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# Questions tagged [mathematical-economics]

The application of mathematical methods to represent theories and analyze problems in economics.

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### Algebraic approach towards convexity

I have a function: $u(x) = x_{1} + x_{2} + \min\{x_{1}, x_{2}\}$. How do we algebraically show if it's convex or not? Also, what would be the general way to show if any given function is convex.
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### Overlapping Generations Model Pension System Question

Part 1 Pension System OLG Model with pension system: Each individual lives up to two periods. The surviving probability at period 2 is p. At period 1, the young household consumes c1, saves s1, and ...
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### Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)

I have question on central bank loss function. We know that the central bank loss function is $$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$ And we know that fisher equation is $$i=r+\pi^e$$...
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### derive value function from utility function

We have the utility function. $$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$ And I am trying to find the value function. $U$ is utility function. $c_t$ is consumption at ...
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### Neoclassical Economic Growth Model Shadow Price for Discrete vs Continuous Time

I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional ...
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### Elasticity of substitution

So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:f(x,y) this function have the elasticity of substitution(EOS): 1/(1-beta). a,...
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### What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy

In my research I am trying to find minimal conditions to guarantee a quasi-equilibrium must always be a typical Arrow-Debreu equilibria in a rather specific production setting. This may be rather ...
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The classic literature refers to the problem where information asymmetry exists between an informed and an uninformed counterpart as the adverse selection problem, but how can we verify what kind of ...
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### Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition

I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the ...
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### What is the the return to scale of per capita production function?

If $Y=f(AL,K)$ is CRS, $a_k+a_L=1$ by the Euler Theorem. However, I wanted to know the return to scale of $y=f(1,k)$ (i.e. $Y$ divided by $AL$). I tried $z=p/AL$ , then $py=f(p,pk)$, differentiate w....
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### Symmetric Cournot equilibrium: suffciency without second order conditon

Let $q_i \in Q = \mathbb R_+$ denote the quantity produced by firm $i \in \{1,2\}$. Further let $\pi_i(q_1,q_2) = (1-q_1-q_2)q_i$ denote the profits of $i$. A Nash equilibrium $(q_1^*,q_2^*) \in Q^2$ ...
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### Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed

Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed. Saw this statement in the textbook but I'm not sure how this is the case when we don't have any restrictions on $u(x)$ such as ...
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### Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?

Quasilinear preference is defined to be: $x \sim y \Rightarrow x+ae_1 \sim y+ae_1$ and $x + ae_1 \succ x$ with $e_1 = (1,0,0,...)$, Given a quasilinear preference, if f $x \succeq y - ae_1$, does ...
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### Supporting Hyperplane Theorem and quasiconcave utility function

My notes says that if $u(.)$ is strictly quasiconcave and differentiable, by the supporting hyperplane theorem, there exists $p >>0$ and $w \geq 0$ such that $x = x(p,w)$ $\forall x$. I am ...
93 views

### Uniform bounds on rate of merging for Bayesian learners

Update. Cross posted at Cross Validated. In a well-known paper, Blackwell & Dubins (1962) show that the posterior probabilities of two Bayesian agents, whose priors agree on events of measure $0$,...
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### Questions regarding 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015

This question is with reference to the paper 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015(See here). At page no. 4 of the paper, the authors describe $v_i$ as the ...
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### Calculating weighted average of operating margin (ROS) in benchmarking analysis

I'm looking for the formula for calculating the weighted average of operation margin data. From all the sources I've found so far it calculated as following: the sum of all EBIT data divided by the ...
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### Difference between social choice function and mechanism outcome function

Mas-Colell, Whinston and Green's Microeconomic Theory (3rd edition) defines the social choice function as the following: Later, the mechanism outcome function is also defined: The relationship ...
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### Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.' However, I couldn't ...
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### Index of an Excess Demand Vector

Mas-Colell, Whinston and Green, in Microeconomic Theory (third edition), postulate the concept of an index for an excess demand vector, which is later used in the Index Theorem: A regular equilibrium ...
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### Rate of convergence and asymptotic dominance in $\Vert x \Vert \gg \Vert(\hat\beta-\beta)\cdot u\Vert$

Let $\Vert A \Vert$ denote the spectral norm of a random matrix. Let $x$ and $u_k$ be N$\times$T matrices. Denote $\beta \cdot u = \sum_{k=1}^K\beta_ku_k$, where $\beta$ is a K-vector and $\beta_k$ a ...
90 views

### What are some applications of Real Analysis in Graduate Economics?

I am interested as to what areas of masters/PhD coursework that learning the fundamentals of Real Analysis would be beneficial for? I am aware of its applications in Econometrics proofs and analysis, ...
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### Applications to Green's Theorem in Economics?

I was wondering about possible of application of integration to economics (other than welfare), more specifically, how might Green's theorem be useful for an economist? Let C be a positively oriented,...
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### Real Analysis and Economics

Is there any application of the Heine-Borel theorem or the Bolzano-Weirstrass theorem to Economics? Also where are the notions of compact sets and elementary Real Analysis used in Economics?
In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, I think it's $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it's some asset with payoffs \$\...