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
684 views

What is my unit of observation for this time series data set?

My professor gave us a data set to run a regression and I have a data set which lists years from 1959-2007, gross private investments (in billions of dollars), and gross private savings (in billions ...
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2answers
24 views

Economics maximization problem linear activity

Consider the vectors: $a_1 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}, a_2 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}, a_3 = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1\end{pmatrix}$ Find a single ...
1
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0answers
29 views

Could math have predicted with certainty the banking crisis? (Why did it happen?) [closed]

The economics I do understand seem like trivial mathematics. Complex proofs are required for engineering involving safety - algorithm running times for pacemakers, strengths of physical structures ...
1
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1answer
47 views

Which derivative is correct?

Consider the following expressions: $$C_{i}=\sum_{j=1}^{N_i}v_{j}, \quad v_j \in \mathbb{R}, \quad N_i \in \mathbb{N} $$ \begin{equation} x_{i}=\frac{C_{i}}{N_i} \end{equation} I want to obtain an ...
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1answer
36 views

Applying Slutsky's Equation

Note: I asked in the Mathematics Meta regarding if it is permitted to ask Mathematics questions of economic nature. I also posted this question in the Economics Stackexchange but have not got any ...
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0answers
46 views

Proving a lower bound on the value of a probability density at the solution to an equation

Assumptions and notation Let $f$ be a twice-differentiable log-concave density function on $[0,1]$, and let $F$ be the corresponding distribution function. Define $x^D$ by: $$\frac{1-2F(x^D)}{f(x^D)}=...
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4answers
67 views

Find the maximum of $U (x,y) = x^\alpha y^\beta$ subject to $I = px + qy$

Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$. I have tried to use the Lagrangian function to find the solution for ...
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1answer
36 views

Expectation of concave transformation of random variable

Suppose I have two different discrete random variables $y>0$ and $x>0$. Now I want to compare two expected values involving these and a nonlinear transformation: When is one larger than the ...
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0answers
8 views

Is this considered a supermodular game?

Is this considered a supermodular game? I have $\Pi_i(x_1,x_2)$ which is a submodular function with decreasing differences in $(x_1,x_2)$. I know that $-\Pi_i(x_1,x_2)$ is supermodular in its own ...
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0answers
12 views

Equilibria in submodular games

Suppose I am solving a Cournot duopoly problem in which $U_i(x_i,x_j)$ is not for certain supermodular in its own strategy. Is the set of Nash equilibria nonempty? Can I find conditions for which the ...
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1answer
42 views

Economics: Nash equilibrium

Two firms, firm 1 and 2 , are competing in prices in two differentiated product markets. The demand for respective firms products are given by the following demand functions; $$ q_1(p_1, p_2)=a-p_1+...
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1answer
60 views

Economics : Game-theory (Nash equilibrium)

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way to solving following question. Question: Six students are going on a ...
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0answers
37 views

The simplest, advanced optimal pricing problem

Consider an arbitrary distribution, $F(x)$, with density $f(x)$ and the property of increasing hazard rate: ${f(x)\over1 - F(x)}$ increasing in $x$. Let $p^*$ maximize $p(1-F(p))$. Show that $p^*$ ...
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0answers
34 views

Applied Rational Choice Theory

I am a programmer researching the application of Rational Choice Theory. I have found many links to the philosophical nature of it. And fewer documents formalizing it's mathematical principles. ( ...
1
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1answer
38 views

Relationship between averages of $x^tx$ and $xx^t$ for column vector $x$

If we have data set $x$ as $m$ of $n \times 1$ vectors, and we know the average over index $m$ of $xx^t$ is $<xx^t> = C$, where $C$ is $ n \times n$ matrix. What is the average of scalar $ x^...
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0answers
15 views

