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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 ...
Bertrand's user avatar
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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 $\...
Michael Greinecker's user avatar
6 votes
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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 ...
tdm's user avatar
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5 votes
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Order relations and preferences using logic

Just a sketch of a more general context, regarding relations in set theory.$^1$ You wrote: I want to understand order relations [...] and what this means for certain results, specifically looking at ...
BakerStreet's user avatar
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5 votes

Order relations and preferences using logic

The definition of $x\succ y$ is: "$x\succsim y$ and not the case that $y\succsim x$". Formally, $$ x\succ y \quad\Longleftrightarrow\quad x\succsim y \;\wedge\; \lnot(y\succsim x) $$ By this ...
Herr K.'s user avatar
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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)...
bomadsen's user avatar
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5 votes
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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\;&...
LudwigNagasena's user avatar
5 votes
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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}...
Michael Greinecker's user avatar
5 votes
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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 ...
Amit's user avatar
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5 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 \...
Durden's user avatar
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5 votes

Why is exponential discounting considered rational?

To see that exponential discounting is (more or less) the only time consistent manner to discount the future, consider a decision maker obtaining a utility (payoff) level at period $t$ and at period $...
tdm's user avatar
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4 votes
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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}{\...
Alecos Papadopoulos's user avatar
4 votes
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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}...
Dayne's user avatar
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4 votes
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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 ...
Michael Greinecker's user avatar
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 ...
Giskard's user avatar
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3 votes
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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 $\sim$ ...
Giskard's user avatar
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3 votes
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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 ...
tdm's user avatar
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3 votes
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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 ...
Patricio's user avatar
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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 $...
Amit's user avatar
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3 votes
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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} \...
Kitsune Cavalry's user avatar
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3 votes
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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, ...
Kitsune Cavalry's user avatar
  • 6,648
3 votes

Is this proof correct (measure theory)?

The first direction is fine, the second has some small problems and some scope for improvement. There is the notational issue that $\cup_{n=1}^\infty \in \mathcal{F}$ should be $\cup_{n=1}^\infty A_n \...
Michael Greinecker's user avatar
3 votes

Intuition of two Measure theory statements

For the first statement, note that $\int_E f~\mathrm d\mu$ is by definition the same thing as $\int 1_E f~\mathrm d\mu$, and the integral of a function is defined as the limit of the integrals of ...
Michael Greinecker's user avatar
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',...
Amit's user avatar
  • 8,976
2 votes
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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 ...
Herr K.'s user avatar
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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.
Giskard's user avatar
  • 29.3k
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 ...
Bertrand's user avatar
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2 votes
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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 ...
Giskard's user avatar
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2 votes

Proving an additive function is sigma additive (Measure Theory)

Sigma additivity requires that if you have a countable collection $(B_i)_{i \ge 0}$ of mutually disjoint sets ($B_i \cap B_j = \emptyset$ for $i \ne j$) then: $$ \sum_{i \ge 0} \mu(B_i) = \mu(\cup_{i \...
tdm's user avatar
  • 12.4k
2 votes

Why is exponential discounting considered rational?

Any discounting that yields time consistent preferences can be considered rational discounting. To check if the intertemporal utility with any discount factor is time consistent, you need to check if ...
1muflon1's user avatar
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