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.

learn more… | top users | synonyms

1
vote
0answers
20 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 $\text{CVaR}...
0
votes
1answer
52 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
26 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
45 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
24 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 \left\{x+[c+p(x)wy]\left[1-\frac{p(x)y}{y^{\...
0
votes
1answer
95 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 ...
1
vote
1answer
36 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
0answers
22 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
49 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
46 views

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

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 ...
2
votes
2answers
128 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 $i\neq j.$ ...
1
vote
0answers
18 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
26 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
39 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
12 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 ...
0
votes
1answer
53 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 ...
0
votes
1answer
18 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
57 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
19 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
15 views

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

Let $f:\mathbf{R}_+\rightarrow\mathbf{R}$ be defined as $f(k_t)=\frac{k_t}{\alpha+(1-\alpha)k_t}$ $p_{t+1}=\frac{r}{n}p_{t}+\frac{\beta}{n}k_t+\frac{e+s-1}{n}f(k_t)$ $k_{t+1}=\frac{\delta}{n}k_t+\...
0
votes
1answer
51 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 ''left-tail ...
1
vote
0answers
16 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
0answers
17 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
87 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 employer-...
1
vote
1answer
72 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 (...
0
votes
1answer
38 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
0answers
31 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 ...
1
vote
1answer
26 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
2answers
41 views

Non-Linear Second Order Differential Equation Regarding Elasticity

What methods can I use to solve the following equation? $$q''(p)=\frac{q'(p)^2}{q(p)}+\frac{q'(p)}{p}$$ I know from wolfram alpha that the solution is $q(p)=c_1p^{c_2}$.
0
votes
1answer
55 views

What is variance of $b$, the OLS estimator of $β$, when $u\sim N(A,σ²I)$?

When $u\sim N(0,σ²I)$ I understand how to determine the Var$(b)=σ²(X'X)^{-1}$ however when $u\sim N(A,σ²I)$ I do not understand how to find the variance. $A$ is $n \times 1$.
0
votes
2answers
33 views

Econometrics Conditional Mean

I have a question regarding linear regression. Suppose we have the following regression model: $$ y_{it}=\alpha+x_{it}'\beta+u_{it} $$ where say $i$ represents individual $i$ at time period $t.$ The ...
1
vote
1answer
54 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 ...
0
votes
0answers
12 views

Relevant information sets in economics

Economists often model the decision-making process of individuals. I'm trying to write down a reasonable definition of the "relevant information set" $\mathcal{I}$ an individual possesses when making ...
0
votes
1answer
117 views

Optimizing number of production runs?

I am having trouble with the following problem: A manufacturer of hospital supplies has a uniform annual demand for $180, 000$ boxes of bandages. It costs $20$ dollars to store one box of bandages ...
0
votes
0answers
55 views

Econometrics: quadratic specification, turning point and time-series

Given the following quadratic specification : ln(yt) = c + beta1*ln(xt) + beta2*ln(xt)^2 where t: represents time Ln: natural logarithmic c: constant yt: dependant variable at time t xt: ...
0
votes
0answers
10 views

Treatment Effect Approach and Selection Bias

Suppose to have the National Health Interview Survey data (NHIS), the health status of the observed people as outcome $y_i$, which has got different potential outcomes on the basis of a treatment ...
0
votes
0answers
30 views

Effective Abortion Rate

I have this equation: Effective Abortion Rate (EAR) for which there is this explanation: "It is critical to emphasize that Donohue and Levitt cannot identify cohorts with these state-year ...
1
vote
0answers
58 views

Unique Nash Equilibrium

If a game has a unique Nash Equilibrium, then does it have a unique Mixed Nash Equilibrium as well, where this MNE is the unique NE? The game I have in mind is the following (but I am more curious ...
1
vote
1answer
50 views

Formulating a game in an economic setting

I'm trying to teach myself Game Theory, and have come across the following question: Suppose that a company, $L$, produces left shoes only, and a company $R$ produces right-shoes. If $L$ charges $...
0
votes
1answer
21 views

Gradient of this function?

I have a function: $e(p_1,p_2,u) = \frac{p_1p_2u^2}{4(4p_1+p_2)}$ and I'm being asked to calculate the gradient vector with respect to p. That is, I want to find: $∇_pe(p_1,p_2,u)$ I understand the ...
1
vote
2answers
469 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
49 views

Find the set of undominated strategies in Cournot duopoly

Consider a version of the Cournot doupoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm $i$ is denoted $q_i$ and ...
0
votes
1answer
38 views

Expectations and Moments Variance

Suppose $E[X|Y=y] = a + by$ and $V[X|Y=y] = C + dy^2$ where $Y$ is normally distributed with mean μ and variance $σ^2$ . What is $V[X]$?
1
vote
1answer
30 views

Economics Application of Rates of Change

The Consumer Price Index ($CPI$) is a statistical estimate of the change of prices of goods and services bought for consumption. It is generally calculated by collecting the prices of a sample of ...
1
vote
0answers
19 views

Quasiconvexity analog for function with an integer domain.

Suppose I have a function that is not quasiconvex, as in the graph below, but would be quasiconvex if we cared only about integer points. That is, $f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$ ...
1
vote
1answer
51 views

In the most basic of terms, what does contraction mapping mean?

So, I have been solving a number of exercises involving contraction mapping. However, I am struggling to understand what exactly contraction mapping is at its most basic. All I really understand from ...
0
votes
1answer
45 views

Microeconomics Tax incident: unit tax imposed to consumer or producer

I come up with a thought that if there is any difference in term of imposing a tax to consumer or producer, I searched for the answer and there is some logic messing up.. let Pd be the consumer price;...
2
votes
1answer
40 views

Under what conditions does a convex objective function have a concave value function?

Suppose that $u:\mathbb{R}^{n} \to \mathbb{R}$ is a continuous, (weakly) convex function. Now define the value function $\phi$ to be: $$\phi(p,w) = \max_{x>>0} u(x)$$$$ \text{ subject to: } p \...
0
votes
0answers
38 views

Mathematical Economics/Environment

Bear with me - this is a long question. But I did do few of the parts, but I'm posting it because it seems necessary that you know what I previously did. Sorry for the "length"! Answers are given in ...