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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2answers
407 views

Interpretation of market completeness: full row rank payoff matrix

Suppose that there are $K$ assets and $S$ states of nature. The assets' payoff is represented by the matrix $$ \underbrace{R}_{S\times K}=\begin{pmatrix} r_{11}&\cdots& r_{K1}\\ \vdots&\...
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1answer
87 views

Optimization to minimize cost using the function C=Tq^(1/a)+F

I was given the function of $C=Tq^{1/a}+F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available (also a ...
0
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1answer
2k views

Maximizing total tax revenue with function Qs+-8+P and Qd=(80/3)-(1/3P)

The supply and demand equations of a good are given by Qs= -8+P Qd=(80/3) - (1/3)P P is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t on the supplier. ...
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2answers
45 views

Chain rule using the expression F=150W^1/3

Suppose the attendence of a baseball game was denoted by W alone in the format F=(150W)^1/3. Is this function (strictly) concave or convex. Explain. To which I answered that it would be strictly ...
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1answer
74 views

Present Worth with Salvage Value

I would like to know how to consider the "Salvage Value" in the following question while calculating the present and future values. here is the question: We are planning to build a new bridge. ...
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1answer
123 views

Partial Derivative Math Homework Help

The attendance (denoted by the variable F , measured in thousands of fans) at a blue Jays home game is approximated by F = 150W^(1/3)P^(2/3) Where W is the fraction of the games they have won so far ...
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1answer
172 views

How to find the price elasticity of demand?

I need help answering if this is demand elastic of inelastic. A policy adviser suggests that in order to improve its balance of trade with china, Canada should lower the price of some heavy ...
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2answers
70 views

Finding the Maximum with Calculus, second order condition.

Question: "At a price of $8$ dollars per icket, a musical theatre group can fill every seat in the theatre, which has a capacity of $1,500$. for every additional dollar charged, the number of people ...
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2answers
392 views

Limit of CES function as $p$ goes to $- \infty$

I am trying to evaluate the limit of the CES Production function: $$Y=(aK^p + bL^p)^{1/p} $$ when $p$ goes to -infinity. It first yields the indeterminate form $0^0$. I tried solving the problem by ...
2
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2answers
256 views

Calculus of optimization help ):

If I need to sell 400 chairs. The price per chair is 90 dollars up to and including 300 chairs. Above 300, the price will be reduced by 0.25$ (on the whole order) for every additional chair over 300 ...
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3answers
248 views

Economics, numeraire, utility, demand, marginal rate of substitution

I typed my question in Microsoft Word and printscreen it instead of typing it, this is because I don't know how to type mathematical questions here, sorry for the inconvenience caused.
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1answer
104 views

Intemporal budget by lagrange

Assume that a representative agent lives forever and receives an endowment, denoted yt, in each period. The entire endowment sequence is known with certainty on date 0. The representative agent ...
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1answer
247 views

Easy (?) application of Lagrange multiplier

I am reading a book about utility theory and there is a exercise (without solution). I can't stop thinking about this, since the normal Lagrange multiplier approach seems not to work. We want to ...
0
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1answer
59 views

Was my inference regarding $u(z)=log(z)$ correct?

I have already solved the problem but would appreciate a clarification in part (b). A has initial wealth $w$ and faces a loss $l$ with known probability $\pi$. Insurance available at unit price $\rho$...
1
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1answer
69 views

Elasticity of demand the idea behind it.

I have been looking at elasticity of demand, but I am struggling to understand the concept. Now I have taken a simple example in hope of beeing able to understand what is going on. Eliacticity of ...
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1answer
60 views

How do I find if this estimator is unbiased and also its variance?

