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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1answer
61 views

quasi rationality, interesting axiom of revealed preferences

So imagine there is a notion of rationality that captures the idea of "thresholds in preference." For example, let $\mathbb{Z}$ be the integers: $\mathbb{Z} = \{\dots, -10, -9, \dots, 0, 1, 2, ...
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1answer
123 views

A simple dual problem in economics: profit v.s. cost

The setup is simple but a bit lengthy. Please bear with me. Suppose that I have a production function $F(K,L)$ that is: constant return to scale; increasing in each factor: $F_K>0$, $F_L>0$ ...
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2answers
65 views

Optimization across markets - How can I solve?

I am unsure how to solve problems involving several markets and optimizing the price across all my markets. Note: I am looking to be pointed in a specific direction of study, not a solution to the ...
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2answers
259 views

If you have two envelopes, and …

Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch ...
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1answer
54 views

Does maximizing an increasing function of two variables in more favorable conditions always increase both inputs?

Consider the problem of maximizing $\sqrt{x}y$ such that $x+y=10$. By basic calculus we can show that the maximum occurs at $x=10/3$, $y=20/3$. If we loosen the constraint to $x+y=12$ then the maximum ...
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1answer
66 views

Verifying a production set is a convex cone

This comes from a paper that I am reading: For $i=1,2$, suppose that $F_i(\cdot,\cdot)$ satisfies the assumption: $F_i(K_i,L_i)$ is defined for all $K_i\geq 0$, $L_i\geq 0$. $F_i(0,0)=0$. ...
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1answer
61 views

Statistica Significane of the Slope Coefficient

Can someone please help me with this? Consider that you are examining the relationship between the height of children and their parents. You decide to collect data from 110 college students, and ...
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0answers
65 views

Convergence of probability

So I am getting ready for my first econometrics exam, and we have a lot of these plim (probability limits). Looking at the definition, I have that as n goes to infinity, ...
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1answer
127 views

Looking for resources for understanding derivation of demand from utility

I am struggling with my homework and would very much appreciate a rundown of the math or pointers to where I can find help otherwise. Quoting from the assigment: There are $n$ sectors in the ...
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1answer
92 views

Econometrics Question

Can someone please help explain this (or provide a website link). I know the answer is (a). To decide whether or not the slope coefficient is large or small, a) you should analyze the economic ...
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1answer
72 views

Matrix question: implication of $\frac{1}{n}X'X\to M$

Suppose $K$ is fixed and consider a matrix $X$ that is $n\times K$ and has full column rank. Assume that we know $$ \frac{1}{n}X'X\to M\text{ as } n\to\infty.\tag{i} $$ That is, as $n$ becomes larger, ...
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1answer
40 views

Show that $\frac{\mathrm{d^{2}}B }{\mathrm{d} A^{2}}> 0 $ if $U''<0$.

Given, $A = W_0 - L_0 + I - qI$, $B = W_0 - qI$, and $EU = p U(A) + (1-p) U(B) = k$, where $k$ is a constant. $\frac{\mathrm{d} B}{\mathrm{d} A}\bigg|_{}^{EU=k} = \frac{\frac{\partial EU }{\partial ...
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1answer
22 views

Inverse of a multivariable function following book derivation

I am trying to follow the text in the appendix, however I get stuck when I come to the part where I need to solve for q1. As Far As I can see I need to find the inverse, which I have seen examples off ...
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1answer
330 views

Good economics textbooks.

I would like a suggestion for the most mathematically fun/interesting mathematical economics textbook, preferably using abstract math. I want to prove theorems to complete my economics minor. I have ...
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1answer
78 views

Lagranges multiplier to minimize function of two variables with two constraints

I have a Cobb Douglas type production function with $K$ and $L$ as inputs; $\alpha$ and $1-\alpha$ as output elasticities and $C$ as efficiency parameter. Now I have to minimize cost $=wL+rK$ w.r.t ...
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1answer
272 views

Impatience and interest rate

I'm having difficulties solving the following problem in economics. I come from a mathematical background, and it's hard for me to get some of the terms: Consider a two-period economy with a ...
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1answer
492 views

How to derive demand function from a utility function without any knowledge of Lagrange Multipliers?

