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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 ...
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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 $\...
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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 ...
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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)...
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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\;&...
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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}...
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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 ...
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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 \...
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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}...
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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 ...
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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 ...
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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 $\sim$ ...
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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 ...
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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 ...
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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 $...
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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} \...
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3 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}{\...
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2 votes
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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',...
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2 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, ...
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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 ...
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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.
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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 ...
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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 ...
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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; ...
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1 vote
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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 ...
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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 ...
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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 - ...
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  • 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 ...
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