Tagged Questions

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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11 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 ...
1answer
34 views

Demand and Cost Functions

Suppose the short run total cost function and demand function of a monopoly are: $T_C = \frac{1}{3} Q^3 – 10 Q^2 + 120 Q + 100$, and $Q=75-0.125P$ Find: (a) Revenue function and maximum revenue ...
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26 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 ...
1answer
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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 ...
1answer
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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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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 ...
1answer
19 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 \hat{\beta}_1 = \frac{\widehat{cov(y,x)}}{\widehat{var(x)}}. ...
2answers
31 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 $q(p)=c_1p^{c_2}$.
1answer
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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 $n \times 1$.
2answers
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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 ...
1answer
37 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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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 ...
2answers
51 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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9 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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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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21 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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22 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 ...
1answer
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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 ...
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 ...
2answers
65 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 ...
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16 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 ...
1answer
33 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 $V[X]$?
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Expectations and Moments

Suppose you invest 10000 in a fund which has expected value of 30000 in two years with a standard deviation of 2000. What can you say about the probability the portfolio value falls between 20000 and ...
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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 ...
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two period one good general equilibrium model

I know this is a math forum but I really need help for this question. Anyone could help? There is only one good in the economy that is consumed for two periods: X1 and X2 2) Two (groups of) ...
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17 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, $f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$ ...
1answer
26 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 ...
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The interim equilibrium levels

supposed I have a function as $$Y_i=(p_i-c+x_i)(1-b(p_i-p_j)+g(k_i-k_j))-hk_i^2-mx_i^2$$ The interim equilibrium levels is $$p_i=\frac{1}{b}+c+\frac{g(k_i-k_j)}{3b}-\frac{2x_i+x_j}{3}$$ I have ...
1answer
24 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 ...
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2nd derivative using rule of implicit differentiation (Economics)

This may well be a stupid question. I'm currently trying to find out whether a production function I have has convex isoquants. I'm aware I can find the derivative $\frac{dL}{dK}$ by using the rule ...
1answer
29 views

Problem with partial derivative in economic payoff function

I was looking through a paper that described a simple payoff function where there is an outcome variable $Y$ that depends on some causation variable $X$ and the payoff is given as some function of the ...