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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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 $Y$ ...
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13 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 ...
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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 ...
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1answer
29 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) }{ ...
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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 ...
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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 ...
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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,\ ...
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21 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 ...
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13 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 ...
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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 ...
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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. ...
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1answer
39 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
24 views

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

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 ...
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2answers
76 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 $i\neq j.$ ...
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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. ...
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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 ...
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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 ...
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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 ...
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0answers
8 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
18 views

General formula for the consumer surplus with linear demand function

What is the general equation for the consumer surplus when the demand is linear? I found a literature states that the consumer surplus equal with $$ C_s = 0.5D^2 $$ where $ C_s $ is consumer surplus ...
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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 + ...
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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 ...
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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 ...
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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 ...
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0answers
40 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 ...
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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 ...
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1answer
30 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 ...
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0answers
26 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 ...
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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)}}. ...
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2answers
37 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 $q(p)=c_1p^{c_2}$.
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1answer
53 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 $n \times 1$.
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2answers
26 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 ...
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1answer
43 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 ...
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0answers
10 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 ...
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1answer
67 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 ...
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0answers
19 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: ...
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8 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 ...
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0answers
27 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 ...
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0answers
29 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 ...
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1answer
47 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 ...
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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 ...