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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75 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
49 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
50 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
58 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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2answers
43 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 ...
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1answer
87 views

Help to solve this problem, the result doesn't seem right :/ silly mistake somewhere probably

Suppose that Coke and Pepsi are the only firms producing cola. Their products are not identical, but are very close substitutes. Let $P_c$ denote the price of Coke and $P_p$ the price of Pepsi. Demand ...
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1answer
44 views

So I have the following question, dont have much info on class notes and not sure how to tackle it, any suggestions, any help?

A seller has a single item for sale (which she values at zero). There are two potential buyers. The seller decides to use the following auction format to sell the object: each bidder submits a sealed ...
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1answer
65 views

Conditions for a positve vector in null space

Precursor to problem: Preparing for an exam in General Equilibrium Theory. One of the important theorems is the No Arbitrage Theorem which states that for an economy with $S$ states of the world and ...
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0answers
37 views

Silly fraction question

I've got $$ MPK'=\frac{1-d}{1+r+x} $$ But I need it in the form $$ MPK'=d+r+x $$ How do I bring those up top?
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1answer
42 views

Horizontal product differentiation problem

It's a version of the Hotelling model (product differentiation). One firm is located at the beginning of a line between 0 to 3, one at the end of the line (so one at 0, one at 3). There are $b/2$ ...
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1answer
441 views

Increasing marginal product implies increasing returns to scale?

Setup Let $f(x,y)$ be twice differentiable in both $x$ and $y$. Assume $\partial f/\partial x>0,\partial f/\partial y>0$ for $x,y>0$. $f$ is said to have increasing marginal product of ...
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1answer
49 views

Utility optimization question

Having trouble with how to put this together. I have an answer key, but the individual steps I am struggling with. Two period economy with a representative consumer that maximizes the utility ...
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1answer
35 views

Economics question. Rehashing the basics of dealing with exponents

Struggling to put two equations together effectively again. I have my income equation: $$ y=zk^\alpha $$ And I'm trying to plug it into my marginal product of capital $$ MPK=\alpha z k^{\alpha-1} ...
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2answers
235 views

Third and average price auction

Third price auction: the winner is the highest bidder but this time instead of paying the second highest bid, he would pay the third highest bid. -assume there are at least 3 bidders. - Average price ...
1
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1answer
146 views

Conditional and unconditional variance in ARMA processes

How can I compute the conditional and unconditional variance of $x_t$ given its past if $\{x_t\}$ is an ARMA$(p,q)$ process. I'm literally struggling over that. Cheers
1
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1answer
548 views

Show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model.

Under the assumptions of the classical simple linear regression model, show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model. Anyone have any ...
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0answers
31 views

Properties of this set of functionals (mixed pairings)

(from the 4th page of http://www.math.toronto.edu/mccann/papers/econ.pdf) Let $X$ be a compact Hausdorff space, and let $\omega$ be a Borel probability measure on $X$. A Borel probability measure, ...
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1answer
197 views

Von Neumann–Morgenstern independence axiom vs. Savage independence theorm

Von Neumann–Morgenstern independence axiom: Savage independence theorem: What is the difference between the two? I'm think Von Neumann is talking about the prizes (outcomes) and Savage is talking ...
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1answer
45 views

Deriving Stochastic Euler Equation

If a consumer has utility function \begin{equation*} u(c_t) = ac_t - \cfrac{b}{2}c_t^2 \end{equation*} and present value budget constraint \begin{equation*} \sum_{j=0}^\infty E_t[\beta^jc_{t+j}] = ...
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1answer
115 views

OLS slope estimate of AR(1) with autocorrelation

I've stumbled upon this question in my econometric textbook and can't work out the right answer. The question: Consider $$Y_t = B_2Y_{t-1} + u_t\\ \\ u_t = pu_{t-1} + \epsilon_t\\ $$ where $ ...
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1answer
37 views

Solving optimisation problem with langrangian method

I have a a question regarding a constrained optimisation problem where there are three constraints. I was wondering if should include lagrangian multipliers for each of the constraints or whether the ...
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2answers
21 views

Variable with an exponent variable

I'm actually dealing with an economics problem, but it seems like the math is always what messes me up. Ignoring what the variables mean, I'm trying to understand how to get from step 1 to step 2. $$ ...
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1answer
53 views

Price and price elasticity problem in Economics

I cannot figure out this problem, if anyone can just point me in the right direction that would be great. The price is $\$20$ and the price elasticity changes from $-2$ to $-3.$ If $\$20$ was the ...
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1answer
12k views

Microeconomics: Calculating Tax Revenue and Tax incidence

Australian Government has imposed a tax on Beer. Assume that the tax on Beer is $20 per unit (a unit is a carton of drinks) Assume the demand and supply functions for cartons of Beers per week are: ...
2
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1answer
255 views

Euler Equation and Marginal Rate of Substitution

I was wondering if someone could help me clarify a result from my lecture notes. I have put them as a picture. It concerns the result on the last slide (the other three slides are included as well ...
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1answer
40 views

Question about plotting indifference curves

So I am doing a last year paper on microeconomics I m asked to find pareto efficiencies , cores and so on within the edgeworth box. So what I wanna know is that whe they give me in this exercise the ...
0
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1answer
111 views

Average cost function and marginal cost function proof

I was wondering if someone could help me prove the following. Let $A(Q)$ be an average cost function and $M(Q)$ the corresponding marginal cost function. Show that if $M(Q)>A(Q)>0$ then $A(Q)$ ...
4
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3answers
331 views

The Historical Importance of Keynes' A Treatise on Probability

A visiting speaker in Economics recently happened to mention that John Maynard Keynes' A Treatise on Probability revolutionized probability theory. I have not heard any such claim before and it struck ...
4
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3answers
502 views

How practically relevant is game theory?

