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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3
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2answers
190 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
59 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 ...
1
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2answers
115 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
vote
1answer
99 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
52 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
74 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
votes
1answer
491 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
votes
2answers
1k 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
votes
1answer
98 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
70 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
56 views

Very simple question about trace

In MSE in econometrics, $$\mathrm{MSE}=E\| \hat{\theta}-\theta \|^2$ $=E(\hat{\theta}-\theta)'(\hat{\theta}-\theta)$$ ...
2
votes
1answer
105 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
182 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
vote
1answer
126 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 ...
1
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0answers
61 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
vote
1answer
52 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
328 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
273 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
votes
1answer
107 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
votes
2answers
92 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
vote
1answer
271 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 $G$ ...
2
votes
2answers
229 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
votes
1answer
228 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
108 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
vote
0answers
107 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
votes
1answer
257 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
84 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
votes
1answer
45 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
votes
3answers
290 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
votes
2answers
299 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
1k 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
votes
1answer
159 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
votes
1answer
215 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
51 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
votes
3answers
131 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
49 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
123 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 ...
0
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1answer
233 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 ...
1
vote
1answer
93 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 ...
0
votes
1answer
22 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 ...
5
votes
1answer
254 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
votes
1answer
217 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
123 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 ...
0
votes
1answer
3k 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
235 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) $P_t = b_1 + b_2.F_t + b_3.S_t + b_4.I_t + u_t$
5
votes
0answers
274 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 ...
1
vote
1answer
337 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
votes
1answer
81 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 ...
0
votes
2answers
706 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) = ...