Questions tagged [economics]
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.
1,602
questions
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61
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Constrained Optimization Insights
I have been experimenting with the following problem paraphrased from Khan Academy:
A manufacturer's revenue is $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the ...
- 1,904
2
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0
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94
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Subgame perfect equilibrium question
There are two neighboring towns, Alfa and Beta. Firm A is located in Alfa, firm B in Beta. The two firms produce and sell identical goods, with fixed marginal costs of c. The two firms simultaneously ...
- 6,458
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0
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42
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Derivation of a parameter in optimization problem
My question
The solution for first part
$$L=(aq+1-a)x1x2+\lambda (y-(a(p1+c)+(1-a)p1)x1-x2p2)$$
As a result after FOCs
$$x1=\frac {y}{2(a(p1+c)+(1-a)p1)}$$
$$x2={y\over 2*p2}$$
The solution ...
- 6,458
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0
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35
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How to show that a selection $b_1(v)$ is always less than another selection $b_2(v)$
I am an economist and need some math help.
My question is simply prove or disprove that $b_1(v) \leq b_2(v)$, which are selections of two maximization problems.
I have two objective functions. $v \...
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2
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0
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222
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Is there a generalization of the Kelly criterion where we add a fixed income/cost?
In the ordinary Kelly criterion the game is that you get to place a bet $B$ and your new wealth would be:
$$W^+ = \begin{cases}
W + cB & \text{ with the probability of }p\\
W - B & \text{ ...
- 16.7k
2
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0
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701
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Game with "pure" Nash equilibrium but not SPNE
after trying to find the answer for this problem for the last few days and crawling through loads of papers and notes, I decided to ask in case someone else has a good example for me.
I am trying to ...
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2
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0
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71
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Calculus of variations - Solving for function inside of a function of an integral
I am trying to solve a problem that I think I should use calculus of variations to solve (maybe optimal control?). I am trying to figure out the correct technique to use, or to figure out whether this ...
2
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0
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190
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Converse of the Central Limit Theorem
I want to know if the following statement holds. It is sort of the converse of the CLT. Intuitively, my guess is that it is true. However, I have no idea on how to prove it.
Suppose that:
$$\sqrt{n}(...
- 1,014
2
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0
answers
135
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Series representation of $({\bf I-A})^{-1}$?
I have a problem of the form
$$\mathbf{x} = \mathbf{A}\mathbf{x} + \mathbf{b}$$
where $\mathbf{x}$ and $\mathbf{b}$ are vectors and $\mathbf{A}$ is an invertible matrix. I can solve this problem for $\...
- 121
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383
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Prove $f(x^D)\geq1$: $f$ is a log-concave density and $\frac{1-2F(x^D)}{f(x^D)}=x^D-\frac{1}{2}$
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)}=...
- 5,704
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1
answer
461
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Find Bayesian Nash Equilibria
A rich, honest, but mischievous father told his two sons that he had placed $10^n$ and $10^{n-1}$ in two envelopes respectively, where n ∈ $\{1,2,3,\ldots,10\}$. The father then randomly handed each ...
- 21
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0
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70
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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 ...
- 21
2
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0
answers
76
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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 $\...
- 133
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50
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Average Cost of Obtaining in game Item
I know this will sound like a trivial maths problem, but recently I've been playing a game in which you can pay 5 in game gems to get a Rare (R), Super Rare (SR), and Ultra Rare (UR) characters ...
- 21
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0
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367
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Utility maximization of n goods
I have a question that involves finding the optimal demand of $n$ goods for a consumer. However, I haven't anything like this before and I'm not sure how to proceed.
The consumer has a utility ...
user131516
2
votes
1
answer
280
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Pricing a riskless asset in the Black & Scholes market
Consider a Black&Scholes Market where a risky asset evolves according to:
$$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ $$S_o=s$$
Riskless asset is associated with risk free rate r. I want to represent ...
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2
votes
1
answer
91
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Modified parcheesi game
A "modified Parcheesi" game starts with the following position:
First $x$ flips a fair coin. If heads he can move two spaces or pass. If tails he can move one space or pass. If he occupies the other ...
user198798
2
votes
1
answer
78
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What are the most recent devopments with applying fractals to economics?
I was researching fractals for my senior mathematics presentation and discovered that one of the most recent pioneers in that section of the field was able to apply fractal mathematics to the field of ...
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1
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315
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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(...
- 10.5k
2
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1
answer
162
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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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2
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1
answer
2k
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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 be ...
2
votes
1
answer
165
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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 ...
2
votes
1
answer
929
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
2
votes
1
answer
181
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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 ...
- 21
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0
answers
858
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Engineering Economics Cash Flow Diagram
I have the following question and solution below. What I don't understand is why is the 100,000 seen as savings/revenue when clearly it is coming out of pocket? Additionally, the monthly loan payment ...
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1
answer
182
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Use Roy's Theorem to prove that ...
I have an advanced microeconomics theory related question:
Use Roy's theorem to prove that $s_i(p,y)= -\frac{\partial v(p,y)}{\partial lnp_i}/\frac{\partial v(p,y)}{\partial lny}$.
This question has ...
user87274
2
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292
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What is the (expected) outcome of this hybrid auction?
A certain hybrid auction can be accurately modelled as follows.
There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$.
Each, except Zerro, has a private ...
- 3,210
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154
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Suggestions for projects in mathematics of finance
I am looking for computational projects related to mathematics of finance suitable for a senior level independent study for a student who has seen the green light, (or the light of the green!). I had ...
