Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
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About infinite strategy sets and $\epsilon$-equilibrium from Game Theory: Analysis of Confilct by Roger Myerson
I am studying infinite strategy sets using Myerson's Game Theory: Analysis of Conflict. On Page 143, he defines an $\epsilon$-equilibrium as follows:
Definition For any nonnegative number $\epsilon$, ...
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Auction with independent private values - An example from Game Theory: Analysis of Conflict by Roger Myerson
I have difficulties understanding the equilibrium analysis of the following auction game:
Suppose that there are $n$ bidders in an auction for a single indivisible object. Each player knows privately ...
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Complex Analysis VS Differential Geometry as signals/preparation for grad school?
I am an undergrad (in math and econ) interested in pursuing a PhD in the latter. I can afford a pure math elective this semester and I am torn between diff geom and complex analysis. Which one is more ...
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Does the payoff in a future option contract include the price of the future contract?
This website has this diagram
which shows the payoff as $0$ for $S_T\leq X$ or $S_T\geq X$ (depending on the position-option combination). But isn't the buyer/seller of the position down/up the ...
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Understanding Duality between Individual and Collective Maximization in Macroeconomic Models
I'm currently studying macroeconomic models, specifically from the book "Recursive Macroeconomic Theory." In Chapter Seven, it is mentioned that some economic models involving firms and ...
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MCQ on income and substitution effects
Person A spends their income only on bread and cheese. Given a rise in the price of bread leaving income constant and the price of cheese constant, the consumer consumes less bread and less cheese. ...
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How do I show this function is a bijection?
Let $\geq$ be a partial order on a set $X$. For each $x\in X$, let $F(x)\equiv \{a\in X:a\geq x\}$ and $\mathcal{F}\equiv\{F(x):x\in X\}$. I want to show that $F:X\to\mathcal{F}$ is a bijection.
First ...
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Equilibrium of Perturbed Dollar Auction Game - An Example from Game Theory: Analysis of Conflict by Roger Myerson
I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties understanding and ...
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Design of a money skew economy (target: museums/libraries loan rotation sharing)
I was thinking about libraries and museums, and began wondering how such institutions might institute what I would describe as a "money skew economy." Skew in the math sense, of two lines ...
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Ramsey no ponzi condition
Please help me understand how to create these functions, sometime they use "t" denote, sometime use "r" denote that make me feel confuse. Function (17) and from (19) to (23)
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Ramsey model problem
Explain the no-Ponzi game condition throughout the function.
Does the constrain function: also reflect include the No-Ponzi condition?
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Question About Stochastic Choice - MWG Exercise 1.D.5
I am studying microeconomic theory using MWG. I got stuck on Exercise 1.D.5, specifically part (c), but I would also like to have my part (a) and (b) checked by someone. Here is the exercise and my ...
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Cost Minimization
Guys in the following optimisation problem why are we not considering the partial derivate of K and L in the Lagrangian in the first two equations
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direction of causality between Increasing returns to scale and economies of scale
Does increasing returns to scale cause economies of scale OR is it the other way around?
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Capital-Output Ratio using Nominal GDP and Nominal GFCF
I have this assignment which asks the following:
"Compute the average capital-to-output ratio in Australia from 1960 to 2022. Explain how to compute
the average ratio. (Hint: use the nominal GDP ...
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Cutoff rule-mechanism question
I was checking on this paper where they define some rule-mechanism that is called cutoff rule-mechanism. I do not really understand why is this useful and how is this defined. Any help would be ...
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What is the most general category-theoretic description of a market?
In the history of economic theories, various formalisms of "a market" or "the marketplace" have been used to explain various facets of economic behavior, starting roughly with Nash ...
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About Proof of Theorem 21.3 in Mathematics for Economists by Simon and Blume
Background Information
I am studying concave and convex functions using Mathematics for Economists by Simon and Blume. I have difficulties understanding their proof of the following theorem:
Theorem ...
