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.
1,542
questions
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Maximization of CD production function
I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem:
$$\max_{L_{t}}=\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\...
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0
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14
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Problem solving a Bellman equation with control variable multiplying continuation value.
The agent problem can be described as follows (for $n\xrightarrow{}\infty$):
\begin{align*}
\max_{\{c_0,c_1,\dots,c_n\}} &\sum_{t=0}^n \beta^t\cdot \left(k-\sum_{\tau=0}^t c_\tau\right) u(c_t)\...
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Need help with this question for undergrad Math Econ [closed]
Find the total differential for y = f(x,z(w))*g(w,x)
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-5
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16
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Stochastic differential equations in economathematics [closed]
Prove that the stock price, governed by the binomial tree, converges to the stock prices of black scholes model.
What kind of assumptions should be made?
Are they unique?
...
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0
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19
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The Method of Quantiles
I am reading a book about quantile regression and at some point, they come to the method of quantiles, which is pretty much like the beloved method of moments. The case is that I do not understand the ...
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How to solve this post? [closed]
How long does it take to download a two-hour HD movie
from the iTunes store? According to Apple’s technical support
site, support.apple.com/en-us/HT201587, downloading such
a movie using a 15 Mbit/s ...
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1
vote
0
answers
22
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Using Beta distribution in a profit maximization problem - replacing e in a system of equations that we'd already solved
I'm working on an economic model that we've already solved using 1-e^(-Investment) as the probability of a stochastic process.
Doing this we get lovely, simple derivatives for our decision rules (ex. ...
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41
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Help understanding notation used in a problem
Can someone help me understand the notation used here? I can't seem to figure out the meaning behind what comes after the "if condition".
If Adeline chooses $T$
\begin{equation}
U_i=v\min\{...
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0
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15
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How to measure the affection of 2 variables on time dimension?
As the title, if in case you have 2 time series sequences A = $[a_{t}, a_{t-1},..., a_{t-n}]$ and B = $[b_{t}, b_{t-1},..., b_{t-n}]$, given $m$ steps back to history $(m<n)$ and assume we have ...
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Are there interior solutions to this constrained optimization problem?
I am getting some confusing results solving this problem:
$$\max_{c_0\geq 0, c_1\geq 0} \ R(1-c_0) [p t_1 + (1-p) c_1^{-\lambda} t_2] \\ s.t. \ c_0+c_1 \leq 1$$
where $\lambda>1$, $R>0$, $p$ is ...
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22
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Proving hemicontinuity.
In my Mathematics for Economics course, we have started studying hemicontinuity. The concept is very clear to me, the problem is that I cannot understand how to prove it. Here is an example, from the ...
0
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1
answer
23
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Utility in case one good is Bad and other is Good
I have a question
Suppose there are $2$ goods, $X$ and $Y$ where, $X$ is Good good, $Y$ is Bad good
Now Can the Utility be
1.) $U = x-y$
2.) $U = \frac{x}{y}$
3.) $U = \frac{\ln x}{y}$
They all are ...
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5
votes
1
answer
181
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How does the Pareto distribution represent the 80-20 rule?
According to the current Wikipedia article:
The Pareto principle or "80-20 rule" stating that 80% of outcomes are
due to 20% of causes was named in honour of Pareto, but the concepts
are ...
- 1,671
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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Representing Submodular Functions As Maxima of Additive Functions
According to this paper, "every submodular function can be represented as a maximum of additive valuations." It gives an algebraic description as well, but I am having trouble internalizing ...
- 1,554
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1
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34
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Finding Nash Equilibria and Best Response in a Strange Game
Consider a two-player game where Player 1 chooses a strategy $x_1=[1,3]$ and Player 2 chooses $x_2=[0,2]$ Let the payoff functions for P1 and P2 be $u_1(x_1,x_2)=\min\{x_1,x_2\}$ and $u_2(x_2,x_1)=\...
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answers
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Distribution of linear combination of normally distributed 2-way censored variable
Consider a censored model with the latent variable given by
$$y^*=\beta_0+\beta_1 x+u,$$
and the dependent variable given by
$$
y =
\begin{cases} \displaystyle
0, & y^* \le 0 \\
y^*, & 0 < ...
