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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112 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
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3answers
150 views

How much is 6% a year in months?

I am currently in high school where we are learning about present value. I struggle with task like these: Say you get 6% interest each year, how much interest would that be each month?
2
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1answer
136 views

What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for ...
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1answer
85 views

Can I embed $\mathbb R^{\mathbb N}$ with a partial order into $^\ast\mathbb{R}$ with the linear order?

Define a relation $\prec$ on $\mathbb R^{\mathbb N}$ as, For all $f, g \in \mathbb R^{\mathbb N} $, $f \prec g$, if for all $n \in \mathbb N$, $f(n) \leq g(n)$, and there exists a $m \in \mathbb ...
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1answer
14 views

Profit Function where total revenue is re-spent on production?

I'm trying to find a function that finds the net profit gain over multiple iterations. For example, if I produce a unit for 5 dollars , and can sell it for 8 dollars, I would have a net gain of 3 ...
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2answers
347 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
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1answer
432 views

Does one necessarily need an MS in Math before taking a PhD in Math? [closed]

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but ...
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0answers
34 views

Quasiconcavity of $g(x)=xf(K-x)$

The function $f(x)$ is strictly increasing, finite, positive and twice continuously differentiable on the compact interval $[0,K]$, and $f(0)=0$. I'm trying to either find a counterexample to, or a ...
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1answer
127 views

Core vs. Strong Core in Housing Allocation Games

I am presently reviewing the course notes for my Game Theory course, and I'm struggling with the concepts of the core vs. the strong core. In the notes, we have three players, with preferences ...
2
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1answer
50 views

Mathematical question about currencies

I've tried to get an answer for this question elsewhere but with no luck, so I would appreciate a mathematical analysis of it. Assume $0\%$ commission. Consider three currencies, let's say the Zong, ...
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0answers
38 views

Average Cost of Obtaining in game Item

I know this will sound like a trivial maths problem, but recently I've been playing a game in which you can pay 5 in game gems to get a Rare (R), Super Rare (SR), and Ultra Rare (UR) characters ...
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1answer
44 views

Endogenous covariate in first-difference panel data model

I have a linear panel data set (murder.dta, standard STATA dataset). First I estimate a first difference model. An assumption from this model is that the first differences of the covariates and the ...
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0answers
94 views

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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0answers
32 views

When the company will stop production?

Given total costs function $C(q)=100+10q-6q^2+3q^3$. For which price the company will stop production given that all of the fixed costs are sunk? Do not know how to approach those type of questions, ...
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1answer
88 views

Given domain and range of a monotone function, what is the maximum slope?

I'm looking for a reference in answer to one of the following questions: Is there a general result out there that will give the maximum possible slope of a monotone function, given its domain and ...
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1answer
46 views

Stochastic dominance characterization

Consider two probability measures on $\Bbb R$ given by $\mu$ and $\nu$. We write $\mu\leq \nu$ if there exists a joint distribution $P$ with the latter marginals such that $P(x\leq y) = 1$. In ...
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1answer
2k views

How to calculate $p_i$ in Blau's Index of Heterogeneity

I came across this: "Therefore, board gender diversity and board racial diversity are calculated using Blau's index of heterogeneity $(1 - \sum p_{i}^2)$, where $p_i$ is the proportion of group ...
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1answer
146 views

From utility function (3 products) to demand function (2 products)

I am struggling with this exercise and would appreciate some help. Consider two goods and a representative consumer whose utility is given by: $U(q_{0}, q_{1}, q_{2})= ...
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0answers
89 views

Utility maximization of n goods

I have a question that involves finding the optimal demand of $n$ goods for a consumer. However, I haven't anything like this before and I'm not sure how to proceed. The consumer has a utility ...
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1answer
47 views

how to solve a simplex with n variables

I don't know how to resolve a simplex with $n$ variables I have this primal problem \begin{cases} \text{min}& z=-x_1 - x_2 -\cdots - x_n\\ &a_1x_1 + a_2x_2 +\cdots + a_nx_n \le 1\\ ...
-1
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1answer
18 views

I need som help with price ratios.

Okay, so I'm doing some econ. homework and I'm supposed to draw a graph where the relative price of $P$ is $4 T/P$. Thing is that my brain stops working anytime I try to think about ratios. I know ...
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1answer
35 views

About uniqueness of interest yield

I am not sure this belong to this site, in case I will post it elsewhere. Let $P$ be the price of a bond, let $C_k$ the promised cash flow in year $k$. Then we define the interest yield $y$ as the ...
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2answers
364 views

dynamical systems applied to economics

I'm ending my undergraduate economics course and I'd like to extend my MA research program to dynamical economic systems. Knowing that my mathematical basis is calculus of 1 and 2 variables, linear ...
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0answers
35 views

Convolution of which distribution will give a uniform distribution?