Determine the system of difference equations for $(k_t,p_t)$

Let $f:\mathbf{R}_+\rightarrow\mathbf{R}$ be defined as $f(k_t)=\frac{k_t}{\alpha+(1-\alpha)k_t}$ $p_{t+1}=\frac{r}{n}p_{t}+\frac{\beta}{n}k_t+\frac{e+s-1}{n}f(k_t)$ $k_{t+1}=\frac{\delta}{n}k_t+\...
0
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2answers
452 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - y)$...
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1answer
43 views

Arrow's impossibility theorem implications [closed]

If we knew that in an unfair voting system (ie: it is known that IIA does not hold) then if (1) holds, how would we go about inferring that (2) holds and then is it possible to show that if (1) and (2)...
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3answers
74 views

economics using differential equations [closed]

I need help with this calculus problem The producer of a certain commodity determines that to protect profits, the price p should decrease at a rate equal to half the inventory surplus $S−D$, where $...
0
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1answer
41 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
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1answer
43 views

(Proof) Representing preorder $\succsim$ as a real-valued function.

I am reading a proof and having a hard time understanding on some parts. In math language, the proof is trying to show the existence of continuous preorder $\succsim$ (i.e. preserved under limits, ...
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2answers
33 views

Marginal revenue of a monopolist [closed]

A monopolist faces a demand function $Q=4000(p+7)^{-2}$. If she charges a price of p, her marginal revenue will be: (a) $p/2+ 7$ (b) $2p+ 3.50$ (c) $p/2-7/2$ (d) $-2(p+7)^{-3}$ Correct answer is ...
1
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1answer
24 views

Production functions total cost

Production function is: $f(L,M)=L^{1/2}M^{1/2}$. L is the number of units of labour, M of machines used. Cost of labour is 9 per unit, whereas the cost of machine is 81 per unit. Total cost of ...
2
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2answers
361 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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0answers
16 views

Two variable integral arising from an application of myersons lemma

We have the following equation. $f:[0,1] \times[0,1]\to [0,1]$ f is monotone. $$(x+y-1)f(x,y)=\int_{0}^{x} f(t,y) dt +\int_{0}^{y} f(x,t) dt$$ further we have the symmetry condition that. $$\...
0
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1answer
42 views

Calculate future price with continuous compounding

A forward contract for 4 months is entered into when a stock index is at 1000. If the risk free interest rate is 3% per year (with continuous compounding) and the dividend yield on the index is 2% ...
0
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0answers
36 views

Normalization in Economics (relative price)

I am reading a textbook in Economics and realized normalization of price is extremely common. (Reference: Microeconomic Theory (MWG, p 24)) Can somebody please help me understand two things: (1) ...
2
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1answer
445 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 ...
0
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1answer
44 views

For which values of $\gamma$ does this inequality hold?

Edited: Just realised my first post was somewhat misleading and not precise. Thanks to the two commetators that pointed it out. I am working on an article and ended up wondering for which values of $\...
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1answer
43 views

Complementarity and Substitutability

I am reading a paper in the international journal of game theory entitled Unequal Connections by Goyal and Joshi (2006) and it has the following sentences: "If strategic complementarity obtains... In ...
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1answer
32 views

Profit after price increase and volume drop

What happens with profit if prices increase by 10 pct and volume decrease by 10 pct? I guess the sales are determined as $$ TR = P \cdot Q, $$ so the changes results in $$ TR' = (1.1 \cdot P) \cdot (...
0
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1answer
36 views

Average cost and the level of output

A firm's average cost function is: $$AC = x^2 - 2x + 50 , \ 0 \leq x$$ a) Find the level of output to minimalize average cost (AC) function b) Find the level of output when average cost equals ...
0
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1answer
35 views

Discrete Uniform Probability: isn't my textbook just wrong?

My textbook is showing me examples of discrete probability distributions, one of them is in the picture: I learned in Calculus that the summation of the series $1/n$ where $n\to \infty$ is ...
1
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1answer
75 views

Does this inequality hold?