How do I derive the demand function for a utility function of, say, $U(x,y)=\sqrt{11x+11y}$ for goods X and Y in terms of $P_x$, $P_y$, and income $I$, with basic mathematics (basic calculus, but no ...
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1answer
1k views

Utility function, plotting indifference curves

I know how to plot indifference curves; simply take the utility function and plot some level curves in $2D$. But how to plot a specific indifference curve, so all bundles on it are indifferent to a ...
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2answers
148 views

Cost per item. Diminishing marginal discount, if you will. (Bigger discount for first few items) Optimal number of units to buy?

The graph above shows price per unit. Say they are cupcakes. When you buy a higher quantity, you get a lower price per unit. Say it levels off like this graph. Obviously, buying 2 nets a nice ...
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0answers
169 views

Finding the competitive equilibrium in an economy with two consumers and two commodities

I am unable to solve the following problem in general equilibrium. Consider an economy with two consumers and two commodities X and Y. $X_i$ and $Y_i$ are the amounts of commodities present with ...
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1answer
57 views

Utility function and preferences?

This might be more of an economical question, but perhaps some of you might've come across it before as math is involved; A person P prefers bundle (x1,x2) over (y1,y2) if (x1*x2) > (y1*y2). What is ...
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1answer
63 views

Utility Max Problem

I have a utility function $U(x,y)=\frac{xy}{x+y}$ and a budget of $200=2x+2y, P_x=P_y=2$. But for the first 50 units of product 1 sell for 2 dollars but for "$x>50$" the price of product 1 ...
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0answers
65 views

Follow-up on solution to Markov process equation

I asked a question here about solving a system related to an absorbing Markov chain. I now have a variation where there are $m$ types (of student, job seeker, etc) each of which applies to ...
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2answers
2k views

Simple differentiation / economics marginal cost question

This seems like a very simple question, so I'm sure I'm doing something stupid here, but I'm not quite getting my head around the following question: I have a total cost function: $C = 5x^2 +15x + ...
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3answers
89 views

Capitalization of interest

I've got quiet a strange problem but a simple one I guess. So I have a starting sum of 10 and I would like to know how many years need to pass to achieve 30000 with 32% interest applied every 2 years. ...
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1answer
184 views

Calculating the chance of something happening over and over again

I'm trying to calculate the probability, and potential cost on society, of people returning to homelessness after going through the system one, two, or several more times. Let's say that someone who ...
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1answer
88 views

Economic Applications of Game Theory

I'm currently looking at this course in economic game theory. However, when attempting this example: In this question you are asked to price a simplified version of mortgage-backed securities. ...
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2answers
125 views

Break Even Point

Suppose that x thousand units of a product will be sold when the price is p(x) = 50 - 1.25x dollars per unit and the cost of producing x thousand units is C(x) = 20x + 100 thousand dollars. a) What ...
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1answer
80 views

Variance of OLS estimator

Iam trying to understand how the variance of the OLS estimator is calculated. here is what i have: $E[\hat{ \beta} \mid X] = \beta$ and $V(\hat{\beta} \mid X) = \sigma^2(X^TX)^{-1}$ where ...
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1answer
93 views

Minimizing the cost of production by choosing between two options with different resource demands

Suppose that the firm has two possible activities to produce output. Activity $A$ uses $a_1$ units of good $1$ and $a_2$ units of good $2$ to produce $1$ unit of output. Activity $B$ uses $b_1$ units ...
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1answer
80 views

Need help explaining how the elasticity was derived [closed]

We had a lecture on calculating elasticity in microeconomics and I need help understanding how the answer was derived. I didn't quite get how we came up with the answer. My classmate said to just ...
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1answer
380 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 ...
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2answers
67 views

Problem understanding an implicit differentiation

Here is a general budget constraint: $p{_1}x{_1}+p_{2}x_{2}=M\Leftrightarrow \frac{p_1}{p_2}x_1+x_2=\frac{M}{p_2}\Leftrightarrow {p_{1}}'x_1+x_2=M{}'$. The main idea is that since prices are given, ...
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1answer
104 views

In Courty and Li (2000) “Sequential Screening”, what justifies the last equation in Lemma 3.2?

