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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14
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12answers
10k views

Which 4 maths courses to take as an Economics PhD student?

I am doing a PhD in economics and I have the chance of taking one subject a semester in the maths department (I would like to do more, but "unfortunately" I have to work on my thesis). I want to have ...
2
votes
2answers
31 views

Maths for economics: finding the level of production that minimises marginal cost [on hold]

Let the total cost function of a firm be given by: $$TC(Q)= 16Q^3 - 72Q^2 + 446Q + 90$$ Find the level of production that minimises the marginal cost of production. (This is basically taking the ...
0
votes
1answer
24 views

Inflation Index

The inflation index of a country $A$ in $1995$ relative to $1990$ was $5$. Meaning that the ratio of dollars spent during $1990$ for goods compared to dollars spent for same goods is $1:5$. In country ...
1
vote
0answers
35 views

Basic optimization question

A teacher put this problem up the other day and I'm confused about how he got to the answer. Can you explain it to me? Job $X$ provides $20$ vacation days and $143,000$ euro annual salary. Job $Y$ ...
1
vote
1answer
541 views

What is my unit of observation for this time series data set?

My professor gave us a data set to run a regression and I have a data set which lists years from 1959-2007, gross private investments (in billions of dollars), and gross private savings (in billions ...
0
votes
0answers
14 views

Limits: Show derivative approaches 0 and infinity.

Consider: $F(K,L) = A[\alpha K^\psi+(1-\alpha)L^\psi]^{1/\psi} $ with $0 \lt \psi \lt 1$ The questions asks to find whether the following two conditions hold: $Lim_{K \to 0}F_k(K,L) = \infty $ and ...
3
votes
0answers
17 views

Calculating the equilibrium price. ( Theory- whith no smooth supply and demands curves)

I was looking for a theory wich explain the following problem: I have the next , demand, supply curve: The book didn't explain how to compute the equilibrium price with this market (only was ...
1
vote
1answer
31 views

Solving for n in Present Value and Annuity formula

I derived the formula for n=.... of the following formula P = $\ 525 $ A = $\ 15 $ i = $\ 0.015 $ Answer: $\ 50 $ P = $\ A [\frac{((1+i)^n - 1) }{ ...
0
votes
0answers
11 views

Engineering Economics

Suppose that a college graduate earns about 78% more money per hour than a high- school graduate. If the lifetime earnings of a high-school graduate average $1,200,000, what is the expected value of ...
0
votes
1answer
26 views

Profit and loss question.

A person bought two bicycles for Rs.1600 and sold the first at 10% profit and the second at 20% profit. If he sold the first at 20% profit and the seconds at 10% profit .He would get Rs.5 more.The ...
0
votes
0answers
38 views

Comparing a maximization to an integration with economics application

This seemingly simple question has interesting interpretation in economics, but I only state the mathematical problem here. Suppose $B(0)=C(0)=C'(0)=0$, $B'(\cdot)>0,\ B''(\cdot)\leq0,\ ...
0
votes
0answers
22 views

Time derivative of a production function

Given the production function is $$Y=F(K,AL) = \left [ \alpha (AK)^{\rho} + (1-\alpha) L^{\rho} \right ]^{\frac{1}{\rho}},$$ and that capital $K$ evolves at time $t$ as expressed by this equation ...
0
votes
0answers
14 views

Envelope theorem for Conditional value at risk

Let $X$ be a Gaussian random variable and suppose $f(p,X)$ is a strictly increasing and continuous function in $p \in \mathbb R$. Conditional value at risk is defined in the following way ...
0
votes
1answer
36 views

For which values of $\rho$ does the CES production function satisfy the Inada conditions

Given $F$ is a constant elasticity of substitution (CES) production function: $$F(K,AL) = \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1}{\rho}},$$ where $\alpha \in \left ( 0,1 ...
-1
votes
1answer
24 views

Mixed portfolio [closed]

Suppose that Ms Lynch can make up her portfolio using a risk-free asset that offers a surefire rate of return of 15% and a risky asset with an expected rate of return of 25% with standard deviation 5. ...
0
votes
1answer
40 views

price and quantity after taxation

Given that demand for a good X is equal to $q_D=393-2p$ and market supply is $q_S=p/4-12$. Find equilibrium price and quantity, consumer and producer surplus and draw a diagram illustrating the ...
1
vote
1answer
23 views

Penalty and minimization of a social cost

It is part of broader question in economics however it is about minimizing the expression (which depicts social cost of a crime): the expression is $$\min ...
0
votes
1answer
24k views

Microeconomics: Calculating Tax Revenue and Tax incidence

Australian Government has imposed a tax on Beer. Assume that the tax on Beer is $20 per unit (a unit is a carton of drinks) Assume the demand and supply functions for cartons of Beers per week are: ...
2
votes
1answer
343 views

How to find the short run and long run cost functions, given the production function?

