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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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Supply/Demand Shift (Find New Equilibrium)

Starting: supply curve $y=x+10$ and demand curve $y = -x+30$ with equilibrium at (10, 20). The problem states that DEMAND shifts leftwards by 3 units and to find the new equilibrium. My professor did ...
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“At what point does another year of age reduce the probability of smoking?”

Say I have something like: $$pSmoke = .656 - .069log(cigprice)+.012log(income)-.029educ + .02age - .00026age^2 - .101restaurn - .026white$$ where $pSmoke$ is the probability of smoking and $age$ is ...
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37 views

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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2answers
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Is an element of a vector a scalar or 1x1 vector or 1x1 matrix?

In econometrics, by Hayashi, they defined the error vector of n observations in a $ (n \times K)$ regression funcntion as: $\epsilon = \begin{bmatrix}\epsilon_{1} \\\epsilon_{2} \\\vdots \\\epsilon_{...
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Hicksian Demand Function

Derive Hicksian demand for - $$u(x,y) = ax+ b\ln(y)$$ Explain in words what they mean? I solved the problem with the Lagrange Multiplier Method and found Hicksian demand for $x$ only. Solution: ...
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What's the application of Leibniz rule in economics? [closed]

I've seen somewhere that Leibniz rule (differentiating an integral) has applications in physics. What might be specific examples of applications of the rule in economics which would make sense to ...
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Real world retail statistical math problem [on hold]

I'm currently working in a relatively large retail company and I've been assigned the task to measure the effect of a new assortment that is being tested in three trial stores. The idea is to compare ...
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1answer
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Nash Equilibrium Grandfather Problem

A grandfather is writing his will and must decide how much money he will leave to his three children: Sean, Tom, and Brad. The grandfather comes up with the following plan. Each of his children is ...
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0answers
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What is the difference between $\min(x_1,x_2)$ and $\min\{x_1,x_2\}$?

This question refers to microeconomic utility optimization where the utility function is Leontief. Are $\min(x_1,x_2)$ and $\min\{x_1,x_2\}$ interchangeable, or is there a difference? edit: I ...
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1answer
46 views

Price elasticity

If $P = 90 – 0.05Q$ is the demand function for calculators in an engineering college. (a) Derive expression for $\epsilon$ in terms of (i) $P$ only, (ii) $Q$ only. (b) Calculate the value $\epsilon$ ...
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43 views

Shadow price not covering costs

In (environmental) econ class we had to solve the following problem: ...
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1answer
33 views

What is the sum of the products of price times quantity divided by sum of quantities

I have a very simple question, what is the meaning, if any, of this? I.e. mathematically what does the following calculation mean? $$ \left(\frac{(Q1\cdot P1)+(Q2\cdot P2)+(Q3\cdot P3)}{Q1 + Q2 + Q3}...
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99 views

Digital Option risk neutral probability

In the $T$-period binomial model, if the asset price is $S$ at any time, the next period's price will be either $SU$ or $SD$. The interest rate per period is $r>0$ and $D^* < 1 < U^*$, ...
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1answer
116 views

Determine fair price of a digital option

A digital option pays one dollar at time $t = T$ if the asset price is above a fixed level (strike) $K$ and is worthless otherwise. Consider the following model, with $r = 0$: \begin{...
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Finding the max of profit

Is the following fourmula right when we have the MC and MR of a certain comodity ? $$P_{max}=\int_ 0^a (MR-MC)dx$$ where $a$ is the root of $MR(x)=MC(x)$ and $[MR-MC]'(a)<0$. I personally think ...
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1answer
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Minimum Rate of Return

Question: A businessman wishes to earn 7% on his capital after payment of taxes. If the income from an available investment will be taxed at an average of 42%, what minimum rate of return, before the ...
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1answer
35 views

Exact Simple Interest Problem

Determine the exact simple interest on 1,000,000 invested for the period from October 24,1987 to January 7, 1990; if the rate of interest is 17%. What I've tried: October 24, 1987 to October 24, ...
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R^2 associated with a restricted LS estimator is never larger than that associated with the unrestricted LS estimator

Prove that the R^2 associated with a restricted least squares estimator is never larger than that associated with the unrestricted least square estimator. So, I tried doing this question but I can't ...
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0answers
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understanding the proof using the mean value theorem in the economic paper.

