Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
464
questions
1
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0answers
35 views
Elasticity of substitution
So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:f(x,y)
this function have the elasticity of substitution(EOS): 1/(1-beta). a,...
1
vote
0answers
22 views
What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy
In my research I am trying to find minimal conditions to guarantee a quasi-equilibrium must always be a typical Arrow-Debreu equilibria in a rather specific production setting.
This may be rather ...
-2
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0answers
37 views
Narrative Economics opinion [closed]
What do you think about it. Are recession substantially caused by narratives. Doesn't this fix the problems with all other recession theories.
0
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1answer
42 views
Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed
Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed.
Saw this statement in the textbook but I'm not sure how this is the case when we don't have any restrictions on $u(x)$ such as ...
3
votes
1answer
41 views
Adverse selection problem
The classic literature refers to the problem where information asymmetry exists between an informed and an uninformed counterpart as the adverse selection problem, but how can we verify what kind of ...
1
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0answers
38 views
Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?
Quasilinear preference is defined to be:
$x \sim y \Rightarrow x+ae_1 \sim y+ae_1$ and $x + ae_1 \succ x$ with $e_1 = (1,0,0,...)$,
Given a quasilinear preference, if f $x \succeq y - ae_1$, does ...
1
vote
0answers
28 views
Supporting Hyperplane Theorem and quasiconcave utility function
My notes says that if $u(.)$ is strictly quasiconcave and differentiable, by the supporting hyperplane theorem, there exists $p >>0$ and $w \geq 0$ such that $ x = x(p,w)$ $\forall x$. I am ...
2
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0answers
41 views
Neoclassical Economic Growth Model Shadow Price for Discrete vs Continuous Time
I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional ...
0
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0answers
21 views
What is the the return to scale of per capita production function?
If $Y=f(AL,K)$ is CRS, $a_k+a_L=1$ by the Euler Theorem.
However, I wanted to know the return to scale of $y=f(1,k)$ (i.e. $Y$ divided by $AL$).
I tried $z=p/AL$ ,
then $py=f(p,pk)$, differentiate w....
0
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0answers
16 views
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 ...
0
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0answers
20 views
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) $$f(x, y) =\left\{\begin{matrix} h(bx^{1-C}+ay^{1-C}),C\neq 1\...
4
votes
2answers
76 views
Set Theory Properties of the Budget Constraint
In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is
Bounded
Closed
Convex
Non-empty
The 1. 2. and 4. are very straight forward and the benefits in terms of ...
0
votes
1answer
32 views
Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition
I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the ...
0
votes
1answer
24 views
Derive the growth rate of an equation
I have the following equation:
$$\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$$
Do ...
2
votes
1answer
24 views
Rules of total differentials [closed]
What are the rules of total differentials? I. e., given an arbitrary expression of which to take the total differential, what rules can be applied to arrive at the desired result, and how does ...
0
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1answer
22 views
Hep with total differentiation of an AD function [closed]
Is there anyone who can help me with a total differentiation exercize. I am starting with the following formula for AD:
$$x=\mu ^{-1}(g+i+e)$$
Where $\mu$ is the Keynesian multiplier.
And have to ...
0
votes
1answer
60 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 ...
0
votes
0answers
29 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' ...
0
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0answers
19 views
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 ...
2
votes
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 ...
1
vote
1answer
73 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?
0
votes
0answers
32 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 ...
1
vote
1answer
24 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 ...
0
votes
2answers
84 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-...
0
votes
1answer
51 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 ...
3
votes
1answer
58 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 ...
0
votes
0answers
11 views
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 ...
1
vote
1answer
15 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 ...
0
votes
0answers
46 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 ...
2
votes
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....
0
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0answers
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/...
3
votes
1answer
99 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(...
1
vote
0answers
30 views
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 ...
0
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0answers
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,~ ...
0
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0answers
38 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 ...
0
votes
1answer
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?
1
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1answer
90 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, ...
0
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0answers
38 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 ...
1
vote
1answer
74 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 ...
0
votes
1answer
41 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 $C(q)=5q-6q^2+3q^3$.
...
2
votes
1answer
40 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 ...
1
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0answers
50 views
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 ...
3
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0answers
73 views
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 ...
-1
votes
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)}...
0
votes
0answers
11 views
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.
0
votes
0answers
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 ...
2
votes
0answers
97 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 ...
3
votes
1answer
55 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 ...
1
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0answers
17 views
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 ...
1
vote
2answers
37 views
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 ...
