Questions tagged [economics]
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.
521
questions with no upvoted or accepted answers
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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 ...
- 153
4
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0
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116
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Derivation of index decomposition analysis
I’m currently reading a paper on index decomposition. The paper is here for reference : https://www.sciencedirect.com/science/article/pii/S0140988315001772
The paper is setting out how it has gone ...
- 5,518
4
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0
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58
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A gap in a game theory derivation
Consider an $n$-player game exerting effort to increase their probability of winning. Let $x_i$ denote $i$'s probability of winning and $e_i$ $i$'s effort:
$$x_i=\frac{e_i}{\sum_{j\in N}e_j}$$
$i$'s ...
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4
votes
1
answer
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A Question about Nested Maximizations
I am working on labor demand models where firms have to choose the optimal level of employment by maximizing profits. In particular, I have faced the following problem:
Maximize with respect to $l$ ...
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4
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0
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409
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How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?
Quite a long title for this:
I'm looking for the general solution of the following difference equation:
$$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$
where $a,b,c$ are real constants and $u_t$ is a bounded ...
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3
votes
1
answer
141
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Homogeneous of degree one functions that are a monotonic transformation of an additively separable function
Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ be continuously differentiable, concave, and homogeneous of degree one. Here, homogeneity of degree one means that for all $...
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3
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66
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Can quasi-concave and monotone functions level curves that are not path-connected?
For $X = \mathbb{R}^2$, does there exist a quasi-concave and monotone function $f : X \to \mathbb{R}$ that has a level curve which is not path-connected?
Secondly, will every level curve necessarily ...
3
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63
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Maximizing expected reward for inhomogenous exponential process
Consider an inhomogenous exponential process
$t \sim \lambda(p(t)) e^{-\int_{0}^{t} \lambda(p(s)) ds}$
where $\lambda > 0$ and monotonically decreases on the reals. Now define the reward function
$\...
3
votes
0
answers
150
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Financial Mathematics Force of Interest, Discount, Accumulation Functions
Mr. Valdez has $10000 to invest at time t=0, and three ways to invest it. Investment account I is governed by compound interest with an annual effective discount rate of 3%. Investment account II has ...
- 49
3
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35
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Geometry of concepts in social sciences (reference request)
When preparing for my research proposal, I read a lot of articles in economics and political science. In the social sciences one often encounters statements such as "the institutions of country X ...
3
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Determine Capacity for Airline Membership Model
I am trying to determine a simple calculation for capacity for one single 747 plane -- there are 200 seats on the plane and the plane flies twice per day, every day of the year.
If there was a ...
- 131
3
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0
answers
67
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calculus of variations with functional being itself an optimum
The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. ...
- 73
3
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0
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114
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Nash equilibrium in Cournot competition
QUESTION:
Assume there are two types of products, labelled $l$ and $n$. Firms compete in the market by choosing which product to sell and then choosing the quantities. Let $Q_n$ and $Q_l$ denote the ...
- 3,256
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42
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Is this random generation of a distribution over the [0,1] interval measurable?
So this is basically from an economics lecture (the lecturer introduced the voting model from myerson 1993).
I have this feeling that the lecturer simplified this model though, so this might not be ...
- 2,035
3
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0
answers
1k
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A car is to be purchased in monthly payments of $19500$ for five years starting at the end of three months. How much...
I was helping my comrade answer some questions when we found this question. It goes like this:
A car is to be purchased in monthly payments of $19500$ for five years starting at the end of three ...
- 1,475
3
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0
answers
684
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Proving upper contour set is convex for preference relation $\succeq (x_o)$
I am asked to prove that the set $\succeq(x_o) = \{x \in X \subset \mathbb{R} : x \succeq x_o\}$ is convex for any $x_o$ given the following Axiom (1):
If $x_1 \succeq x_o$, then $tx_1 + (1-t)x_o \...
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3
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0
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89
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Need help solving a limit as x approaches infinity
I have to take the limit of the first derivative of:
$$ U(x_i) = \bigg( \sum_i \alpha_i x_i^p \bigg)^\frac{1}{p} $$
as $x_i \rightarrow \infty$.
The first derivative is
$$ \dfrac{\partial U}{\...
- 43
3
votes
0
answers
129
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Rate of convergence for law of large numbers
You pass a street performer who offers you the following gambling deal:
You have 1/3 chance of winning 3 USD and 2/3 chance of losing 2 USD. However you may only play one game.
The street performer ...
