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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Very simple question about trace

In MSE in econometrics, $$\mathrm{MSE}=E\| \hat{\theta}-\theta \|^2$ $=E(\hat{\theta}-\theta)'(\hat{\theta}-\theta)$$ ...
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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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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 ...
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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) ...
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2answers
113 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 ...
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1answer
107 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 ...
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111 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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49 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 ...
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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 ...
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2answers
220 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 ...
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1answer
209 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, ...
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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 ...
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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)
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1answer
166 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 $G$ ...
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1answer
149 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 ...
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1answer
123 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 ...
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0answers
82 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 ...
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0answers
89 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), ...
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1answer
139 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 = ...
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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] ...
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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 / ...
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3answers
195 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 ...
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2answers
218 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 ...
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2answers
200 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 ...
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1answer
143 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 ...
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1answer
139 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 ...
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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 ...
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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 $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, ...
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3answers
125 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 ...
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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
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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 ...
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1answer
188 views

Factoring a constant into a graph's edge weights for triangular arbitrage

I wrote a program which finds negative weight cycles in a graph to find triangular arbitrage opportunities, using the bellman ford algorithm. The basic principle is this, given three currencies ...
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1answer
79 views

Economic Elasticity: where elasticity-equation come from?

I know the equation for economic elasticity is: $$\varepsilon = \frac{\%\,\Delta Y}{\%\,\Delta X}\frac{X}{Y} = \frac{\partial Y(X)}{\partial X}\frac{X}{Y} = \frac{\partial \log(Y)}{\partial ...
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1answer
21 views

Change in the price of an item based on a group's need for the item.

Okay so this is likely to be confusing to read, please bear with me. The base price of an item is 1440. There are only 6 of these items and there is a population of 20. 10 of the population want or ...
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1answer
173 views

When do $\epsilon$-Nash equilibrium strategies converge to Nash equilibrium strategies?

Suppose I have a game on $n$ players and a sequence of strategy profiles $(s_1^{(1)},\dots,s_n^{(1)}), (s_1^{(2)},\dots,s_n^{(2)}), (s_1^{(3)},\dots,s_n^{(3)}), \dots$. Each ...
3
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1answer
119 views

Existence of asymmetric equilibria in the dollar auction game

Consider a game in which an auctioneer sells one dollar to the highest bidder. The high bidder wins the dollar, but every bidder pays their bid. Concretely, assume that there are two bidders ...
3
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1answer
115 views

Wealth indicator function for bidder agent logic

I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others). Given: Total number of competitors $n$ Amount of all ...
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1answer
2k views

L'hopital's rule in deriving Cobb-Douglas function from CES production function

$$ \ln(Y) = \ln(A) + \frac{\ln[\alpha K^\gamma + (1-\alpha) L^\gamma]}{\gamma}$$ can be taken to the limit by applying l'Hôpital's rule: $$\lim_{\gamma\rightarrow 0} \ln(Y) = \ln(A) + ...
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0answers
159 views

Econometrics Simultaneous equation Indirect Least Squares and Two Stage Least Squares

I still can't figure out this problem. PLEASE HELP! (1) $F_t = a_1 + a_2.C_t + a_3.P_t + e_t$ (2) $P_t = b_1 + b_2.F_t + b_3.S_t + b_4.I_t + u_t$
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224 views

Open Problem in Fixed Point Theory [Prize]

This open problem appeared on the bulletins of Evans Hall at Berkeley this week. I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post ...
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1answer
169 views

Random walk serial correlation

Given a model $$Y_t =b_0 + b_1 \cdot X_t + b_2 \cdot Z_t + e_t,$$ where the error term $e_t$ follows a random walk form of serial correlation $e_t = e_{t-1} + u_t$. Further assume $u_t$ has zero mean ...
3
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1answer
73 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
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2answers
352 views

Endowments & Utility Function to get Demand Function

We have two people, $A$ and $B$, A has $200$ units each of both good $X$ and $Y$ and $B$ has $100$ units each of both good $X$ and $Y$. $A$ has tastes providing a utility function such that $u(X,Y) = ...
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1answer
47 views

How to build (and maximize) this equation

I'm trying to solve an economics problem but I cannot figure out how to build the equation system, or how to find the maximum in a piecewise function. A simplified version of the function would be ...
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1answer
61 views

Find the transaction cost-adjusted expected return of the stock

Let $W^b_i$ denote the weight of stock i in the existing portfolio and $W^a_i$ denote the weight of stock i in the new portfolio to be created. Let $c_i$ denote the transaction cost of stock $i$. If ...
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2answers
207 views

Geometric series to calculate price

I decided to add my extension to this question as a new question here. I am trying to represent the following as a geometric series equation: ...
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1answer
90 views

Using limit argument with non-continuous social-choice functions

This question is related to another question of mine Invariance of strategy-proof social choice function when peaks are made close from solution, and it revolves around the use of limit arguments with ...
6
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1answer
118 views

Invariance of strategy-proof social choice function when peaks are made close from solution

A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ...
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2answers
153 views

Solving for steady state in macro model, probably just simple calc problem…

I am building a macroeconomic model and I am having trouble calculating the steady state. GDP in the model is determined by Y(L,B,K) = x*L+y*B+z*g*K where (x,y,z) are known constants, L is the ...
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1answer
22 views

Finding the most profitable option

Right now, I have 1€. And I know exactly how to invest that euro to make profit, but I have 3 options: Invest the euro, and win exactly 1€ (I'd have 2€ then) Invest the euro, and earn some money ...