1
vote
2
answers
54
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A Basic Question of Continuous Time Macroeconomic Model (Variation of Constant to Solve an ODE)
I am going through the continuous time macro slides by Ben Moll (link is: https://benjaminmoll.com/wp-content/uploads/2019/07/Lecture2_ECO521.pdf), when deriving New keynesian model in continuous time,...
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17
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Inflation Calculations Skewed by low prices.
I am currently calculating inflation metrics for a retail store. I am simply using their POS data, to calculate the Year-Over-Year price change of each item.
I then weight this on the sales, sum it up,...
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0
answers
20
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What are some common distributions assumed for approximating a demand curve?
A demand curve is a function $d(p)$ such that at price $p$ the demand for the good is given by $d(p)$. This means that you are making a sales of $p\cdot d(p)$ at price $p$. For typical goods, demand ...
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A put option and a call option with identical exercise price are both marketable or neither is.
I am doing Exercise 1.13 in Introduction to Mathematical Finance: Discrete Time Models by Pliska.
Exercise 1.13 Suppose the interest rate $r$ is a scalar, and let $c$ and $p$ denote the prices of a ...
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votes
1
answer
82
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Is there any book "economics for mathematicians"? [closed]
If you search economics book there are plenty of them on the market. We may say the say about "mathematics for economists" or "application of mathematics in economics". But I've ...
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1
answer
40
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Engineering Economics Question: Uniform series present worth question
The cost of a fence that can detect poacher intrusion into a National Wildlife Preserve is $3.3 million per mile. If the effective life of the fence is 10 years, determine the equivalent annual cost ...
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1
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29
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How do I normalize/standardize a data set to a certain set value (not average, not min, not max)?
I know the formulas for standardizing, normalizing, and mean normalizing (if we take the wikipedia's definition for these terms). What I want to do is to normalize to a set value. For example, if I ...
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1
vote
1
answer
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Intuition behind the Compensating Variation!
In Economics, we can calculate the compensating variation (CV), which (to my understanding) is the amount of money we would need to give back to a consumer to keep them at the same level of Utility ...
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vote
1
answer
46
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Minimization of the Pollution Problem
I study Economic Analysis.I came accross the following minimization problem
I tried to show that M is quasiconcave but i failed.
I tried using the definition of quasiconcavity or using the min ...
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1
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26
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How can elasticity of $f(x)$ be expressed as a percentage change in $f(x)$ per $1$% change in $x$?
In answering the above question, we took the formula of elasticity:
$El_xf(x) = (x/f(x))*(∂f(x)/∂x)$
We can transform it in the following way:
$El_xf(x) = (∂f(x)/f(x))*(∂x/x)$
Now at this point, the ...
1
vote
1
answer
40
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0.35 log points to percentage
What is 0.35 log points in percentages? I have no knowledge of this subject at all. From what I've gathered, it should be Percentage change = (e^log points - 1) * 100, so 42%. Is this correct?
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1
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Need help identifying what sort of differential equation this is
I'm reading an economics paper, and have stumbled upon this differential equation system:
$\dot x = -\beta xy^\alpha$
$\dot y = \beta xy^\alpha - \frac{1}{\sigma}y$
I haven't taken any classes on ...
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1
answer
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Equilibrium interest rate
There are two countries in the whole world: Bolivia and Peru.
The aggregated investements in Bolivia is $I^B$ And in Peru they are $I^P$ (They are both exogenous).
The savings for Bolivia, where $B,b &...
0
votes
1
answer
36
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Trying to find why solutions to both maximization and minimization problems exist before even solving it [closed]
There is a problem which I have been thinking about but can't come to a conclusion. And Final exam is tomorrow:
enter image description here
My attempt: To find solutions to both maximization and ...
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0
answers
26
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Why is the GST for the wholesaler based on the marked price? We don't know if that is the cost price.
The price of a sewing machine is Rs.40000.The wholesaler allows a discount of 10% on it to the shopkeeper. The shopkeeper sells the sewing machine to a customer at a discount of 5% on the marked price....
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1
answer
45
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Taylor Expansion for the Return averaged over k periods? [closed]
this is my first question here. I need help to understand the Taylor Expansion which gives the (2.2.5) equation (see the pictures). Thanks
(pictures from: Schmidt - Quantitative Finance for Physicists....