I usually don't care too much about the practical relevance of nice mathematics :-) But this time, as I am looking to find some areas where I can apply maths and possibly collaborate with ...
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0answers
23 views

Optimize profit given complete market information

Assume there are $N$ market participants (on the order of several hundred), and $M$ items (several thousand) being bought and sold on a market. For each participant/item pair, you know how many units ...
2
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1answer
58 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
0
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1answer
70 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
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0answers
36 views

Introductory book about economic models with deterministic chaos

I'm looking for introductory textbook about economic models (micro/macro/finance) which incorporate deterministic chaos. Models could be with or without random noise. By introductory I meant master ...
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2answers
37 views

what are some typical systems of equations generating from practical problems?

I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc. Some examples or research articles would be good. Specifically, I am ...
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1answer
86 views

We are interested in price of a commodity, traded at regular intervals. Why it is reasonable to take $a$, $c$, and $d > 0$ and $b < 0$?

We are interested in the price of a commodity which is traded at regular intervals. We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. ...
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1answer
92 views

How to differentiate this expression with respect to t

I don't understand how to differentiate $f(tx_1,\cdots, tx_n)$ with respect to t.
2
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3answers
341 views

Precise definition of a “game of incomplete information” (Game Theory)

Question: In game theory, what is the precise definition of a "game of incomplete information"? What I've found so far: In the standard first year graduate economics textbook on microeconomics ...
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1answer
732 views

Sequence of learning mathematics from basic algebra to calculus.

What would be a step by step sequence of learning mathematics from basic algebra to basic calculus? I pose this question because I am in the process of self-learning mathematics as a preparation for a ...
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1answer
130 views

Equilibrium level of income question

Question: In the 2012 tax year, Consumption= (85% of income after tax) + (Autonomous Consumption) Taxation rate= 20% Autonomous consumption= 175 million Investment= 485 million Government ...
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2answers
325 views

Cobb Douglas production function problem

My question is related to the Cobb-Douglas production function: $Y= A \cdot L^\alpha \cdot K^\beta$ Assumptions: constant-returns to scale, meaning that when $L$ and $ K $ increase with a factor ...
3
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1answer
95 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
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0answers
199 views

Put-Call-Parity of Asian Options

I could need some help with deriving the put-call-parity for asian options. Let $S_t$ be the price of the underlying asset at time $t$ and set $Y_t = \int_0^t S_t dt$. Then the payoff of an asian ...
2
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1answer
40 views

Arbitrage opportunity for call price set on avarage

I have the following problem. Let C(K) be the market price of a Option Call with respect to the strike K. Let $C(100) = \frac{C(110)+C(90)}{2}$, then show that there exists an arbitrage opportunity. ...
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2answers
62 views

Point between left and right limits of a CDF

This is from the Chapter 15 text of Gourieroux and Monfort's Statistics and Econometric Models II: Set Up: Suppose that there are 2 possible parameter values $\theta_0$ and $\theta_1$ from which ...
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1answer
97 views

Welfare analysis after unit tax is imposed

image of welfare after tax This is what the lecturer drew in my class. $S_T$ is supply curve after a tax of $t$ dollars per unit is imposed .But I don't understand several things. 1. He said the tax ...
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2answers
59 views

Simple algebra, economics

Using some algebraic manipulation the expression $\displaystyle\frac{P-W_1P_x}{W_2}$ is made into $\displaystyle \frac{1-W_1 \frac{P_x}{P}}{W_2}$. It says it is simply multiplied by $\displaystyle ...
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2answers
2k views

Strictly Convex and Strictly Monotonic Preferences

On the assumptions for a well behaved preference, I read recently that a preference must be complete, transitive, continuous, strictly monotonous (if the vectors x≥y and x≠y, then x is strictly ...
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1answer
104 views

Optimization: Finding line connecting non-pareto-optimal allocation in Edgeworth Box to PO allocation

Two people, A and B, with respective utility functions of: $$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$ Total $X$ (that is, $X_a+X_b$) is fixed at $36$. Total $Y$ ($Y_a+Y_b$) is fixed ...
2
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1answer
692 views

The implications of Completeness and the Continuity axiom for utility representation

Completenes means that every basket of goods in some set previously defined is comparable with the use of a complete preference. Now, with the additional assumption that the preferences are ...
2
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2answers
115 views

Understanding basic stochastic differential equations

This is from a physics course in economics, the literature provides a bare minimum of mathematical explanations. I am trying to understand how to work with stochastic differential equations given in ...