For questions regarding the mathematical analysis of economic models and problems. This includes questions about the formulation or solution of models from microeconomics or macroeconomics.

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1answer
30 views

When do $Y$ and $r$ increase in the given economy?

This question is from "Mathematics for Economists" by Simon and Blume. IS curve: $[1-c_1(1-t_1)-a_0]Y+(a+c_2)r=c_0-c_1t+I^*+G$LM curve: $mY-hr=M_s-M^*$ The parameters $c_1$,$t_1$ and $a_0$ are ...
2
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2answers
99 views

Why not n=2 in Arrow's theorem

Why in the statement of Arrow's impossibility theorem we omit the case n=2? I will appreciate it if you can explain it in easy words. I'm by no means an expert in the area (I think it's very much ...
2
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2answers
119 views

Ideas about an Ordinary Differential Equations research work (University level)

Good afternoon to everyone, I need some ideas about a Ordinary Differential Equations research work. It is for the ODE subject that I am doing at my Mathematics degree in my University. They asked me ...
2
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1answer
72 views

Use Roy's Theorem to prove that …

I have an advanced microeconomics theory related question: Use Roy's theorem to prove that $s_i(p,y)= -\frac{\partial v(p,y)}{\partial lnp_i}/\frac{\partial v(p,y)}{\partial lny}$. This question has ...
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0answers
188 views

Stone-Geary utility function

$$ u(C_{t})=\frac{(C_{t}-\underline{C})^{1-\sigma }-1}{1- \sigma};\sigma > 0 $$ Does anyone know how to solve the utility maximization problem here and how consumption varies depending on ...
0
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1answer
61 views

Density income and Total Income.

Suppose that the density income function is $f(r) = a*exp(r^2)$ with $r \in(0; 10)$ (hundred thousand euros) and $a=exp(10)-1$. Assume that the total number of people in this economy is 1 million. ...
0
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1answer
135 views

Alternatives to Monte-Carlo simulation

Imagine I have a model of economy of a region, which consists of several companies, importers and population. Let's assume that all local companies in question produce food and agricultural ...
3
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2answers
185 views

Functions minimized at the median of their arguments

I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : ...
0
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1answer
86 views

Production function (economy)

A company produces one good using two factors of production factors. If $x$ and $y$ denotes the units of the factors used by the company, the technology function is given by $F(x; y) = xy^2$. In the ...
3
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2answers
153 views

Substituting total derivative d for partial derivative \partial

In economic models it seems to be commonplace to substitute a total derivative derived from one equation, say $\frac{d k}{d \tau}$, for the partial derivative derived from another equation, say ...
0
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1answer
54 views

Statistical inference and t-stats?

I have this linear regression model with an intercept(b0) and 3 variables(b1,b2,b3). Then they drop b2 and b3 and they give a new regression line with a new b0 and b1 and consequently new standard ...
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2answers
82 views

Using derivative to estimate change overestimates change between two states

I have a profit function in which revenue is given by the state of a variable y at each point in time t: R=y[t]-y[t]^2 and costs depend on the change in y from the previous state so that large changes ...
1
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1answer
89 views

Properties of concave,two-parameter function

I already showed that the function $\psi(\mu,\sigma)=\mathbb{E}U(X)$ is concave in $(\mu,\sigma)$, where $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$. $U$ is a nice concave ...
0
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2answers
44 views

Average and aggregate values with a distribution function

I'm reading an economics paper and I'm trying to understand if a statement made by the author is an assumption or the consequence of a previous definition. The part I don't understand is the ...
0
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1answer
64 views

Statistics - Covariance and variance question

Please fill in the intermediate steps $$\sum_{i=1}^nx_i(x_i-\bar x)=\sum_{i=1}^n(x_i-\bar x)^2$$ and $$\sum_{i=1}^nx_i(y_i-\bar y)=\sum_{i=1}^n(x_i-\bar x)(y_i-\bar y)$$
0
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1answer
177 views

