Questions tagged [mathematical-economics]

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

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Meaning or Interpretation of the denominator in the implicit function theorem

Consider $p=g(p,y)$. Rewrite this equation as $p-g(p,y)=F(p,y)=F(P(y),y)=0$, where $P(y)$ is the implicit function of $p$. By the implicit function theorem, we obtain $\partial P(y)/\partial y=-F_y/...
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Cobb Douglas production function without capital [on hold]

is there a possibility that a production function of cobb Douglas type without capital? How it look like?
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53 views
+50

Maximising a partly concave and partly convex function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable and strictly increasing function. Suppose that we are searching for the numbers $x_1$, ..., $x_n$ that maximise $$\sum_{i=0}^{n}{f(...
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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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26 views

Equilibrium price determination in a 2 commodity framework

Following are the set of equations describing the demand and supply of two goods X and Y: Demand functions: $$X_d = a_1 - b_1P_x + c_1P_y$$ $$Y_d = a_2 - b_2P_y +c_2P_x$$ $a_1,~ a_2,~ b_1,~ b_2,~ ...
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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 ...
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Intensive and extensive margin of labor demand [on hold]

Before the question, first a little of background. The model is as follows: How can i identify the extensive and intensive margin in the labor demand equation?
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27 views

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?
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70 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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Is a combination of Political economics and Game theory possible and beneficial?

By Political economics I do not mean the "economical" advice given by some people(see the Wealth of Nations by Adam Smith) but rather the heavily mathematized subfield of Economics studying and trying ...
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50 views

Is there a proof for composite commodity theorem?

I have been reading Economics and Consumer Behavior by Angus Deaton and John Muellbauer, specifically reading up on Composite Commodity Theorem, which states: if prices move in parallel to each ...
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37 views

Industrial Economics

I am currently struggling trying to find the short-run equilibrium price, output per firm, and profit per firm if $190$ firms supply the market. I am given $p=102-1/2Q$ and $C(q)=5q-6q^2+3q^3$. ...
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A weaker definition of local non-satiation can also imply indifference “curve”

Let $u$ be a continuous utility function on $\mathbb R^2_+\setminus\{0\}$. Consider the following three conditions: Local non satiation says that for any $x \in X$ and $\epsilon > 0$, there exists ...
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Independence Axiom for Linear Utility - Who proved this first?

Who first proposed the following axiomatization of linear utility using Independence? I remembered that it was Debreu but I am not so sure. What was the first paper proving this? Consider a ...
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Linear Homothetic Utility

A Homothetic Utility is where $$ \forall x,y, \forall a \in \mathbb{R}_+: \ u(ax,ay)=au(x,y) $$ (or its monotonic transformation). A linear Homothetic utility is defined as $$ \forall x,y, \forall ...
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Maximization when parameters are unknown

I would like to know if my understanding about how to find a maximum of the function when some parameters are unknown is correct. Consider the following maximization problem. $\max_{x}V=\int_0^{a(x)}...
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How do I calculate quarterly Sharpe Ratio USING geometric averages?

I am looking to calculate the Sharpe ratio using Geometric average. The issue is I don't understand how to find the standard deviation term. Any help would be greatly appreciated.
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India as center region of foreign powers for storing nuclear and space weapons

India have recently imported many billion dollars worth weapons from a number of foreign countries which are against sanctioned loans.will it not adversely affect India's economy?It seems India is ...
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What is the difference between a growth shock and a productivity shock

In a RBC context, what I mean by growth shock is a total factor productivity shock that follows a stochastic trend $ln(a_t) = ln(a_{t−1}) + \gamma + \epsilon^a_t$. I.e. the technology $a_t$, has a ...
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Derivation of demand function

Hello. I'm graduate student in Japan. This time, what I want to ask is how to solve the profit maximization problem using the image production function and derive the demand function. This ...
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49 views

Envelope Theorem in Hopkins and Kornienko (2010)

This is from Hopkins and Kornienko (2010). In this model, $x$ is investments, $s$ is status, and $y=z-x$ is leisure, where $z$ is endowments. $x(r)$ is the optimal investment, and the relative ...
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What experiment could I run in order to test the 'Independence of Irrelevant Alternatives' axiom?

I need some help in designing an economics experiment to test the IIA axiom. My understanding of it is if you rank A above B then if you introduce C or D, you must still always rank A above B in any ...
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How to calculate payback period of the project?

In order to finish a technological product, it was spent 50000 dollars (for materials, salary). 500 dollars per month is needed for support of the project (will be spend for servers). This product ...
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On complements and substitutes with a CES function

Define the CES function $q : \mathbb R_+^n \to [0,1]$ by \begin{align} q(x) = \left[\frac{1}{n}\sum_{j=1}^n{x_j^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1} \end{align} where $x \in \mathbb ...
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Maximization problem FOC and Euler equation

Can someone please help me with the Lagragian and the derivation of the following objective function ? Beneath I provide the objective function, the constraint and the Euler equation that results from ...
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Economic Metrics (ROI, PP, NPW)

So I am doing this project, which includes in calculating the Economic Metrics for a profitability analysis. I have calculated Return on Investment (ROI), Net Present Worth (NPW) and Payback Period (...
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WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement?