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3
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0
answers
79
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Globally stable with derivative less than 1
I am reading Acemoglu's intro to modern economic growth. But I am having trouble understanding his proof to a theorem related with stability. Here is the theorem:
And here are some related ...
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3
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238
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Calculating the equilibrium price. ( Theory- whith no smooth supply and demands curves)
I was looking for a theory wich explain the following problem:
I have the next , demand, supply curve:
The book didn't explain how to compute the equilibrium price with this market (only was named). ...
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3
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144
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Auction Design : Multiple lots, one win max per bidder, not regret
This is a real life game theory problem.
I have to organize an auction.
There is a finite number of lots, which are not equivalent.
There is a finite number of bidders; the number of bidders is ...
- 41
3
votes
2
answers
807
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Percentage change relationship
Given $Z=X^{\alpha}$ how can one prove that the percentage change in $Z$ is simply $\alpha$ times the percentage change in $X$?
This was given as a 'simple mathematical rule' in an economics text ...
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3
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191
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Personal Experiences with Probability Simulation
Simulations methods are increasingly used in theoretical and (especially) applied probability. Personally, I have used simulation for purposes that range from recreational Q&A to applications of ...
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3
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answers
247
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Optimal strategy in a VCG auction with partial collusion?
Suppose you control the bid prices in a multiple-item VCG auction for a partial coalition of bidders. Each bidder is only allowed to win one item out of the set of multiple items, which are all ...
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3
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171
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Optimal tax rate
Suppose you have two countries A and B, with a tax rate $T_A$ and $T_B$, respectively. The tax is redistributed to all people equally. Hence if you live in A and you make $I$ as income then you will ...
- 31
2
votes
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38
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Fair apportionment of a single resource over time
Suppose there is a single, indivisible resource that can be used or enjoyed by just one of two parties at a time. You can imagine a field of crops that can be plowed by just one of two farmers in any ...
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2
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0
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82
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A question about "welfare independence"
I am working on some applications of measures of inequality (economic inequality). Upon reading a paper by Kolm (Kolm, S. C., Unequal Inequalities I, Journal of Economic Theory, 12 pp 416-442, 1976) I ...
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2
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0
answers
44
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Bayesian Nash equlibrium
Assume we have 3 types ($A$, $B$, $C$) each assigned probability $\frac{1}{3}$ and two players in a Bayesian game.
Player 1 (Pl1) only knows if $A$ is played or not, Player 2 (Pl2) only knows if $B$ ...
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2
votes
0
answers
76
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Is there a category of markets?
A market is an ill-defined game-theoretic structure where multiple agents negotiate some agreements by wagering some stakes. I have a very open-ended question:
Is there a good definition of markets ...
- 323
2
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0
answers
43
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Proving local non-satiation in arbitrary metric space
I have a pure exchange economy where every consumption set $X_i$ is non-empty and convex and every preference relation $\succeq_{i}$ is strictly convex. I am asked to show that preferences are locally ...
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2
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0
answers
175
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Handling Reserve in Vickrey Auction
I'm trying to learn more about auction theory and after studying Vickrey and first price auctions I wanted to try including a reserve price (in a Vickrey auction), but something seems to be going ...
- 353
2
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0
answers
68
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Equilibrium for Implicit Best Response Functions
I am dealing with a problem in economics, specifically game theory, where $n$ agents have a best response $x_i$ given by an implicit function, as described below.
We have $ i \in \{ 1, 2, ..., n \}$, $...
2
votes
0
answers
80
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Seemingly simple convex analysis problem
A function $v: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$ is is said to be a valuation function if
The value of function $v$ at $x = \textbf{0}$ is $0$: $v(\textbf{0}) = 0$
$v$ is continuous on the ...
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2
votes
1
answer
121
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Maximise the function subject to constraint
So I basically need to find
$$\max \ \left\{u(x) = \left(\frac{1}{n} \sum_{i=1}^{n} x_i^\rho\right)^{1/\rho}\right\}$$ subject to $p_1x_1 + p_2x_2 + \dots + p_nx_n = W$
Can I still use Lagrangian? And ...
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2
votes
1
answer
202
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Textbooks of economics for mathematicians
There was an analogous discussion 6 years ago, but I open this discussion hoping there any some options on the market.
I'm looking after textbooks expecting mathematical maturity from the reader (or ...
