Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
437
questions
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10 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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0answers
20 views
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?
2
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1answer
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
$$\sum_{i=0}^{n}{f(...
1
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0answers
19 views
Index of an Excess Demand Vector
Mas-Colell, Whinston and Green, in Microeconomic Theory (third edition), postulate the concept of an index for an excess demand vector, which is later used in the Index Theorem:
A regular equilibrium ...
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0answers
26 views
Equilibrium price determination in a 2 commodity framework
Following are the set of equations describing the demand and supply of two goods X and Y:
Demand functions:
$$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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31 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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0answers
17 views
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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1answer
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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1answer
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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25 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
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1answer
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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1answer
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$.
...
2
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1answer
30 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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0answers
41 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 ...
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0answers
69 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 ...
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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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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.
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14 views
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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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
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0answers
79 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
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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0answers
16 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 ...
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2answers
36 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 ...
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0answers
49 views
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 ...
0
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1answer
54 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 ...
0
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1answer
20 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 (...
2
votes
1answer
58 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 ...
4
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0answers
29 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 ...
1
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1answer
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 $(q_1^*,q_2^*) \in Q^2$ ...
0
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1answer
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 ...
1
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1answer
43 views
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 ...
2
votes
2answers
225 views
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 ...
0
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1answer
33 views
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)^...
4
votes
1answer
66 views
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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13 views
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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1answer
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 ...
2
votes
1answer
137 views
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 ...
4
votes
1answer
48 views
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.
...
3
votes
1answer
32 views
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 ...
1
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1answer
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 ...
0
votes
1answer
29 views
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 ...
3
votes
2answers
81 views
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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0answers
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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17 views
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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1answer
46 views
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?
0
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1answer
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 ...
0
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0answers
83 views
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 ...
0
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1answer
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}\...
4
votes
2answers
100 views
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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1answer
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} \...