This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the ...
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1answer
33 views

non-negative matrix satisfying two conditions

A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$. I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)...
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96 views

Risk-neutral probability measures

Consider a market with $\Omega= (\omega_1, \omega_2, \omega_3)$, $r = 0$ and one asset $S$. Suppose that $S(0) = 2$ and $S$ has claim $\bar S = (1, 3, 3)$ at time 1. Find all the risk-...
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1answer
86 views

Sure Thing Arbitrage

Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10\%$ \begin{array}{|c|c|c|c|} \hline n&S_n(0)& S_n(1,\omega_1) & S_n(1,\omega_2) & S_n(1,\...
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1answer
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equivalence of two conditions on non-negative square matrix

Let $A$ be a nonnegative square matrix (every entry is non-negative, $A\ge 0$). Look at these two conditions: (1) $A^n\to0$ as $n\to\infty$ (2) $(I-A)^{-1}\ge 0$ I can show that (1) implies (2). In ...
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1answer
101 views

Is there arbitrage?

An economist writes a 1-period expectation model for valuing options. The model assumes that the stock starts at S and moves to $2S$ or $\frac{1}{2}S$ in 1 year's time with equal probability. ...
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1answer
25 views

Find $\frac{\partial}{\partial p}$ and $\frac{\partial}{\partial q}$ of $U(f(p) + q^2, g(q)^2)$

Here's my question: Suppose $U(x_1, x_2)$ is a differentiable function. Suppose $f(p)$ and $g(q)$ are differentiable functions that depend only on $p$ and $q$ respectively. Find $\frac{\partial}{\...
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How to show that the labour supply curve will not be negatively sloped?

Given $$U(c,1-h)=log\left (c-\frac{Bh^{1+\gamma}}{1+\gamma} \right ),$$ where $B>0$,$\gamma \geq0$, $h$ denotes the household's optimal labour supply and $c$ denotes consumption. The budget ...
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21 views

What is the equation for calculating an investment with compounding interest and periodic contribution?

I am a begginner C# programmer working on a investment calculator as a personal project. I have not done much math in the last 15 years, please help. The parameters are: Principal - the original sum ...
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1answer
116 views

Market quotes and arbitrage argument

DUPLICATE ON HOLD Use an arbitrage argument to construct a formula relating the price of the European to the price of an American. Good day, I wanted to ask for help with a question from one of my ...
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1answer
94 views

Calculate 3 month future of Oil

This question is from an exercise sheet for financial mathematics. Calculate the $3$-month future on Oil given the following information: Spot Oil is trading at $90$ USD per barrel. USD ...
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1answer
91 views

Fair Strike of a 1 Year Forward

Good day, I have been tasked with this question and I am unsure if my understanding and work so far is correct. On January 1st 2018 AAA shares were trading at $30USD$ per share. The shares ...
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How to Calculate Blau's Index with categories

I want to calculate age diversity with categories. The categories are: 1= 25 until 35 2= 36 until 45 3= 46 until 55 4= 56 until 65 5= more than 65 years please help me answer my question, I've been ...
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1answer
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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 $e(\theta)$, ...
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Solving a question with more unknowns than equations

I am not sure if I am thick in the head or not, but I am trying to replicate this paper: https://www.nber.org/papers/w19086.pdf. The equations characterising the equilibrium are given on Page $22$. ...
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1answer
21 views

Why do you need to substract initial investment in NPV formula?