The production function of car is given by $f(x_1,x_2,x_3) = \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}$ (assume competitive input and output markets). Find the short run cost function (let input 3 ...
1
vote
1answer
21 views

Government's intervention - price and quantity after taxation

Given that demand for a good X is equal to $q_D=393-2p$ and market supply is $q_S=p/4-12$. Find equilibrium price and quantity, consumer and producer surplus and draw a diagram illustrating the ...
0
votes
1answer
18 views

Public goods - 2 people buying something

Bob and Ray are thinking of buying a sofa. Bob's utility function is $U_B(S,M_B)=(1+S)M_B$ and Ray's utility function is $U_R(S,M_R)=(3+S)M_R$ where $S=0$ where S=0 if they do not get the sofa and S=1 ...
0
votes
0answers
21 views

Change in variables that depend on each other in a system of equations

This is an economics question, but I have been referred to math stack exchange: We have the following equations: \begin{align}Z &= C + a + b \\ C &= cY, \\ Y &= Z.\end{align} Easily ...
0
votes
0answers
42 views

how to calculate expected utility for probability decision problem?

consider a decision problem: classifying $x$ as belonging to one of two classes $C_1, C_2$. there are prior probabilities for each class, $p(C_1), p(C_2)$ and likelihood probabilities for data given ...
2
votes
2answers
78 views

First mover advantage in a Stackelberg game

I am considering a simple game with two firms. Each firm faces the following demand function \begin{equation*} q_i(p_i,p_j)= a- b p_i + cp_j, \end{equation*} where $i,j\in \{1,2 \}$ and $i\neq j.$ ...
0
votes
1answer
107 views

Microeconomics : Total revenue for unit elasticity

I recently read in my microeconomics book that when elasticity = 1, total revenue remains the same. However, I am unable to follow this. I have two questions on this concept: Question#1- ...
0
votes
1answer
43 views

Application of Integration on Investments

A small business expects an income stream of $\$300$ per month for a period of $9$ years. The income will be invested at an annual interest rate of $17\%$, compounded continuously. How much interest ...
1
vote
0answers
16 views

Difference between Sequential and Weak Sequential Equilbria

This is in reference to the Game theoretic concepts as Nash equilibrium refinements. Sequential equilibrium are often defined as satisfying two conditions: consistency and sequential rationality. ...
0
votes
1answer
25 views

Recursive utilities in a repeated game

I am trying to set up utilities for an infinitely repeated game and I am having some trouble figuring out how to write the correct functional form. This game has a stochastic component where a ...
1
vote
0answers
32 views

Kuhn Tucker condition is sufficient for a global optimum?

$L$ is the variable and $s,r$ are parameters. The question asks to solve $max_{L\geq0}rf(L)-wL$ where $f(L)$ is twice continuously differentiable, strictly increasing and strictly concave. Then how ...
1
vote
2answers
35 views

Max price of a share

Company is planning to pay a dividend of 5\$ per share (dividend for previous year). Investor that wants to buy a shares of this company assumes that dividend will be stable (Thus will not change in ...
0
votes
0answers
9 views

Are distributed lag (DL) models with i.i.d. error always stable?

I am currently doing an econometrics course for which there is no textbook available, and cannot find the answer for the above question. I understand that a distributed lag model DL(q) can also be ...
1
vote
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\\ ...
0
votes
1answer
17 views

How to determine loan payment and total price of a loan

Let's say, that we have borrowed a $100,000 for a 5 years, with 6% p.a. interest rate. How can one determine the value of a loan payment, if we are making payments every quarter and at the beginning ...
0
votes
4answers
56 views

What are the restrictions such that $f'(x) = f(x)/x$?