Total differential Economics Application

Suppose we have a revenue function: $R= P*Y$ where $P=$ price and $Y=$ output and is a function of $P$ and $C$, $Y= Y(P,C)$. How could we write the total differential of $R$ with respect to $P$ and ...
2
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2answers
456 views

derivative with respect to $\log(x)$

I have a dynamic equation, $$ \frac{\dot{k}}{k} = s k^{\alpha - 1} + \delta + n$$ Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate ...
3
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1answer
85 views

Constraining estimated linear regression coefficients over several regressions

I'm trying to run a series of simultaneous linear regressions, and I want to constrain the regression coefficients. For the standard ordinary least squares regression, the specification of the ...
1
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0answers
62 views

what is a connection between two simple yet important economics and math formula: elasticity

what makes it interesing to define them in mathematics? what is a connection between two simple yet important economics and math formula: elasticity? Something interesting to read: ...
0
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1answer
52 views

Very simple question about trace

In MSE in econometrics, $$\mathrm{MSE}=E\| \hat{\theta}-\theta \|^2$ $=E(\hat{\theta}-\theta)'(\hat{\theta}-\theta)$$ ...
0
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0answers
36 views

Optimizing revenue clicks CPC

I am quite new to optimization procedure and would like to get some hints here: I am work for a small business that offers a service of niche selling items where subribers pay for every item they ...
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0answers
10 views

How to reflect weighted or inertial increase / growth

If I make 1 dollar for every law I mow, then after mowing 1 lawn, I have 1 dollar. After 2 lawns I have 2 dollars which is an ...
2
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1answer
85 views

Reformulation of the Weak Axiom of Revealed Preference

This question about foundations of mathematical economics. Let $X$ be some set, $\mathcal{B}\subset 2^{X}$ and $C:\mathcal{B}\rightarrow 2^{X}$ such that for all $B\in\mathcal{B}$ we have 1) ...
1
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2answers
116 views

How do you call $\succsim$?

actually, I study economics, not math. As some of you may know, there is a sign for comparing goods: $\succsim$. My professor read $x\succsim y$ like "x is at least as good as y". I asked her if ...
1
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1answer
110 views

Two traders don't trust each other; what transactional equation optimises reward and minimises risk?

Years ago while on a Wikipedia browsing binge, I read a maths article about how two (or more) mistrusting parties can reach an transactional equilibrium, but I've wracked my brain and I can't remember ...
0
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0answers
117 views

Linear regression model in econometrics and error's variance/standard deviation

In econometrics, there is a one-dependent, one-independent-variable linear regression model that goes like: $b_0 + b_1x +\epsilon = y$ where $b_0$ and $b_1$ are to be constant, and $\epsilon$ is error ...
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0answers
52 views

$\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
1
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1answer
46 views

Integration of a sum of production functions

I'm reading an economics paper in which technical knowledge at time t ($A_t$) is the function of past production ($y_{i,t-1}$) for each individual and the learning done by one individual affects the ...
4
votes
2answers
227 views

What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility?

Let $A$ be the set of all possible states of the world, let $G(A)$ be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over $A$. Now consider an individual with ...
2
votes
1answer
212 views

How can a social welfare function be a linear combination of von Neumann-Morgenstern utility functions?

The von Neumann-Morgenstern axioms were an attempt to characterize rational decision-making in the presence of risk. The von Neumann-Morgenstern utility theorem says that if someone is vNM-rational, ...
4
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1answer
82 views

Shapley value: an alternative representation

It is my belief that the more common representation of the Shapley value is given by $$ \phi_i(v)=\sum_{S\subseteq N-i} \frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup\{i\})-v(S)) $$ where $v \in ...
0
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2answers
89 views

Matrix Operation

Let $x$ be a $n \times 1$ vector whose jth element is $x_j$. Show that $A = xx^{T}/x^{T}x$ and $B = I_n - A$ are symmetric idempotent matrices. Note that $x^Tx$ is a scalar (real number)
1
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1answer
177 views

SPNE of infinitely repeated game

Let $G$ be a game with finitely many players and $\underline{v}= (\underline{v}_i)$ be the minmax payoff profile. Denote by $G_{\infty}(\delta)$ the infinitely repeated game whose stage game is $G$ ...
1
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1answer
152 views

The 10.4 Problem on Benassy's *Macroeconomic Theory*

This a repost. In my first try at the problem here I was not very clear with my question and I was also troubled by my lack of knowledge of MathJax. And I probably lost the attention of you guys while ...
0
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1answer
133 views

What is the definition of the slope of a linear function in the context of economic graphs?