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The sufficient condition for unique interior solution in utility maximization problem
Suppose the utility function is continuous, differentiable, strictly increasing and strictly quasiconcave. Whether the utility maximization problem has unique interior solution? If not, is there any ...
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Determine Whether A Preference Relation Satisfies The Continuity Axiom - from Exercise 1.1 in Game Theory: Analysis of Conflict by Roger Myerson
I am self-studying game theory using Game Theory: Analysis of Conflict by Roger Myerson. Here is an exercise from the textbook. I tried it myself, but I am not sure if it is correct. I would really ...
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Quasiconvexity of the indirect utility function for Cobb-Douglas utility
I've recently started Mas-Colell's, Green's and Whinston's Microeconomic Theory. In section 3.D, the authors define the indirect utility for a price vector $p$ and wealth $w$ as the utility derived ...
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Ramsey model condition
Why are transversality conditions and no-ponzi game conditions needed in the Ramsey model? And why do we assume that is a CIES utility function?
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What is the relation between Blackwell's order and Stochastic Dominance order?
In Kamenica and Gentzkow (2017) as well as in Bergemann and Morris (2016) the notion of Blackwell comparioson of experiments is used to compare different information structures. I am trying to find ...
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Consumer theory with subproblem
Say the agent's problem is
$$\max_{c,\{h\}, N}\{U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))+\lambda(w N-c)\}$$, subject to $\sum_{i=1}^{I} h_{i}+N \leq 1, \quad N \in \mathcal{N}$.
Assume $U(c, v(\...
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How often (if ever) do currency exchange cross rates differ from actual rates?
Do major currencies' cross rates ever differ from their actual ('bilateral') exchange rates, and if so, how often does it happen in practice?
I guess it would never happen (beyond, say, the fourth ...
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About Characterization of Homothetic Function - Mathematics for Economists by Simon and Blume Chapter 20 Exercise 18(3)
I am studying homothetic function and got the following problem:
Problem
Determine whether the function $x^3y^6 + 3x^2y^4 + 6xy^2 + 9$ is homothetic or not.
Here is my attempt.
My Attempt
I did it ...
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what to learn to write a comprehensive strong paper?
to write an excellent comprehensive paper, what topics and techniques must an economist learn? One of my former professors wrote a paper I read, there are theories, calculations, and empirical tests. ...
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Economic models that are useful to understand macroeconomics
I started to study economics. Since I majored in Mathematics, I'd like to learn the principles of macroeconomics through the analysis of a simple economic models (using equations). That's why I ...
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About Proof of Theorem 20.8 in Mathematics for Economists by Simon and Blume
I am studying homothetic functions using Mathematics for Economists by Simon and Blume. I am reading their proof of the following theorem:
Theorem$\quad$ Let $u: \mathbb{R}_+^\mathbf{n} \to \mathbb{R}...
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About The Bayesian Conditional-Probability Systems in Myerson's Game Theory: Analysis of Conflict
I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his Bayesian conditional-probability system. The Bayesian conditional-probability ...
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Showing that reward function is bounded (dynamic programing)
I have the following dynamic programming problem:
$$\max_{\{x_t,y_{t+1}\}_{t=0}^\infty}\sum_{t=0}^\infty \beta^tu(F(x_t)-y_{t+1})\;\;\;\;\;\text{s.t}\;\;\;\;y_{t+1}\in\Gamma(x_t)$$
where $\Gamma(x)=\{...
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The Solow growth model
I wonder why at the beginning sf(k(t)) is steeper than δk(𝑡), but at some point, it starts flatter than δk(t)?
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Solow Model in disrectly and continuously
what is the differences between solow model in disrectly time and continuously time?
Why in disrectly time function use the equation: k(t+1)=(1-δ)k(t) + sf(k(t)) and in continuously time function use: ...
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CRS assumption in solow model
Can you explain why does Solow growth model assume "constant returns to scale"?
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About Theorem 1.1 in Game Theory: Analysis of Conflict by Roger Myerson
I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his proof of Theorem 1.1, the Expected-Utility Maximization Theorem. The Theorem goes ...
