...
- 121
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