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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.

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Solution to a system of nonlinear equations with certain conditions

I am working in a model and I found a problem relating a nonlinear system of equations. Let $\mathbf{D}(\mathbf{Q})\in \mathbb{R}^N$ for $\mathbf{Q}\in \mathbb{R}^N$ be a continously differentiable ...
Tan1278's user avatar
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4 votes
1 answer
73 views

Characterize the mixed strategy Bayesian Nash equilibria for this game. [closed]

A two-player game where Player 1 can choose either U or D and Player 2 can choose either L or R. Player 1 is either cooperative with probability $P$ or uncooperative with probability $1−P$. Player 1 ...
Toshani Singh's user avatar
2 votes
0 answers
21 views

Max-min and min-Max relationship in ZeroSum Games

In the attached Zero Sum Game, I have solved for two Mixed Nash Equilibrium, $(l,m)$ and $(m,r.)$ In $(l,m)$, the payoff is $8/5$ to $P_1$, and $-8/5$ to $P_2$. Here, $P_1$ mixes between Top and ...
CormJack's user avatar
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1 vote
1 answer
53 views

Link between summing orthogonal projections and Projection Matrix $A(A^TA)^{-1}A^T$

Assuming the usual inner product $\langle x, y\rangle = x^\mathsf{T} y$ on a real vector space $V$, I believe we can define the orthogonal projection $\mathbf P_W\colon V\to W$ as $$\mathbf P_W(v) = \...
CormJack's user avatar
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1 vote
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Isoelastic utility functions, how to calculate risk boundaries.

I am trying to replicate a calculation I found in a paper that calculates the level of risk aversion for an individual to choose one option over the other, but I am struggling to solve it. Ideally, ...
anona's user avatar
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1 answer
22 views

A concrete example with Arrow-Pratt coefficient of absolute risk aversion

Let $u_1$ and $g$ be increasing strictly concave functions from $\mathbb{R}$ to $\mathbb{R}$. Let $u_2:=g\circ u_1$. If we regard $u_1$ and $u_2$ as utility functions of two players, this is saying ...
No-one's user avatar
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2 answers
36 views

Determining whether a housing allocation is in the Core

I have recently been thinking about the housing allocation problem where we have a set of players and a set of houses where players have strict preferences over the houses. I am aware of the Top ...
Finn's user avatar
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What is meant by the inverse of a CDF? Logit vs. Logistic Regression.

There is seemingly a difference whether one approaches the "logit" from statistics / econometrics: $$F^{-1}(\pi) = \log\left(\frac{\pi}{1-\pi}\right) = \text{logit}(\pi)$$ (I) Or from the ...
Marlon Brando's user avatar
1 vote
1 answer
109 views

How to deduce an expression of a specific conditional expression

The problem occurs when reading Bombardini et al., 2023, "Did US Politicians Expect the China Shock?", American Economic Review, Vol.1, PP174-209. The authors define $\xi_{it}$ to be a ...
zyy's user avatar
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2SLS and Instrumental Variables

I don't understanding the estimation of IV thoroughoutly. The following is my puzzle. In the first stage, we have $$D_i = \alpha + \beta Z_i + u_i$$ then we get predicotrs $\hat{D_i}$. In my mind, $\...
HSINSHUO's user avatar
1 vote
1 answer
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Determine the values at which the point elasticity of demand is maximum

Given the demand equation $$p=\frac{240}{q+15}$$ Where $15 \leq q \leq 105$, for what value of $q$ is $|\eta|$ a maximum? for what value is it a minimum? Well, i try the following. Note that: $$\eta=\...
Wrloord's user avatar
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Bolzano-Weierstrass theorem in the neoclassical growth model

I have a doubt regarding the use of the Bolzano-Weierstrass theorem in a dynamic optimization problem. Exactly, the problem is the neoclassical growth model (or Ramsey model). In the website that I ...
Ibai's user avatar
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Estimate the current elasticity and the current value

Consider the statemen A manufacturer of aluminum doors is currently selling 500 doors per week at a price of \$85 each. If the price were lowered to \$80 each, an additional 70 doors per week could be ...
Wrloord's user avatar
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Maximizing a function with binary indicators

I am an econ undergrad trying to understand how to maximize this payoff function, which includes binary components. I want to solve this equation using backwards induction, so I want to maximize the ...
trynacode's user avatar
1 vote
3 answers
71 views

Attempt at creating a formula relating debt, payments and interest

I tried writing down a formula relating a given debt and interest to the periodic payments and number of payments. So let's say someone starts off with a debt of $D$. The periodic interest is $r$ (for ...
HappyDay's user avatar
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Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?

Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable. ...
opre's user avatar
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Closed-form solution for polynomial recurrence relation.

Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$, where $\alpha \in (0,1)$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it ...
Arseniy  Samsonov's user avatar
1 vote
0 answers
16 views

Discrepancy between the average annual return and the actual stock price

I would really appreciate it if somebody can help me with the math on this one. I feel the need to apologize in advance if this errs on the dumb side. If somebody could walk me through this and make ...
Magus's user avatar
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1 vote
0 answers
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Exercises on Edgeworth Box

I don't know if this is the right forum to ask these questions (if so, tell me where to ask them). I have my microeconomics exam soon and I would like to know if I solved a couple of exercises ...
Marco Di Giacomo's user avatar
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23 views

Quasi-Concave Function $u(x,y)$ has Decreasing Marginal Rate of Substitution

Let $u:\mathbb{R}^2 \to \mathbb{R}$ be $C^2$, strictly increasing in both arguments, and strictly quasi-concave: $\{(x,y)\vert u(x,y) \leq \bar{u}\}$ is strictly convex for all $\bar{u}$. By the ...
Smithey's user avatar
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1 answer
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Why does the Paraeto Principle actually work? (The 80-20 Rule) [closed]

So, in economics, (and now expanding to other fields too, most significantly marketing tho I suppose commerce is but an extension of economical thoughts), there is a very widely held principle that 80%...
SuperSexyTrash's user avatar
1 vote
0 answers
45 views

Finding optimal economy path with best value [closed]

This problem is based on Warcraft 3 Legion Tower Defence custom map. First level problem: Let we say that from the beggining of the game we have infinite gold. We have 1 builder from the start. He ...
Stanislav Shchyrba's user avatar
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0 answers
31 views

How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?

I am really struggling to grasp how the Hamiltonian Function and Pontryagin's Maximum Principle work in the context of Optimal Control Theory (Maths for Economics) course. I am given the following ...
astute-hoplite's user avatar
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0 answers
32 views

Approximating the half-life of a shock to a system?

I found the following statement in here regarding the effect of twice lagged differences of CO2 ($\Delta C$) in the atmosphere on the once lagged values, i.e. $$\Delta C_{\text{ @ }t=-1}= 0.83 \times ...
JAP's user avatar
  • 553
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0 answers
10 views

Dummy variable with multiple criteria - how to

Dummy variable is simple when it is true or false. What happens if the 'true' has multiple criteria that needs to be met? For example, there are 4 criteria in total. And 3 out of 4 must be true for ...
MLux's user avatar
  • 1
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25 views

How to show the existence of a Bellman Equation Solution?

consider the Bellman Equation \begin{equation*} V(\alpha)=\max_{\beta} f(\beta,\alpha)+A(\beta,\alpha) V(\alpha)+B(\beta,\alpha) V'(\alpha) \end{equation*} How can I show the existence of the solution?...
Isn't Adobe Acrobat the best's user avatar
1 vote
0 answers
38 views

Solving a System of (In)Equalities FAST

I am given a system of (in)equalities (see below). There is a continuum of possible solutions and I just have to find one of them. My question is whether there is a faster way to find one of the ...
asdf1234's user avatar
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33 views

How was the return on investment calculated

This result was found in What Would You Pay for an Extra Year of Life?. But how was the 5.8% calculated? Here is the excerpt: What’s the Return on Investment? I know there’s a lot more involved in ...
Daniel's user avatar
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0 votes
1 answer
30 views

Showing that $C(\theta)$ is strictly increasing in $\theta$ and the relationship between $X$ and $A$

Given $U''<0<U'$ and $f'>0$, $f'(k)k=\alpha f(k)$ and this equation below $$\left[ f(k)-f'(k)k \right]\theta U'(C) = 1$$. How do I show that C is strictly increasing in $\theta$? If I totally ...
OGC's user avatar
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0 votes
1 answer
40 views

Total Differential w.r.t. 2 Endogenous Variables

I am given two terms and the solution of a total differential based on these terms with respect to two endogenous variables. However, I am not sure how they arrived at this result? The setting (...
asdf1234's user avatar
1 vote
1 answer
55 views

Utility function and bundles

I want to isolate $x_2$. Is this correct? Lets say I have $(x_1,x_2)$ = (2,4) and the utility function $v_1(x_1,x_2)=g(x_1^2x_2)$ $$v_1(2,4)=g(2^2\cdot4) = g(16)$$ $$g(x_1^2x_2) = g(16) \...
Yonathan's user avatar
1 vote
0 answers
23 views

Find variance in a two-period model

I am currently reviewing in-depth materials from Svensson (2007) and trying to retrieve equation (2.6) myself. To put it simply, I have such equation: $$\pi_{t+1} = \gamma \pi_{t} + \alpha x_{t} + \...
GaëtanLF's user avatar
2 votes
1 answer
46 views

CCR model in DEA - proof of dual linear program

I am studying Data Envelopment Analysis and the CCR model from Cooper, W. W., Seiford, L. M., Tone, K., & Cooper, W. W. (2006). Introduction to data envelopment analysis and its uses : With DEA-...
MattTct's user avatar
  • 23
0 votes
1 answer
156 views

Help with solving a GDP rate question?