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1
vote
1
answer
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Regression relation to casual relationship
If the correlation coefficient of two variables is 0, can there still be a causal effect between them? And can the causal relationship between these two variables be studied by regression analysis?
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Many questions about a loss distribution
Long question, but mostly set up:
I'm trying to understand the capital requirements formulas of Basel III. They come from the Merton-Vasicek model for the value of a company's assets. Assume we have $...
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Definition of transition density
Sorry if this is a basic question but I have a hard time grasping the definition of transition density in a financial setting.
From what I understand, transition density is the expected (future) value ...
1
vote
0
answers
57
views
Expected utility
The figure below describes John's preferences over lotteries for three prizes- $1, $2, and $3, where p1 is the probability of winning $1, and p3 is the probability of winning $3. Assume that John is ...
1
vote
0
answers
75
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What does it mean to "solve forward/backward" a differential equation?
I know what it means to solve forward/backward difference equations. What I have not been able to wrap my head around is how that formally makes sense in continuous time. To make things concrete, here ...
4
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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
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0
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33
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How do you allocate based on historical outcomes when these can be both positive and negative?
I would like to allocate an expected yearly return based on the monthly outcomes of last year. However, these monthly outcomes can be both positive and negative.
For example (year 3 in the image), the ...
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votes
1
answer
25
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Combining Math and Reading Test Scores presented in standard deviation units from the mean (mean not given)
I am working with a large dataset for regression purposes and am attempting to predict test scores using various societal / demographic factors. There are two scores, math and reading, that I am ...
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1
answer
38
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How to prove $\{(x_1,x_2)\in\mathbb{R}^2\mid u(x_1,x_2)\geq u^*\}$ is closed? [closed]
I have the following problem:
(continuous function)
Suppose that we consume non-negative amounts of a good $1$ and good $2$, namely, $x_1$ and $x_2$. The utility from consuming $(x_1,x_2)$ is defined ...
0
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0
answers
38
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Basic log-linearization
In the first equation I can't figure out where the $1/x$ part comes from...
When using first-order taylor approximation arount $x_t = x$, I just get
$ln(1)*(x_t-x) = (x_t-x)$
Thank you for your help!...
0
votes
1
answer
47
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How to prove that any preference relation on (countable) X has a utility representation with a range (0,1)?
Theorem: If $X$ is countable, then any preference on $X$ has a utility representation with a range $(0,1)$.
The stated proof in Rubinstein's lecture notes:
Proof: Let $\{x_n\}$ be an enumeration of ...
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0
answers
16
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Pair Matching Algorithm Ideas?
I am trying to work out a small task where I need to match up a sequence of events that have offsetting scores and see if there is a better theoretical framework to solve this in a more efficient way
...
0
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0
answers
20
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How to represent probability in the matrix payoff?
I need to build a 2x2 matrix of the game payoff.
There are 2 players A and B that can play 1 or 0 simultaneously.
I have a rule that in the case of both players playing 1, the award equal to X may be ...
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1
vote
1
answer
45
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First order condition (consumer problem) [closed]
Can you help me find the first-order condition of the following problem with respect to α:
$max[u(C_t) + δ *E_t[u(C_{t+1})]]$, such that:
$C_t=e_t-αp_t$, and
$C_{t+1}=e_{t+1}+αX_{t+1}$
For context ...
0
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0
answers
29
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Solve optimization problem using KKT method
Question:
Solve the UMP $${Max} \quad u(x,y) = ln(x+1) + ln(y+1)$$
$${s.t.}\quad p_1x + p_2y \leq w$$ $$x,y \geq 0$$
Then the Lagrangean function is
$$L = ln(x+1) + ln(y+1) +\lambda(w - p_1x - ...
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1
vote
0
answers
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Prove that the stochastic process $s_t$ follows a normal distribution where the mean and the variance are functions of time in each case.
The two basic models of finance are the following:
$\textbf{The Samuelson SDE (aka Black - Scholes - Merton model):}$ Suppose that $Z=\left(Z_t, t\in\mathbb{R}^{+}\right)$ is a Wiener process (aka ...
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0
answers
15
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Economics Question - Days Forward Coverage Formula
I am trying to replicate a built-in function in our application that calculates Days Forward Coverage (How many days can your supply fulfill demands in a given horizon).
I know how to arrive at the ...
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