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2
votes
1
answer
68
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Dimensional analysis of a very simple utility function
Suppose we have a utility function $u:\mathbb{R}_+\rightarrow\mathbb{R}$ defined by $u(x)=x$, where $x$ stands for quantity of apples. Suppose we measure the quantity of apples in kg.
How can we ...
2
votes
0
answers
97
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Dynamic Programming and Hamiltonian problem
Consider the following infinite-horizon optimal control problem
for a firm in continuous time. At any moment $t \geq 0$, let $s(t) \in [0, 1]$ be the relative size of the market for the firm’s product....
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2
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0
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87
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How to recognise connections/relationships to solve difficult problems?
As an undergraduate (I don't study maths primarily, but employ these in the study of econometrics/economics), I've come to realise that solving difficult problems in fact hinge on one's ability to ...
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2
votes
0
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86
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proof of conditional expectation given n i.i.d. random variables
This is another question from my self-study of Hayashi's Econometrics.
How do we show in mathematical proof that given:
$X = \begin{bmatrix}x_{1}' \\x_{2}' \\\vdots \\x_{n}'\end{bmatrix}$ where $x_{...
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2
votes
1
answer
47
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How do you solve the following differential equation (Proof of Lemma 3 of Hermalin, 1998)?
$(e(\theta) - s \theta)e'(\theta) = s(1-s)\theta$
The solution to the differential equation is given by:
$e(\theta)=\frac{1}{2}(s+\sqrt{4s-3s^2})\theta$
Here, the dependent variable is $e(\theta)$, ...
2
votes
0
answers
54
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Constrained Optimization Insights
I have been experimenting with the following problem paraphrased from Khan Academy:
A manufacturer's revenue is $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the ...
- 1,554
2
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0
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89
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Subgame perfect equilibrium question
There are two neighboring towns, Alfa and Beta. Firm A is located in Alfa, firm B in Beta. The two firms produce and sell identical goods, with fixed marginal costs of c. The two firms simultaneously ...
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2
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0
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38
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Derivation of a parameter in optimization problem
My question
The solution for first part
$$L=(aq+1-a)x1x2+\lambda (y-(a(p1+c)+(1-a)p1)x1-x2p2)$$
As a result after FOCs
$$x1=\frac {y}{2(a(p1+c)+(1-a)p1)}$$
$$x2={y\over 2*p2}$$
The solution ...
- 6,356
2
votes
0
answers
35
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How to show that a selection $b_1(v)$ is always less than another selection $b_2(v)$
I am an economist and need some math help.
My question is simply prove or disprove that $b_1(v) \leq b_2(v)$, which are selections of two maximization problems.
I have two objective functions. $v \...
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2
votes
0
answers
220
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Is there a generalization of the Kelly criterion where we add a fixed income/cost?
In the ordinary Kelly criterion the game is that you get to place a bet $B$ and your new wealth would be:
$$W^+ = \begin{cases}
W + cB & \text{ with the probability of }p\\
W - B & \text{ ...
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2
votes
0
answers
635
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Game with "pure" Nash equilibrium but not SPNE
after trying to find the answer for this problem for the last few days and crawling through loads of papers and notes, I decided to ask in case someone else has a good example for me.
I am trying to ...
- 33
2
votes
0
answers
70
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Calculus of variations - Solving for function inside of a function of an integral
I am trying to solve a problem that I think I should use calculus of variations to solve (maybe optimal control?). I am trying to figure out the correct technique to use, or to figure out whether this ...
2
votes
0
answers
181
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Converse of the Central Limit Theorem
I want to know if the following statement holds. It is sort of the converse of the CLT. Intuitively, my guess is that it is true. However, I have no idea on how to prove it.
Suppose that:
$$\sqrt{n}(...
- 1,036
2
votes
0
answers
92
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Series representation of $({\bf I-A})^{-1}$?
I have a problem of the form
$$\mathbf{x} = \mathbf{A}\mathbf{x} + \mathbf{b}$$
where $\mathbf{x}$ and $\mathbf{b}$ are vectors and $\mathbf{A}$ is an invertible matrix. I can solve this problem for $\...
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2
votes
0
answers
370
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Prove $f(x^D)\geq1$: $f$ is a log-concave density and $\frac{1-2F(x^D)}{f(x^D)}=x^D-\frac{1}{2}$
Assumptions and notation
Let $f$ be a twice-differentiable log-concave density function on $[0,1]$, and let $F$ be the corresponding distribution function. Define $x^D$ by: $$\frac{1-2F(x^D)}{f(x^D)}=...
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