- 3,702
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2
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75
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Creating indices
Is there a "proper" formula for creating indices? I need to compute series of numbers into a KPI that can be tracked over time.
Example dataset is like this:
...
- 25
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0
answers
170
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Constructing and understanding stock-flow model
Suppose that $\textbf{x} = A\textbf{x} + B\dot{\textbf{x}}$ where
$\textbf{x}$ is vector of economic output level, $A$ is input-output
matrix, $B$ is stock-flow matrix. The system represents ...
- 105
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1
answer
219
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proof using (fixed point theorem)
I am seeking to solve for a Nash equilibrium in pure strategies $(d_2,d_2)$ involving two players, $1$ and $2$. Given that $h'(.)$ is s strictly decreasing and continuous function, $\Phi(d_1-d_2)$ ...
- 315
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1
answer
3k
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Unable to find Nash equilibria in mixed strategies
Here is the strategic form game:
Player 2
Left Middle Right
Top 2,2 0,0 1,3
Player 1 Middle 1,3 3,0 1,0
...
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2
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0
answers
171
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Can These Types of Curve be Differentiable in their Parameters?
This question arises from economic research attempting to model and estimate demand curves relating the quantity consumed $Q$ of goods to their price $P$. Assume $P > 0$, $Q \ge 0$, and as $P$ ...
- 4,197
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0
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214
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economic modelling asked for
I am putting together a proposal for a system of economic exchange which links a transaction of money through a period of time, and subjective relationship evaluation between two people. How can it be ...
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2
answers
172
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Help with understanding bond duration.
I'm currently studying for a pure maths degree, and so have no background knowledge of bonds. I'm reading through some material regarding actuarial work, and came across the following definition of ...
- 451
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1
answer
971
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Is "Real Analysis with Economic Applications", by Efe A. Ok, a good book for a self-teaching beginner in real analysis? [duplicate]
I'm interested in self teaching myself Real Analysis with the primary purpose of using it for economics, for this I decided to use Real analysis with economic applications by Efe A. Ok
Now what I was ...
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2
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A loan is to be repaid quarterly for five years that will start at the end of two years. If interest rate is $6$%..
I was helping my comrade answer some questions when we found this question. It goes like this:
A loan is to be repaid quarterly for five years that will start at the end of two years. If interest ...
- 1,495
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vote
2
answers
5k
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} $$
when $p$ goes to -infinity.
It first yields the indeterminate form $0^0$. I tried solving the problem by ...
- 63
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3
answers
2k
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why is this economics parabola sideways? switch x and y axis
Why is this "parabola" sideways?
Normally, f(x)=y, would be that x is graphed horizontally, and you would write that "wage rate is a function of hours worked", which is how I would normally think ...
- 165
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3
answers
196
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My brothers share from income.
My brother is driving a limo with his partner who told him that they will go $50-50$ on income and also $50-50$ on gas.
So if my brother earns $\$1000$ by spending $\$200$ on gas, what will be my ...
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2
answers
2k
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Von Neumann–Morgenstern independence axiom vs. Savage independence theorm
Von Neumann–Morgenstern independence axiom:
Savage independence theorem:
What is the difference between the two? I'm think Von Neumann is talking about the prizes (outcomes) and Savage is talking ...
- 115
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2
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2k
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Absolute value and sign of an elasticity
In my microeconomics book, I read that when we have $1+\dfrac{1}{\eta}$ where $\eta$ is an elasticity coefficient, we can write $1-\dfrac{
1}{|\eta|}$ "to avoid ambiguities stemming from the negative ...
- 305
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2
answers
506
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what does $\min()$ and $\max()$ mean in a function?
Could someone please explain to me what is the difference between these equations how it would affect the result and the way they are solved?
$$Q2(K,L) = A\min(BK,CL)\text{ s.t. }A,B,C > 0$$
$$Q2(...
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3
answers
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Inspiration for exam [closed]
To complete college (or what the equivalent in Denmark would be) I will have to write a text about a topic that I choose.
I have chosen to write about mathematics in business economics, and ...
1
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3
answers
3k
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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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1
answer
3k
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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^{...
- 9,306
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1
answer
423
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Normal distribution probability
just a quick question dealing with probability. The annual returns on stocks and treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal ...
- 127
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2
answers
170
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Why does compound interest usually have an $\frac{r}{n}$ term?
I assume discrete time. Suppose I start with some amount of money $P_{0}$. Then, using a simple rate of interest $r$ for a given period of time, I would have
$$P_{t}= P_{0} (1+r)^{t}$$
Suppose $t$ ...
- 4,939
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2
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161
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Lagrange multiplier with line constraint
So I am attempting to minimize the 2-dimensional plane equation
$$p(x,y)=ax+by$$
with the constraint
$$g(x,y)=x+y=c.$$
By the method of Lagrange multipliers I get
$$\nabla p=\lambda\nabla g\iff\...
- 25
1
vote
2
answers
48
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Show that for $f(x)=x^\alpha$, $f'(x+\frac{1}{2}a) - \frac{1}{2}f'(x+a) - \frac{1}{2}f'(x) \leq 0$
I am trying to set up a simple (economic) model to illustrate an argument.
I have an increasing concave production function $f(x)=x^\alpha$ with $\alpha\in(0,1)$ and want to show that
$f(x+\frac{1}{...
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