The gross domestic product (GDP) of a country A is growing at a constant rate. In 2021, the GDP was 125 billion dollars, and in 2023 it was 155 billion dollars. At what percentage rate will the GDP ...
Isaac Yang Hao Tung's user avatar
2 votes
1 answer
100 views

Convex Independence

I have a difficulty in understanding a given definition for convex independence: A set of beliefs $\beta$ of an agent $i$ satisfy convex independence if beliefs of no type $t_i$ can be represented as ...
asdf1234's user avatar
0 votes
0 answers
32 views

How to integrate a function when one of the limits of the integral is a stopping time?

I have just took my first course of stochastic calculus and I am having trouble. The problem is the following: Consider a firm that pays a coupon $\delta$ until the cashflow $d\pi_t = \mu \pi_t dt + \...
Tobías Martínez González's user avatar
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0 answers
19 views

Some questions about the number of moments for random variables

I'm reading a paper about econometrics and meeting some questions, and I'm not sure I can describe them perfect. $s_t$ is a scaled score function, and $f_t$ is a stochastic time-varying parameter, and ...
Wing's user avatar
  • 1
0 votes
0 answers
24 views

Estimating Present Bond Value without YTM using yield curve rates

Suppose I have a bond where I know the par value, coupon rate, and maturity date as well as the daily Yield Curve Rates given here How can I go about estimating the ytm needed to determine the present ...
ccj242's user avatar
  • 21
1 vote
0 answers
32 views

Determining the strategies and finding subgame perfect nash equilibria.

I am trying to find the strategies and the subgame perfect nash equilibria of the following dynamic 3 player game: While I think I managed to find the strategies (see below), I am struggling to find ...
l337n00b's user avatar
  • 475
2 votes
1 answer
106 views

Linear Algebra - Economic Models

Suppose the coal and steel industries form an open economy. Every \$1 produced by the coal industry requires \$0.15 of coal and \$0.20 of steel. Every \$1 produced by steel requires \$0.25 of coal and ...
halcyon's user avatar
  • 23
2 votes
1 answer
79 views

What make a function become a utility function?

I am an engineer who is working in networking and I have a question about the stuff that is called "Utility Function". I have been researching over some scientific papers in my field and see ...
Tuong Nguyen Minh's user avatar
0 votes
0 answers
42 views

Mixed strategy Nash equilibria for an all-pay auction between two players?

I am actually unsure if this game would be considered a minimum effort game or an all-pay auction, so please forgive me if I'm misusing these names for the title. In a game where 2 players each choose ...
rwbycwbe's user avatar
0 votes
0 answers
43 views

Why use Price Elasticity of Demand rather than just $\frac{dR}{dp}$?

I'm helping a family friend with an introductory business mathematics course; I have no background in economics, so I've been previewing the applications covered in her textbook, and just read about ...
A.J.'s user avatar
  • 3,892
0 votes
1 answer
70 views

Meaning of a negative subscript of a Variable.

I have the following maximization formula of an adapted Monti-Klein model of Banking constructed by Borio et al. (2015): \begin{equation} \max_{\{L_{j},D_{j}\}}\pi_{j} = [l(L_{j}+L_{-j})-\tau]L_{j}-...
Fabio's user avatar
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2 votes
1 answer
106 views

Very complicated differentiation

I am trying to solve all steps in a economics paper , but after spending two days with the same differentiation Im losing faith. Can someone out there help me? The problem: Differentiate: $$ \begin{...
Johan's user avatar
  • 23
0 votes
0 answers
41 views

Is my Hamiltonian correct?

To solve the following optimization problem, $$ \max _{\left\{\ell_t\right\}} \int_0^{\infty} e^{-\rho t} u\left(c_t, N_t\right) d t $$ subject to $$ \dot{N}_t =\left(\bar{b}\ell_t-\delta\right) N_t, ...
Anpline Z's user avatar
1 vote
1 answer
46 views

Derivative w.r.t exogenous variables

Suppose $f(x_{1}, x_{2})$ defined on a compact set, which guarantee the existence of extrema, $x_1*, x_2*$, and also given the Hessian matrix of $f$ is strictly concave. Suppose there is also a ...
Doublehappy Tough's user avatar
0 votes
1 answer
57 views

Calculating the value of this asset

calculate the cash value of a financed asset that is paid in the following way: an initial payment of 500,000, in month 2 a payment equal to half its value, in month 7 a payment equal to a third of ...
MrJonesBones's user avatar
0 votes
0 answers
48 views

Consistency OLS without intercept

I have this model without intercept $y_i = \beta x_i + u_i$ where $\widehat{\beta} = \frac{\sum x_i y_i}{\sum x_i^2} = \beta + \frac{\sum x_i u_i}{\sum x_i^2}$ I want to verify the consistency of $\...
Constanza Zepeda's user avatar
0 votes
0 answers
33 views

Lending under VAR constraint

After working on this problem for several hours I still can’t wrap my head around how I would determine the lending volume taking into account probability of a solvency of 95%. I know how to calculate ...
Miro Sarkkinen's user avatar

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