Let $A$ be a menu and $R$ be a complete and transitive binary relation. Define choice correspondence generated by $R$: $$c_R(A)=\{x\in A|| xRy \ \forall y\in A\}.$$ Theorem (from Kreps 1988): for ...
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Afriat theorem for negative goods

GARP and Afrait theorem assume that the alternative $x\in\mathbb R_+$ is always positive. In some economic contexts, such as financial choices, the attribute can be negative. I wonder if we can ...
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64 views

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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65 views

What mathematical structure or formalism can be used for modelling strategic investment decisions of agents in the context of competition?

I'm trying to analyze the decisions of agents in investing scarce resource like time and money into developing their product/service/offering in the presence of competition. I need to know what kind ...
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Probability of the event knowing that I received no informations

First I want to thank you if you pay attention to my post. I apologize if it seems elementary to you, note that I searched a lot an answer before posting. I have a particular informational framework ...
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Is it possible to give, in economics, an example of a relation ( set of ordered pairs) that is not a function?

In mathematics, some relations ( sets of ordered pairs) are not functions. I know economists make use of functions. But do they also consider relations that are not functions. In which branch of ...
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Financial economics: Bond Price

I know that if the coupon rate on a bond is larger than the yield-to-maturity, then the price must be higher than the par value. Yet I have the bond price equation $P=\frac{c}{y}\left(1-\frac{1}{(1+y)^...
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Is First Order Stochastic Dominance (FOSD) relation convex?

A convex relation is that $x\succeq y$ implies $\alpha x+(1-\alpha)y\succeq y$. Let $>_{FOSD}$ be $\succ$, is the FOSD convex? Intuitively it seems convex.
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Is there such a thing as resonance in economic underliers?

In physics the occurence of resonance is explained and widely understood in its linear form and subject to research in nonlinear resonance. Example for instance are resonant frequencies of objects. ...
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130 views

Average employment length

From Macroeconomics 7th edition by Gregory Mankiw, p166. Here’s a numerical example. Suppose that 1 percent of the employed lose their jobs each month ($s$ = 0.01). This means that on average jobs ...
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Need a math help for the Cagan's model in macroeconomics

From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw. To keep the math as simple as possible, we posit a money demand function that is linear in the natural ...
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Convex games: equivalence of definitions

Let $N \subset \mathbb N$ denote the set of players and $v : 2^N \to \mathbb R$ , $v(\emptyset) = 0$, the characteristic function. We call $(N,v)$ a cooperative transferable utility (TU) game. ...
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Existing metric for personal productivity hours needed per life hour?

With about 50 hours of productivity a week, including work, cooking, etc. I can complete the tasks and pay the expenses necessary to live about a week. Subtract maybe 5 hours of labor that goes into ...
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125 views

Utility maximimization for unusual Leontief utility function

The problem is basic utility maximization subject to a budget constraint with $$u(x,y) = min\{x+y,4\sqrt{x},4\sqrt{y}\}$$ $$p_x = 1, p_y = 1, M = 4$$ I will have to first plot the Indifference ...
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Here's a question about the solow model [closed]

At first, I was confused by how y keeps increasing after depreciation exceeds savings, then I finally found what was truly bothering me. The fact that when depreciation exceeds savings, capital per ...
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Mechanism Design: Proving that the expected utility is differentiable

Given a direct mechanism, we define a buyer's expected utility $u(\theta)$ conditional on her type being $\theta$ by $u(\theta)=\theta q(\theta)-t(\theta)$, where $q:[\underline{\theta},\bar{\theta}]\...
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52 views

Equivalence of definitions of upper contour sets

Mas-Collel, Winston and Green's Microeconomic Theory (3rd editions) offers two definitions of the upper contour set: How can the equivalence between the two definitions be proved?
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What model to use when defining a 20-year growth path in a developing country?

What economic growth model is commonly used for formulating a 20-year national development plan in a small open developing country?
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Constrained Optimization using Lagrangian method

The stationary points that we derive by solving the first order conditions of the Lagrangian are those points global optimum points or local optimum points?
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27 views

What is the reason behind the demand function of a perfect complement good?

so I know that usually the income curve is equal to: $$x_1p_1 + x_2p_2 = m$$ if we rearrange this equation we get that the demand for good one ($x_1$) is equal to: $$x_1 = \frac{m-x_2p_2}{p_1}$$ None ...
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What are eigenvalues and eigenvectors used for in economics?

In economic and econometric literature, i see references to eigenvectors and eigenvalues. Having taken my fair share of linear algebra and econometrics courses I have not really seen how exactly they ...
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19 views

What does the sun product of price times quantity divided by average price

I have a very simple question, what is the meaning, if any, of this? I.e. Economically what does the following calculation mean? $$ \left(\frac{(Q1\cdot P1)+(Q2\cdot P2)+(Q3\cdot P3)}{Q1 + Q2 + Q3}\...
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2answers
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Purpose of a monotonic transformations in utility functions

Based on my economics book, monotonic transformations for a utility function can look something like this: $f(u) = u + 17 $ or even like this: $f(u) = u^3$ That being said what it purpose in the ...
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38 views

Can I normalise the Dixit-Stiglitz price index to 1?

I've built a model that I am trying to solve in Matlab. The model has Dixit-Stiglitz preferences and $m$ goods, so admits a price index of the following form: $ P = \bigg[ \sum_m p_m^{1-\varepsilon} \...