The initial cash flow(s) of a project will be negative because of the investment. Why do you need to substract the initial investment in order to calculate NPV? Doesnt't the initial investment ...
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2answers
42 views

Price Calculation Paradox: How to cover tax and fees when these values depend upon one another

I have a real-world math problem pertaining to a pricing formula, a paradox. In this formula, two adjustments are needed, but both depend on knowing the result of each other first. I need to apply ...
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1answer
50 views

Minimising a black box function

I have a function with 8 inputs, which yields a single output. I do not know what the function does and so cannot use any derivative-based method to minimise said output. Currently this is done by ...
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1answer
25 views

Paying back for shared costs among a group

After celebrating the new year, me and my pals wanted to split the costs of our mischief. I started wondering about the best way to toss money around, i.e. a way to systematically minimise the amount ...
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39 views

question about differential equation from the economic paper

I am reading the economic paper: "Hopkins, E., 2018. Inequality and risk-taking behaviour. Games and Economic Behavior, 107, pp.316-328." I have one question about the step of one of proofs in this ...
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1answer
25 views

Subgame Perfect Equilibria in a one-stage game

Consider the following one stage game, with two players A and B. There is a pie which is to be divided between the two players. A can offer B any fraction of the cake, which B can accept or reject. ...
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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 ...
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1answer
37 views

Static Optimization problem assumptions

Consider the following Constrained Optimization Problem: $$max: -2(x+y)^2 +5x-y$$ Subject to: $$x≤B$$ $$-x+2y≤3$$ $$y≥0$$ with$$ \beta = 0$$ I have found the solution that solves the problem:$$x=\...
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2answers
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System of differential equation stability, with initial condition

Consider the following system of differential equations: $$ \dot x(t) = 3x(t) - 2y(t) + 3$$ $$\dot y(t) = 2x(t) - 2y(t) - 1$$ (a) Find the steady state of the system and determine its stability. (...
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1answer
84 views

Paasche price index: cost of living adjustment (economics)

I am trying to figure out the following question involving the Paasche price index: two goods are perfect substitutes (let's assume for simplicity that indifference curve has slope of -1) relative ...
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Finite Horizon Dynamic programming optimization (consumption-savings problem)

I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction. Maximize $\sum_{t=0}^{T}u(c_{t})$ subject to $w_{0}>0, c\in [0,w],w(t+1)=(w_{t}-c_{t})(1+...
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1answer
44 views

Finding the Bayes-Nash Equilibrium for First-Price Auction with 2 bidders

Let $\sigma_i$ be the strategy profile for bidder $i$ that indicates how they should bid based on their value. As we know, if there are 2 bidders both with their values $v_1,v_2$ on $U[0,1]$ in a ...
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20 views

Exchange rate, interest rate problem

I am supposed to solve this: czech investor has 1 000 000 Czech crowns available. Exchange rate is 27 Czech crowns / 1€. Interest rate in a czech bank is 3% and 6% in a German bank. Investor expects, ...
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1answer
39 views

Prove $(b-a) \cdot \int_a^b \alpha(x) \beta(x) dx \ge \int_a^b \alpha(x) dx \cdot \int_a^b \beta(x) dx$.

Suppose $\alpha: [a, b] \to (-\infty, \infty)$ and $\beta:[a,b] \to (-\infty, \infty)$ are both nondecreasing. Then, $$(b-a) \cdot \int_a^b \alpha(x) \beta(x) dx \ge \int_a^b \alpha(x) dx \cdot \...
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1answer
24 views

Interest rate problem

I am supposed to solve this problem: mexican interest rate is 28%, interest rate in the US is 7% and exchange rate is 7.5 Mexican pesos for 1 US dollar. If interest rate parity applies and we don't ...
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1answer
19 views

Percentage change of a quotient

This was line in my Econ notes that was "by definition". If anyone can clarify this I would really appreciate. We have a relationship: $APL = \frac{Y}{L}$ Now, define $\Delta x$ as "percentage ...
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1answer
37 views

Looking for economics research article that applies concepts from real analysis.

As a final project for my real analysis/advanced calculus I am being asked to write a discussion/critique of a paper in my field that applies concepts from real analysis. This could include anything ...