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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1answer
30 views

Compound Interest similarity

You have just purchased a car with a $£20,000$ price tag. The dealer offers to let you pay for your car in $5$ equal annual instalments, with first payment due in a year. If the dealer finances your ...
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0answers
35 views

The correlation between alpha and beta

Consider the following 2-variable linear regression where error $e_i$'s are independently and identically distributed with mean 0 and variance 1; $$ y_i=\alpha + \beta (x_i - \bar {x}) + e_i$$ where ...
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2answers
55 views

Effect on Minimizer of Tightening Constraints

The Statement of the Problem: Consider the minimization problem $f(x,y)=14x+20y$ under the constraints $x+2y \ge 4 $, $7x+6y \ge 20$, and $x,y \ge 0$. Don't use the simplex method! (i) Draw the ...
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0answers
55 views

Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in ...
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1answer
29 views

A risk neutral individual chooses among a pair of gambles.

I'm a little confused about the following lecture slide, which is written as follows. A risk-neutral individual chooses among pairs of gambles ...
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0answers
102 views

How CES function with integral becomes min function in the limit

I wonder how a CES function over a continuum of goods, $$\left(\int_1^\infty c(\theta)^\delta g(\theta) \mathrm{d}\theta\right)^\frac{1}{\delta}$$ where $c(\theta), g(\theta)>0 \forall ...
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1answer
25 views

Convex combination and convex set

From where does $tx + (1-t)x'$ originate from? I am selfstudying an economists book, and this is popping up all of a sudden. I get that it's a line between $x$ and $x'$, but why? And is $tx' + (1-t)x$ ...
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0answers
26 views

Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
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153 views

Cournot competition: profit maximizer vs. market share maximizer

Today during an informal conversation with an established business researcher, I learned such a fact: In the classical Cournot competition model, if one player is a profit-maximizer, the other ...
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1answer
30 views

Econometrics/Statistics, variance and means

here's the problem I can't figure out on my own: The weight of a randomly selected student, (W), has a mean of $170$ and variance of $10$. Defining the new random variable ($Y$): the total weight of ...
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0answers
17 views

Gross Substitutes under continuous perturbations

Let $v_i(S)_{i \in [n], S \subseteq G}$ be a collection of Gross Substitute valuations. I am wondering if I can add a small perturbation to each valuation and still get Gross Substitute valuations. ...
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116 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
164 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?
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1answer
143 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 ...
1
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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
392 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
433 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
35 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 ...
4
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1answer
130 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
95 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
47 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
148 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\\ ...
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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
385 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
237 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
136 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
183 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 ...
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0answers
911 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
80 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
202 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
35 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
126 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 + ...
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1answer
61 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 $$C_1 \geq X_1$$ $$C_2 \geq X_2$$ ...
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1answer
49 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 ...