Regarding the article "Sequential Screening," in Review of Economic Studies, 2000 by Courty and Li: In Lemma 3.2, the last equality states that ...
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2answers
47 views

Rounding a real number w.r.t. a given amount of steps

Let $x$ be a real number, $x \in [0,1]$. Suppose a system can only provide a noisy signal about the value of $x$, given the granularity allowed by the system, $N \in \mathbb{N}^*$. I'm looking for an ...
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1answer
114 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- ...
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0answers
67 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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0answers
176 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
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2answers
788 views

Pure mathematics in our society

Is there some book or essay which deals with the sociological and economical justification of doing and funding pure mathematics? I'm looking for a modern version of Hardy's A Mathematician's Apology, ...
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2answers
55 views

income distribution from N, min,max and mediansalary, and total compensation

My goal is to obtain a reasonable approximation of the Gini index of a company (UBS). I need to obtain an estimate of the salaries distribution from publicly available data: Nuber of employees=60205 ...
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5answers
445 views

A mathematical approach to economics

Are there books or papers where economics is formalized and studied very rigorously? I am very interested in this topic. I would preferably like free online books and/or papers, but that is not ...
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1answer
60 views

Profit Model from Ellison Glaeser 1997

Can someone help me understand the following profit maximization problem (from Ellison Glaeser 1997)? Manufacturing plants choose where to locate each of $N$ plants in one of $M$ locations. The ...
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1answer
396 views

Expected revenue obtained by the Vickery auction with reserve price $1/2$

I would like to prove that the expected revenue of the Vickery auction with reserve price $1/2$ is $5/12$ when there are one item and two bidders the distribution of valuations are uniformly between ...
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143 views

preference relation.

In the exercise below I need to check whether the relation below is a preference relation ( need to be transitive (if $x>y$ and $y>z$ then $x>z$) and connected ). But I cannot find an ...
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1answer
70 views

Compactness and existence of Pareto-efficient cake partitions

I am trying to understand a fundamental statement in the theory of cake-cutting. BACKGROUND: There is a certain "cake" $C$ (a subset of $R^n$). The cake is divided among two agents, 0 and 1. Each ...
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1answer
96 views

Counterexample in axioms of expected utility theory

This is an exercise problem. Suppose $X_1=200$ with probability $1/3$, $0$ with probability $2/3$. $X_2=200$ with probability $p$, $0$ with probability $1-p$. $X_3=200$ with probability $1-p$, ...
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1answer
67 views

Expected utility representation

I am stuck on some question on utility theory. The question is as follow: Consider $A=[0,+\infty)$, and $Q=${F-cumulative distribution function on $A: \int^{+\infty}_0 x dF(x)<\infty$}, the set of ...
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2answers
55 views

Autocorrelation and var-cov matrix

$$Y_t=\beta_1+\beta_2 X_{t2}+\dots +\beta_k X_{tk}+\epsilon_t \qquad (t=1,\dots,T)$$ $$\epsilon_t=\rho \epsilon_{t-1}+v_t, \qquad v_t \sim \mathrm{i.i.d.}(0,\sigma^2_v)$$ GLS estimation under AR(1) ...
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0answers
60 views

Finding criteria for a household financial budget falsification

I’m working on a financial problem about budget of households. Households in a state fill a form about their net budget in every year and our insurance company investigate their financial status and ...
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50 views

$\operatorname{Var}(\hat{\beta_1})=\frac{\sigma^2}{\sum(X_i-\bar{X})^2}$ : how to derive this?

$\hat{\beta_1}$ is an OLS estimator for parameter $\beta_1$: $Y_i=\beta_0+\beta_1 X_i+\epsilon_i$, So $\hat{\beta_1}=\frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{\sum(X_i-\bar{X})^2}$ and ...