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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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 ...
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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)}}. ...
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2answers
40 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
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 $n \times 1$.
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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 ...
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1answer
52 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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11 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
113 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
48 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
29 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
55 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
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 ...
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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 ...
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2answers
389 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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0answers
37 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 ...
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1answer
37 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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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 ...
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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, $f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$ ...
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1answer
47 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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1answer
42 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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1answer
39 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 ...
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0answers
37 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 ...
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0answers
64 views

Value at Risk, Confidence level, infimum

Given an uncertain future loss L which is modelled as a random variable with cdf Fl , the value at risk (VaR) at confidence level α is defined as $VaR_\alpha (L) = \inf\{l \in R | Fl(l) \ge \alpha\}$. ...
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1answer
46 views

Why is the derivative used to represent marginal cost instead of the difference?

Marginal cost is informally defined as "the change in the total cost that arises when the quantity produced is incremented by one unit." And given a total cost function $C(q)$ that's differentiable, ...
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2answers
40 views

Utility Maximization with a transformed min function

I was just wondering what the steps one would take to maximize the utility of a function of the form U(X,Y) = min{X,Y} + X subject to income I = $p_x$X + $p_y$Y where $p_x$ is the price of X and $p_y$ ...
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0answers
19 views

Deriving the formula for Fixed-rate mortgage using z-transform

So, using z-transform, one could easily derive the formula for fixed-rate mortgages. However, when I tries this, I noticed one thing I'm very uncertain of. The relationship for fixed-rate morgages ...
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0answers
22 views

Marginal Cost confusion, possible error in book.

I'm going through Morris Kline's "Calculus, an Intuitive and Physical Approach". One of the problems is about Maginal Cost, and it states (page 126, ex 11): If the cost C of producing $x$ units of an ...
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0answers
42 views

Proof on Preference Relations

trying to work out this problem, please find the theorem, the problem and my attempt below. Theorem: Let Z be a finite set and let X be the set of all non-empty subsets of Z. Let $ \ge $ be a ...
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0answers
39 views

Flat Tax vs. National Income Tax Average

If the national income tax average under a progressive system is 30%, will tax revenue change if the progressive system is changed to a flat tax also at 30%? In other words, if the mean average ...
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1answer
74 views

Solving a system of equations with Cobb-Douglas production function

I have two equations and two unknowns in the following: $$p \alpha x_{1}^{\alpha-1}x_{2}^{\beta}-w_{1}=0,$$ $$p \beta x_{1}^{\alpha}x_{2}^{\beta-1}-w_{2}=0.$$ After solving I am supposed to get my ...
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3answers
118 views

What does this notation mean as a production function: f = min [t, 1.05k]

I'm attempting to work on an econ problem, but I'm having trouble understanding the notation that the production functions are given in to even begin to attempt to solve the problem. This is what is ...
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1answer
70 views

Marginal products of labor and capital (cobb-douglas) [closed]

Given the cobb-douglas function $Y=1.01*K^{0.25}L^{0.75}$ $Y=$output; $K=$capital; $L=$labor How can I determine the marginal products of labor and capital? Not used in this equation. Thanks for the ...
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0answers
37 views

Local Non-Satiation Proof for utility functions

I have been having trouble with how to go forward with a proof for about three days now. I know the basic structure of the proof, but can't seem to construct it. Basically, I am trying to do a proof ...
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1answer
62 views

Net profit involving Capital Recovery Factor

I'm having trouble understanding the last bit of the solution to this problem: An industrial juicer costs \$45 000. It will be used for five years and then sold to a remarketer for $25 000. If ...
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1answer
30 views

Deriving Preference Relations

I have actors A and B who have a preference relations $\succeq_A$ and $\succeq_B$ on a set $X$. Both are complete and transitive. Actor A will report Actor B's preferences as his own if Actor B ...
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1answer
22 views
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0answers
37 views

Dynamic Optimization on Matlab

I have been working on the following (quite long but easy) dynamic optimization problem: $$ \max_{C_1} 2\ \sqrt[]{C_1} + \beta V_T(W_2)$$ Where $W_2$ is defined as follows: ...
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1answer
46 views

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 ...
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1answer
38 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 ...
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0answers
89 views

How to Prove that a Choice Structure Satisfies WARP?

Would you please help me to prove the following statement? If R is a rational binary relation on a finite set X, then $(B, c_R)$ is a choice structure satisfying WARP. Note: Only for clarification, ...
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0answers
52 views

If a choice structure satisfies WARP, then the underlying binary relation is rational.

Would you please help me to prove the following proposition: If $(P(X), c_R)$ is a choice structure that satisfies WARP, then $R_c$ is a rational binary relation? Clarification: For any nonempty ...
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1answer
71 views

$(P(X), C_R)$ may be a choice structure even if $R$ is not a rational relation.

Would you please give me an example to show that $(P(X), C_R)$ may be a choice structure even if $R$ is not rational (i.e., complete and transitive). Clarification: For any nonempty set $X$, let ...
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1answer
37 views

Equivalency of two definitions of WARP (Weak Axiom of Revealed Preference)

I have two definitions for WARP as follows. How can I prove they are equivalent? First Definition: $C(A) \cap B \neq \emptyset \Rightarrow C(B) \cap A \subset C(A)$ Second definition from ...
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0answers
42 views

How to differentiate without knowing the function, only the inputs?

Given a function $f(p, \alpha w)$, where $\alpha > 0$, how do I differentiate with respect to $\alpha$ ? The answer is supposedly $ w D_w f(p,w) $ (it says that it differentiated with respect to ...
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0answers
25 views

Implicit function theorem in microeconomics

Consider the problem $f(x_1,x_2)-w_1x_1-w_2x_2 -> max$, where the function $f(x_1,x_2)$ is twice continuously differentiable and whose Hessian matrix is negative definite. Show that $x_1$ can be ...
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0answers
24 views

Calculating the curve traced out by minimum points of short run average cost curve

$C(K,Q) = \alpha K + \beta\dfrac{Q^2}{\sqrt K}$ The Long run average cost curve is calculated by differentiating with respect to K which gives you K at the minimum long run cost, which you plug into ...
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0answers
21 views

Q Production function w/constant returns (Economics)

I recognise the production function experiences constant returns to scale, however I do not understand how to answer the question without knowing the value of $Y$. I have fiddled around attempting ...
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1answer
47 views

Show that a recurrence relation is bounded

I'm studying the following recursive equation as part of my Economics class: $$k_{t+1}={k_t}^\alpha - c_t$$ For $t=0,1,2,...$, $0\leq \alpha<1$, and $k_0>0$ and $c_t\geq 0$ for every $t$. I ...
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1answer
24 views

Given the utility function $U(F,G)=FG^2$, find the MRS

Intermediate Micro. My understanding for MRS conversions is that you find the derivative of each variable and place the former over the latter. That then in this case yields $\frac{1}{2G}$? Is this ...