I only ask this because of the fact that economists tend to plot the dependent variable on the horizontal axis and the independent variable on the vertical, which is opposite to the "normal" way of ...
1
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0answers
83 views

Optimal auction for risk averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...
1
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0answers
94 views

Math model - constrain GDP given different growth rates of industries

ideas needed to model national GDP given different sector growth rates subject to some contraints Given: GDP equations for $n$ industries depend on growth rates and time i.e. $g(r_1,t), g(r_2, t), ...
0
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1answer
143 views

Mathematical Economics - Utility maximization

I am thankful to any hints: What I have: Simple log-utility form: $u = \log c_1 + \beta \log c_2$ Budget constraints: $c_1 + s \leq w$ $c_2 \leq R\; s$ Problem: For utility maximization: $s = ...
0
votes
1answer
77 views

Portfolio which replicates given payoff

Consider the following payoff function: $$p(S_T) = \begin{cases} 0 & \text{if } S_{T} \leq 70 \\ S_{T}-70 & \text{if } S_{T} \in (70; 100] \\ -S_{T}+120 & \text{if } S_{T} \in (100; 120] ...
0
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1answer
41 views

How to determine the MU in economics?

I currently have a table like so: Hours spent on Activity X | Total Utility 120 220 300 360 396 412 I know that Marginal Utility is calculated use slope formulate (delta Y / ...
0
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3answers
201 views

Arrow impossibility theorem and social choice.

I have read the Arrow impossibility theorem in Foundations of Mathematical Economics(Michael Carter). It is just too difficult to understand. So, does Arrow'theorem mean that there is always a ...
3
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2answers
223 views

Exercise in Mechanism Design

I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution. Exercise. Use the characterization of incentive compatible direct-revelation ...
1
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2answers
231 views

What is the equation representing a constant elasticity of 1?

I'm reading the chapter in my textbook about the price elasticity of demand, and it was pointed out that most demand curves do not represent a constant elasticity of demand - even linear curves like ...
0
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1answer
145 views

$\log(0,05)$ is minus, but $\log(0,04999\ldots)$ is plus?

How is this calculated, and why is this? We're calculating fixed-rate mortgage, with following formular: $$ n = 1-\frac{\log(\frac{L\cdot x}{y})}{\log(1+x)} $$ Where: $L$ is the loan size, $x$ is ...
2
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1answer
146 views

generalized method of moments and the case when solving linear regression with two error conditions

So, I am slowly getting introduced to generalized method of moments (GMM), but I am getting confused over some issues, and this is one of them: I heard that GMM solves the problem that an estimator ...
0
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0answers
38 views

How to Taylor expand $\ln{1-\exp{-i_t}}$ around i?

my question here is how to Taylor expand around $i$ $\ln{(1-\exp{(-i_t)})}$ to the first order? $i_t$ is a time series variable, $i$ is its steady state. Could anyone show me how to expand it ...
1
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0answers
45 views

How is “index” at an Walrasian equilibrium proved? (in relation to Hopf-Poincare theorem)

So, the index of an (Walrasian/general equilibrium) equilibrium point is determined as the sign of $(-1)^{L-1} \times \det M$ where $M$ is a matrix and $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, ...
4
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3answers
126 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
0
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2answers
42 views

Distance between convex set and non-convex set?

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can ...
2
votes
1answer
109 views

function with a continuum of inputs (economic application)

In economics, we often use real-valued functions of the following type: $$U (x_1, x_2)$$ $x_1$ and $x_2$ are the quantities of two goods (real numbers). It is straightforward to work with this kind ...