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Prove Any Strictly Increasing Function is Equivalent to A Homogeneous Function of Degree 1 - Mathematics for Economists (Simon & Blume) Exercise 20.14
Question
Proof that any function $f: \mathbb{R}^1 \to \mathbb{R}^1$ with $f' > 0$ everywhere is equivalent to a homogeneous function of degree one.
My Attempt
Note that all strictly monotonic ...
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Quasiconvex and quasiconcave utility function
I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can ...
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Convexity preferences
What is the difference between convexity and strict convexity preferences? What is the difference between quasi-concavity and quasi-convexity? And is MRS still true in concave preferences?
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Samuelson Consumption loan paper and Overlapping Generations
The famous Samuelson paper on consumption loan and interest rates is a bit odd in my opinion. I wanted to understand it a bit better. The paper outline 3 generations.
Youth that produce 1 unit.
The ...
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When could value functions in Bellman equations be calculated explicitly?
Given the simplest form of a Lucas model, i.e., a Bellman equation given by
\begin{align}
J(x_t) & = \max_{c_t, x_{t+1}} \{ u(c_t) + \beta E_{\pi} [ J(x_{t+1})] \} \\
& \textrm{ s.t. } ...
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Continuity of preference
"A preference is continuous if for any $a,b\in X$ with $a\succsim b$ there are some neighborhoods $N_{\varepsilon}(a)$, $N_{\delta}(b)$ around $a$ and $b$ such that for every $x \in N_{\...
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Moral Hazard in Teams. Deriving Nash Equilibrium with bonus paying sheme
I have a few mathematical problems with the paper Moral hazard in teams, by Bengt Hölmstrom 1982. Theorem 4
Denote the conditional distribution of $x$, given the action vector $a$, by $F(x, a)$ and ...
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Value of information in the Moral hazard in teams context
I have a few mathematical problems with the paper Moral hazard in teams 1982 by Bengt Holmstrom, especially in chapter 3. (sufficient statistic) page 330 following
The welfare problem can be stated as:...
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Generalization of Debreu's additive utility function $\sum_nu_n(x_n)$ with infinite number of commodities
I want to generalize: $\sum_nu_n(x_n)$.
Here $x_1,x_2,..,x_n,...$ are commodities. There are infinite number of commodities: $n\in\mathbb N$ or $n\in \mathbb R_+$
The following not a candidate: $\...
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Estimation of learning model with finite horizon forward-looking individuals
I am stuck with the estimation of learning model where individuals are forward-looking in a finite horizon.
Specially, a user is watching a TV program containing many episodes, and he doesn't know the ...
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Integrate a sufficient statistic
I have a few mathematical problems with the paper Moral hazard in teams 1982. How do I get from (2) to (1) by integrating and why is the qualification necessary "for almost all"
$$g(y,a)=h_i(...
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Habit formation ala Constantinides (1990)
Consider the following problem, from Constantinides (1990).
\begin{align}
V(W_0, x_0) \equiv \max_{c, \alpha} \mathrm{E}_0 \int_0^\infty e^{-\rho s}\gamma^{-1}[c(s) - x(s)]^\gamma \mathrm{d}s,
\end{...
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Question About Non-Degenerated Constraint Qualification (NDCQ)
I am studying constrained optimization using Mathematics for Economists by Simon and Blume, and I have some difficulties understanding the Non-Degenerated Constraint Qualification (NDCQ). I would like ...
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Frameworks and models in economics
A framework is not a model of a specific system,
but a way of formulating and studying a
variety of systems. Classical mechanics,
quantum mechanics and statistical mechanics are all really frameworks. ...
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Does this utility function work?
I'm reading over some models and I found a paper in progress that uses this model
$$U_i(c_i, n, \theta_i) = \log(c_i) + \log(n) + \theta_i$$
where $c_i \text{ and } n \geq 0$ , but when you consider ...
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