Suppose there are two IID random variables x1 and x2. What should be the distribution of these random variables so that the distribution of x1-x2 is a uniform distribution?
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1answer
236 views

Reference Request - Introductory book on Mathematical Modelling in Economics and Business

I have to take a compulsory course named Mathematical Modelling in Economics and Business this semester and have absolutely no background on the subject. I also noticed there is no post on this site ...
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1answer
35 views

Find the OLS estimator $β_1$ when a new variable is added to the regression

Suppose $y_t = \beta x_t + u_t$, where $t = 1, 2, ..., n$. We know, in this case, the OLS estimator is $\hat{\beta} = ∑x_t y_t /∑x_t^2$. Now suppose one more observation $x_{n+1}$ is added. At the ...
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1answer
132 views

Optimization of utility function with Lagrange multiplier

Let u: ${\mathbf R^n_+ \rightarrow \mathbf R}$ be a utility function of n goods which you buy in quantities $x_1,…,x_n$ to the prices $p_1,…,p_n$ under the budget K. So maximize $u(x_1,…,x_n)$ subject ...
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1answer
44 views

State the likelihood function and the appropriate objective function which must be maximized for maximum likelihood estimation

Consider a simple exponential employment duration model where $t_j$ is length of the employment and $x_j$ is a suitable economic explanatory variable. The distribution of $t_j$ is assumed to be ...
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1answer
177 views

Simple PPF question

Econoland and Free Enterprise are two economies each producing two goods: airplanes (A) and grains (G). Currently the two economies do not trade with one another. The table below provides ...
4
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0answers
890 views

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I ...
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0answers
76 views

Implicit differentiation: Differentiating function with respect to integral

I am stuck on a simple problem and would highly appreciate your opinion. I have a optimization problem over $x$ with the objective function $$F=aG(x,y)+ (1-G(x,y))(1-x)$$ So the first order condition ...
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1answer
69 views

Hypothetical scenario with economics

You have been assigned to purchase a new molding machine. One vendor offered a machine that will cost $200,000$, with an estimated installation of $10,000$. The machine has an expected life of $10$ ...
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1answer
196 views

cash flow diagram, in/outflow series

I have a econ midterm coming up soon and stumbled upon this question. I know this is the math section but it appears not many use the finance one. My approach is: ...
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0answers
24 views

House allocation with existing tenants

In a house allocation with existing tenants model using the TTC mechanism, consider the incentive of an agent to misreport his/her preferences. Can it ever be that misreporting the true preferences by ...
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0answers
33 views

length plus width equals price, factoring?

im trying to grasp what this means as I usually work with areas (L x W = A) or perimeters (2L + 2W = P)... This problems was presented to me by a colleague and i'm just trying to wrap my head around ...
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1answer
123 views

How can I find the sum of an infinite series of products?

Background Today in my macroeconomics class my teacher taught us three concepts. The first is very simple: consumption $c$ is a linear function of national income $y$. Mathematically, $$c = My + ...
2
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1answer
57 views

Pricing a riskless asset in the Black & Scholes market

Consider a Black&Scholes Market where a risky asset evolves according to: $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ $$S_o=s$$ Riskless asset is associated with risk free rate r. I want to represent ...
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0answers
32 views

What is the solution to this maximization/minimization problem.

How would i go about solving the problem where i want to find $$max/min \quad \sum X_i$$ Where i have constraints which are $$M = P_1 X_1 + P_2 X_2+ P_3 X_3$$ and $$C_1 \geq X_1$$ $$C_2 \geq X_2$$ ...
0
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1answer
48 views

Understanding the proof of Gibbard-Satterthwaite theorem

Let $n$ be the number of voters and $A$ be the set of alternatives. For voter $i$, we denote by $a \succ_i b$, if $i$ prefers $a$ to $b$, where $a,b \in A$. Let $L(A)$ denote the set of all strict ...
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1answer
140 views

Manipulating a summation expression for Future Value (Annuities)

I've been given the following expressions for regular payments with regular annual compounding (Annuities): (1) $$ F - (1 + r)F $$ (2) $$ F = d \sum^T _{t=1} (1+r)^{T-t}$$ where: ($F$) is a Future ...
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2answers
122 views

Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
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2answers
40 views

Adding interest and saving to principal

I know the formula $$ K_n = K_0 \cdot (1+r)^n $$ to get the balance after $n$ years with an interest rate $r$. What if I'm adding 5 dollars to the principal each day. How can I find the balance ...
2
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1answer
91 views

Second price auction, page 82-84 of Osborne's An Introduction to Game Theory

Consider the second price auction defined and discussed on pages 82-84 of Osborne's An Introduction to Game Theory $($pages 80-82 here in this online draft version of the textbook: Martin J. Osborne, ...
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1answer
174 views

Deriving demand from quadratic utility function

How do you derive the demand for utility $u(x_1, x_2) = x_1^2 + x_2^2 $ and initial endowment is $\omega = (2,2) $? I believe this demand has three cases. Thank you
2
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1answer
51 views

Modified parcheesi game

A "modified Parcheesi" game starts with the following position: First $x$ flips a fair coin. If heads he can move two spaces or pass. If tails he can move one space or pass. If he occupies the ...
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1answer
63 views

Consumer Surplus

I am working on an economics problem where my solution is not correct, and I'd like to know why. Below is the question and my subsequent work/solution: The demand function for a particular ...
1
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1answer
342 views

Pure and mixed strategy in Nash Equilibria with n player

I got confused when I see the following problem: There are n staffs and they want to raise their salary, if any one or more than one of these staffs suggest their boss they want raise salary, all ...
0
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1answer
174 views

What is the pure strategy Nash Equilibria of asking your professor to cancel class?

Each student in a class has the option to remain silent or ask the professor to cancel class. If any students asks to cancel class, all students get a payoff of $r$. However, the student that asks ...
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2answers
931 views

How to make your own supply/demand graphs on a computer?

This is an econ question but I think you guys might have the solution. I am writing an economics paper and I need to show, in my theory, shifts in the supply and demand that I predict. I don't have ...
3
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3answers
57 views

How does this derivative notation work?

Elasticity of substitution = $\dfrac{\mathrm d \ln(k/l)\;\;\;}{\mathrm d \ln(f_l/f_k)}$ This is type of notation I haven't really worked with before. Is this read as "The change in the natural log of ...