I am currently writting a paper and ended up with an expression that looks like the following: $$\frac{\sum_{i=1}C_{i}}{\sum_{i=1} {N_i}^\gamma {C_{i}}^{1-\gamma}}$$ for $i=1...K$, $C_i>0$, $N_i ...
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0answers
12 views

Econometrics, forecast, in sample and out of sample regression

Is there anyone who can explain me the difference between an in sample regresion, out of sample expansion regression and an out of sample rolling regressions? I have time series data from 1990 to 2016/...
0
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1answer
16 views

Elasticity of the function

Please find the elasticity of the function f(x) in x0: $$f(x) = \frac{ax^2} {1+ x^2} $$ $$ x0=10$$ Could you please check is done correct ? $$E_xf(x_0) = \frac{f'(x_0)·x_0} {f(x_0)} $$ $$f'(x)=\...
0
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1answer
50 views

Maximalize profit

Please advise on how to solve this exercise A firm’s profit is given by the function: $$\text{Profit} = 1200 - 6x^2 - 8x + 4xy - 10y^2 + 96y$$ where $x$ denotes output and $y$ advertising ...
1
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1answer
29 views

The level of output to maximize profit

Could you please give me any advices how to solve this exercise? Would be grateful for any help:) A firm's market demand function is $AR = 150-3x$ , and its total cost function is $TC= 0.1x^3 - 3x^2 +...
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2answers
33 views

Maximize firm's profit

Please give me any advices how to solve this. I'm good in maths, but I just don't understand these economical issues A firm's profit is given by the function: $PROFIT(L,K) = 50 L^{0.2} K^{0.8}$, ...
0
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1answer
26 views

Find change in gross profit after price change

How do you solve this question? A company has 3 products. They contribute to 30%, 30% and 40% of sales respectively. They have profit margins of 15%, 30%, and 50% respectively. If the ...
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2answers
22 views

Replication a options portfolio

I'm not too sure I'm going about answering this question correctly. I have used the following to derive my answer. I see that the portfolio above is a combination of a short-call(strike price 20) ...
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2answers
66 views

help for an integral

I need help calculating this integral: $$\int_0^x \frac{2(e^{\gamma u}-1)}{(\gamma+\kappa)(e^{\gamma u}-1)+2\gamma} du$$ I tried with the integration by parts but the situation seems to get ...
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1answer
163 views

What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for ...
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1answer
29 views

Counterexample to each Arrow's axiom

I am trying to find ways that combine group preferences which hold for 3 out of the 4 Arrow's axioms. Here is what I have come up with so far: Unrestricted Domain Not sure what scheme would ...
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1answer
32 views

Population and production in a dynamic system [closed]

I' trying to solve this problem but would need some help: Let $N=N(t)$ denote the size of a certain population, $X=x(t)$ the total product and $y(t)=X(t)/N(t)$ the product per capita at time $t$. ...
1
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1answer
16 views

cost minimization knowing the cost of labor and capital

The production function is $f(L,M)=4L^{0.5}M^{0.5}$ , where L is the number of units of labor and M is the number of machines used . If the cost of labor is \$100 per unit and the cost of ...
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0answers
31 views

Why is maximising expected monetary value not plausible for large amounts of money?

My teacher asked us which of these gambles do you prefer: $g_1=£500,000$ $g_2=0.1£2,500,000 \oplus 0.89£500,000 \oplus 0.01£0$ The EMV for each gamble is £500,000 and £695,000 respectively, ...
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1answer
20 views

Depletion Engineering Economy

An engineer sold his machine for $17,000$, after using it for 5 years. He bought a new machine worth $75,000$ with an expected life of 12 years, and a salvage value of $2,000$. The operating cost ...
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0answers
14 views

Which of these influence diagrams are not valid?

Which of these are valid influence diagrams? For those that are not, state the exact reasons why The squares, circle and diamonds represent decision, chance and value nodes respectively. (1) - ...
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0answers
27 views

Decision tree probabilities

I am solving the following decision problem. I formulated the problem into the tree OK but am unsure of what the probabilities along the branches are. i.e. how do I go from, for example, $P(H_1 \cap ...