I need to find if the estimator $\tilde{\beta } _{2} = \frac{(y_{n}-y_{1})}{(x_{n}-x_{1})} $is unbiased given that i) $E(u_{i}\mid x)=0$ ii) $E(u_{i}\mid x_{i})=0$? I also need to calculate its ...
1
vote
1answer
626 views

Finding the optimal combination for the Cobb-Douglas function given a budget

I am trying to figure out to find the optimal combination of the Cobb-Douglas function given some budget. An example question is: Output can be produced with labour and capital according to $Q = L^...
1
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1answer
183 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: $r(t)=F(0,t)+\int^{t}...
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0answers
28 views

A sufficient condition for a good to be normal

Context: there are $2$ goods with prices $P_1$ and $P_2$ and the decision maker has the utility function $U(C_1,C_2)$. Denote $U_j=\frac{\partial U(C_1,C_2)}{\partial C_j}$ for $j\in\{1,2\}$. A good $...
13
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1answer
392 views

Equilibrium existence proof

Problem: Let $J$ be an integer and let $I$ be an integer multiple of $J$. Let ${\cal I}= \lbrace 1,2,\ldots, I\rbrace$ and ${\cal J}= \lbrace 1,2,\ldots, J\rbrace$. The set $X_{I,J}$ of all $\frac{I}...
0
votes
1answer
77 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let $P(t,...
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1answer
65 views

Non strictly convex “singleton” preferences

A relation $\succeq $ over a vector space $X$ is rational if it is transitive and complete. We say $x\succ y$ iff $x\succeq y$ and NOT $y \succeq x$ Moreover $x\sim y$ iff $x\succeq y$ and $y \...
6
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2answers
455 views

Applications of information theory in economics?

What are some direct applications of information theory in economics theory and/or finance? Any relevant articles, surveys, or book references are appreciated (especially if they are targeted to ...
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1answer
276 views

Marriage Market Proof (Alternative Proof of Rural Hospitals Theorem)

How do I get (a) + (b) + (c) $\implies$ (d) $\implies$ (e)? (a) Show that for each $m \in M$, if $\mu(m) = \emptyset$ for some stable matching $\mu$, then for the woman-optimal matching, $\mu_W$, $\...
0
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1answer
67 views

Spotting mistake: unnecessary given condition

I have solved the following problem without using a given premise. Could someone please spot whether I have done something wrong? Suppose we have a relation $\geq$ that is transitive, but not ...
0
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1answer
38 views

Is there a mistake with this national income model Mik. Wisniewski Intro to Math Methods in Econ pg. 61

I can't figure out the steps to this equation for the national income model. It seems simple, but I don't see why in step 4 I don't get Y-bY+tY I thought it was a mistake in the book at first, but ...
0
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4answers
41 views

Can someone walk me through how this expression simplifies to y/x?

I am just wondering how this equation comes to be: it is from an economics problem involving marginal utilities. I have my two variables, $x$ and $y$. Intuitively, how does $$\frac{0.5\times x^{-0.5}...
0
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1answer
33 views

Confused by informal math: total differentiation

I'm reading these notes that say: total differentiation gives $$ P=a_LW+a_KR\implies dP=a_LdW+a_KdR+[Wd(a_L)+Rd(a_K)]\tag{i}. $$ Please let me explain the notation: we can think of $R,W$ and $P$ as ...
1
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2answers
65 views

How to analyze convergence of non-linear difference equation (recurrrence relations)

I've a couple of functions, such as: $Y(t+1)=2-\ln(Y(t))$ $Y(t+1)=(Y(t))^{-2}$ $Y(t+2)=e^{-Y(t)}$ and I need to analyze stability and convergence. No problem with stability, but I can't figure out ...
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1answer
139 views

Functions of 2 variables and applications to economics

Given the production function $Q := \sqrt K + L^2$, determine the optimal level of production and the relative demand of the two inputs capital $K$ and work $L$. The cost of a unit of capital ...
1
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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: $\mathbb{Z} = \{\dots, -10, -9, \dots, 0, 1, 2, \dots\}$...
3
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1answer
126 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 ...
4
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2answers
267 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 ...
0
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1answer
55 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
70 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 $K_i\geq 0$, $L_i\geq 0$. $F_i(0,0)=0$. $F_i(...
1
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1answer
63 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 ...
0
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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, $$P(|X_n-Q|>\epsilon)\...
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1answer
128 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
95 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 ...
3
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1answer
75 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, ...
2
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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 ...
0
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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 ...
2
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2answers
362 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
276 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 ...
0
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1answer
512 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 ...
0
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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
157 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
177 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 ...