Let $C(y) \geq 0$ denote some cost function. Let $MC(y) = C'(y)$. Let $AC(y) = \frac{C(y)}{y}$. I am considering the economic case where $$MC (y)= AC(y)$$ This boils down to a simple math problem as ...
1
vote
1answer
15 views

Microeconomics competitive equilibrium interest rate determination

I've got a microeconomics question that involves rearranging an equation with summation, where the only constant are $1$ and $r$. Firstly this is the equation stating that across individuals $i=1$ to ...
0
votes
0answers
9 views

Determine the system of difference equation for $(k_t, p_t)$

Now I have $ f : \mathbf{R}_{+} \rightarrow \mathbf{R} $ be defined as $f(k_{t}) = \frac{k_{t}}{\alpha + (1-\alpha)k_{t}}$ $ p_{t+1} = \frac{r}{n}p_{t} + \frac{\beta}{n}k_t + ...
1
vote
1answer
201 views

Economic Analysis / Minimization Problem

I am studying and going through some old exams for a microeconomic analysis class. I am just looking for some clarification regarding one of the answers given. The question is as follows Suppose ...
1
vote
1answer
33k views

Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$. Derive the ...
0
votes
1answer
50 views

Tail driven inequality [closed]

In this empirical model, Lj is a measure of left-tail driven inequality and Rj a measure of right-tail driven inequality. It represents what in this article? What exactly is the meaning of ...
1
vote
0answers
14 views

R squared conceptual question with respect to number of observations.

The following statement is true. However, I have difficulties to understand why. I would appreciate if someone could explain it conceptually or perhaps with or without reference to any formula. In a ...
1
vote
1answer
55 views

Kelly Criterion and mean variance optimization

I noticed that the Kelly Criterion resembles a ratio between the mean and variance in a continuous probability distribution. Now the mean and variance are important values in portfolio optimization ...
1
vote
0answers
13 views

consistency of variance MM estimators residuals

How can I prove with $Var(\hat{u})_t= E(\hat{u}^2_t)= (1- h_t)\sigma^2_0$ that MM estimator $\hat{\sigma}^2 \equiv \frac{1}{n} \sum_{t=1}^n \hat{u}_t^2$ is consistent? I can may assume that a LLN ...
0
votes
0answers
41 views

Game theory question inflation and macro

Suppose the Federal Reserve can fix the inflation level ˙p by an appropriate choice of monetary policy. The rate of nominal wage increase W˙, however is set not by the government but by an ...
1
vote
1answer
44 views

Log-linearizing $Y_t=\int_0^1 F(X_{it}) di$

I want to prove that log-linearizing the expression $Y_t=\int_0^1 F(X_{it}) di$ yields: $$Yy_t \approx F'(X)X\int_0^1 x_{it} di$$ Where: $\{X_{it}\}_{i \in (0,1)}$ is a continuum of strictly ...
1
vote
2answers
155 views

Find revenue, maximum revenue?

A manufacturer of tablet computers, after extensive research established the following price-demand, and cost functions: $p(x)= 360-20x$ $c(x)= 300+95x$ where $p(x)$ is the wholesale price in ...
0
votes
0answers
27 views

Finding the equation of a compound discount curve using two points.

I have a problem that goes beyond what I am capable of resolving. Basically I have two Net Present Values at different discount rates for a series of UNEVEN cash flows. As a reminder this is the NPV ...
0
votes
1answer
32 views

Game Theory Mixed Strategy Nash Equilibrium

I have been trying to solve this particular game in terms of mixed strategies, but I am unable to find the strategy using expected payoffs. Is there a way to solve this particular problem? There are ...
0
votes
1answer
1k 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 ...
1
vote
1answer
24 views

Understanding convergence of OLS estimator

From a linear regression with one explanatory variable, $ y = \beta_0 + \beta_1x+e$, the OLS estimator can be written as \begin{equation} \hat{\beta}_1 = \frac{\widehat{cov(y,x)}}{\widehat{var(x)}}. ...
0
votes
1answer
1k views

Maximizing total tax revenue with function Qs+-8+P and Qd=(80/3)-(1/3P)

The supply and demand equations of a good are given by Qs= -8+P Qd=(80/3) - (1/3)P P is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t on the supplier. ...