Questions tagged [mathematical-economics]

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

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Questions regarding 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015

This question is with reference to the paper 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015(See here). At page no. 4 of the paper, the authors describe $v_i$ as the ...
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Modeling risk aversion. Expected utility function

I want to model an Expected Utility Function for risk aversion but my problem is uncertainty in itself. I want a function(a special case) $$f(x, y) =\left\{\begin{matrix} h(bx^{1-C}+ay^{1-C}),C\neq 1\...
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What are the different approaches to represent and analyse an economic model/system with scientific rigour [on hold]

My objective is to test the said model's stability or a lack thereof, along with factors that cause such. I have read about certain techniques like using partial differential equations (chaos theory) ...
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Set Theory Properties of the Budget Constraint

In Microeconomic theory, the budget constraint is defined by 4 distinct properties: It is Bounded Closed Convex Non-empty The 1. 2. and 4. are very straight forward and the benefits in terms of ...
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1answer
25 views

Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition

I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the ...
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1answer
22 views

Derive the growth rate of an equation

I have the following equation: $$\mu =\left [s_{\pi }-v(s_{\pi }-s_{W})+\zeta \right ]$$ And I have to derive its growth rate, which is: $$\dot \mu =-\frac{v}{\mu } (s_{\pi }-s_{W})\dot v$$ Do ...
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1answer
22 views

Rules of total differentials [closed]

What are the rules of total differentials? I. e., given an arbitrary expression of which to take the total differential, what rules can be applied to arrive at the desired result, and how does ...
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1answer
22 views

Hep with total differentiation of an AD function

Is there anyone who can help me with a total differentiation exercize. I am starting with the following formula for AD: $$x=\mu ^{-1}(g+i+e)$$ Where $\mu$ is the Keynesian multiplier. And have to ...
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1answer
58 views

Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)

A specialized question for those, who excel in monopolistic competition and modern trade theories. I am interested in a derivation of an aggregate demand function for a model of monopolistic ...
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28 views

In this price model, employment grows with higher wages. Makes no sense

Using a Sraffian price model we have that, $p=(1+r)(pA+wl)$ where w (nominal wage rate), r (average profit rate) are scalars; p (market prices), l (labor input coefficients) are vectors; A (producers' ...
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Understanding lognormal distribution of stock prices

I am trying to understand how to correctly use the lognormal distribution. Let's say I have a list of Adjusted Price of a stock, I know that its daily returns is ...
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1answer
50 views

Abraham (1987) Simple Job Market Matching Model

I have a question about a derivation in Abraham (1987)'s simple job market matching model (equations 3 through 7): She begins by writing down tautologies: J - V = L - U = E where J is the number of ...
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1answer
70 views

How do I derive Hessian of this function and check for concavity?

The function is $f(K,L)= AK^{a}L^{b}$ on the set of points $(K,L)$ with $K\geq 0$ and $L\geq 0$, assuming $A>0$ How do I find the Hessian and check for concavity?
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31 views

Moral hazard, bubbles, and economies of scale in limited companies and organizations

http://eprints.lse.ac.uk/100058/1/Goodhart_CEPR_DP13494.pdf What would be the disadvantage of limiting the liability to the product of the proportion of the company and the debt(a person with 2% of a ...
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1answer
23 views

The notation cl co (A)

I stumbled upon this notation, (cl co(A+C), while reading "Set Optimization—A Rather Short Introduction" by Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff and Carola Schrage. Is it ...
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80 views

Log linearising (Gali textbook)

In Gali (2015)'s textbook Monetary Policy, Inflation, and the Business Cycle, variables in levels are denoted with capital letters, logged variables with lowercase letters. However, when a log-...
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49 views

Mathematical framework for modelling the relationship between price and sales of a product

In my job as a data scientist, I am required to model the relationship between the price of a product and the sales or number of unit sold. I am trying to build a simplistic model, the assumptions of ...
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56 views

Direct revelation mechanism's sets of strategies and types

Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows: The employed notation is the following: $θ_i$: Player i's type $Θ_i$: Set of types for ...
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Calculating weighted average of operating margin (ROS) in benchmarking analysis

I'm looking for the formula for calculating the weighted average of operation margin data. From all the sources I've found so far it calculated as following: the sum of all EBIT data divided by the ...
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1answer
11 views

Difference between social choice function and mechanism outcome function

Mas-Colell, Whinston and Green's Microeconomic Theory (3rd edition) defines the social choice function as the following: Later, the mechanism outcome function is also defined: The relationship ...
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39 views

Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.' However, I couldn't ...
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1answer
33 views

Meaning of assumptions regarding continuum of agents in economics models

I am reading an economics paper which contains a model with households and banks. In the model, banks are owned by households. The authors make this assumption when they discuss households (section 2....
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15 views

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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1answer
96 views

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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31 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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35 views

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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33 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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84 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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37 views

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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62 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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40 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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1answer
35 views

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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1answer
42 views

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 [closed]

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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22 views

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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89 views

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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1answer
54 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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2answers
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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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72 views

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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1answer
21 views

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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1answer
96 views

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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32 views

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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1answer
65 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 ...