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.
1,540
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Deriving Hamiltonian with Lagrange multiplier method
This is probably a basic concept, but I am having a hard time understanding why in deriving the Hamiltonian with the Lagrange multiplier method 1) we introduce a time-varying multiplier for ...
- 191
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25
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ROR/IRR Analysis of mutually exclusive alternatives
I've got this homework problem, and I get two different answers solving it two different ways. I'd like to know which is the right answer and, if it's easy to explain, what is the flaw in the method ...
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1
answer
96
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Recursive economy in two islands
There are two economies A and B with infinitely many identical households. Since there are infinitely many identical households in each economy, they are all price taker. Maeasure of agents in A and B ...
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123
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Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies
Question from M.W.G-8B4
Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies.
My proof attempt
I assume $G_n$ the ...
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Every Nash equilibrium is subgame perfect Nash equilibrium
Every Nash equilibrium is subgame perfect Nash equilibrium
This statement is wrong. But i could not find a counter example
What are your suggestions?
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Optimization of convex function - proof replication
Trying to solve:
$$ \max_d \pi = \frac{(V^+ +2d_u + V^+\, d_u)^2}{16(V^+ - V^-)} \;\;\;\text{s.t.} \frac{4P_u}{1-d_u}>b-1 $$
And where $P_u = \frac{1}{8} (d_u-1) (V^+ d_u-V^++2
d_u)$
This is a ...
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1
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Cobweb recurrence relation
Consider the recurrence relation given by $2P_t + P_{t-1} = 3, P_0 = 0$. Does the graph of $(Q_t, P_t)$ look like a Cobweb where $Q_t = 4-2P_t$?
My calculation tells me that $(Q_t, P_t) = (2 + 2(-0.5)^...
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highly monotonic preferences and indifference curves
Show that if in $ R^2_{+} $ are continuous, highly monotonic preferences then indifference curves (i.e. classes of abstraction of relations ∼) are graphs of continuous, decreasing functions from the ...
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What is my mistake here? I was trying to find F/G factor here
I was trying to derive an expression for F/G factor here for homework. I have attached my working in this picture. I always end up with a (1+i) in the numerator which is not in the actual formula. ...
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21
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Finding cost function
The production function is as follows
$$f(z)=(z_1+z_2)(z_3+z_4)$$
Find the cost function?
What I did is as follows. But I am not sure about my solution. How do you solve it?
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Walrasian equilibrium with quasi linear function
There is a two-person exchange economy
Each agent has the following utility $u_i(x_i,y_i)=v(x_i)+y_i$ for agent $i=\{A,B\}$
Assume that $v$ is strictly concave and increasing function that has a ...
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1
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Representation of an indicator variable conditional on means
I am just starting out with math in uni for my research so this is a really simple question. I calculated the fraction of number of school enrollments during a policy period from 1991 to 1993 over the ...
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1
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The expectation of the product of an estimator with its transpose
I hope you are well. I have a problem that it seems to be really easy but I just don't know how to find the solution.
So, this is the question ; I have to compute E(Ŷ'Ŷ). For that, I have some ...
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51
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Iterated elimination of dominated strategies
I have a simple question that I am struggling to think about,
If a strategy survives IEDS does this it must be a best response?
For example the following game:
...
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1
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1
answer
25
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Asympotic Distribution
Question:
Let $Y_1,\cdots,Y_n $ ~$_{i.i.d} N(\mu_0，\sigma^2_0)$, now consider following 3 estimators for $\theta_0 = \mu^2_0$:
$\hat\theta_1 = \bar Y^2, \hat\theta_2 = \bar Y^2 - \frac{\sum_{i=1}^n(...
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1
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How do I substitute an expression into a summation term that includes a fraction?
I'm working on some economics homework and there is a problem I'm finding particularly difficult. The first question asks us to find the function that a firm will use to determine how much CO2 to emit ...
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1
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39
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Most Efficient Way Saving Money (Compound Interest)
If I have four children and I want to ensure each have £8,500 come there 18th birthday, assuming that I put money in each month, and gain 10% interest per year (taking into account compound interest). ...
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1
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123
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Elasticity of scale from cost function
I have a problem that I just dont seem to figure out how to solve.
Find the elasticity of scale when the following cost function is given:
3(n)^(2/3)(m)^(1/3)y^(4/3)
Where n and m are input prices of ...
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3
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47
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When differentiating $4x^2 + 3y^2 -3xy$ with respect to $x$, why does $3y^2$ 'disappear'?
When differentiating $4x^2 + 3y^2 -3xy$ with respect to $x$, why does $3y^2$ 'disappear'?
Not too sure where this $3y^2$ goes, or why $3xy$ can turn into just $3y$. Would love an explanation behind ...
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1
answer
40
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Annuity future value, getting extremely low value
The question states:
Relative to your current age, the present value at age 67 found in question #1 becomes
the future value you need to obtain given your current age now, How much money do
you need ...
- 169
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1
answer
46
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Suggested reading after course in game theory
I would like to hear some of your recommendations on books/papers or topics to research after having learned the basic vocabulary/mathematical framework of game theory.
I would like to gain some ...
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Dual Problems in Economics - Conceptual Question
Assume that I am solving a typical consumer problem in Economics in which I want to maximize the utility of an agent that consumes good $x$ and good $y$ and has a utility function $U=xy$ and a budget ...
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104
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How to find Marshallian Demand?
If the individual's utility function is given by:
$U(x,y)=(X)^{1/2}+(Y)$
With constraint:
$M=p_1X+p_2Y$
Find the Marshallian Demand functions for this individual.
So far I can:
Set the lagrangian: $...
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votes
1
answer
67
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Asymptotic Cone of Product Sets [closed]
We denote the asymptotic cone of a set $S$ by $A(S)$: given $S\subset \mathbb{R}^n$ and a natural number $k$, let $\Gamma(S^k)$ be the smallest closed cone at $0$ containing $S^k=\{x\in S: ||x||\ge k\}...
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Nash equilibrium in insurance pricing
The insurance market is considered to be a competitive market, so in order to study competition to determine a competitive premium, game theory seems to be a useful tool for studying those situations.
...
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1
answer
29
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Profit Max Cartel
2 Companies both selling the same luxury good. Company A cost to produce < Company B cost to produce, price for the good is the same (P). How do I address the issue of profit maximization if the ...
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0
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13
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Price elasticity over years, semesters, quarters, etc - Midpoint method
I'm trying to calculate price elasticity using the midpoint formula over different years, semesters, quarters, etc.
Taking in consideration the formula:
Midpoint method
Let's say we have this data:
...
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0
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14
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Economic indexing, how to use relative index to compose a ranking method?
I want to rank regions based on the quantity and "quality" of the jobs.
The following example will clarify mini research question:
Suppose for this months there have been registered 5000 new ...
0
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1
answer
82
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Question on binary relationship.
Condition A:
Given x, y in X such that $yRx$ then it follows that
$\lambda y +(1-\lambda)xRx$ for all $0< \lambda<1$
Condition B:
Given x, y in X such that $yPx$ then it follows that
$\lambda y +...
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answers
38
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Fair apportionment of a single resource over time
Suppose there is a single, indivisible resource that can be used or enjoyed by just one of two parties at a time. You can imagine a field of crops that can be plowed by just one of two farmers in any ...
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Find Pareto optimal allocations
There is an exchange economy with two people and two goods.
Utility functions are
$u_A(x_A, y_A)=\max\{x_A, y_A\}$
$u_B(x_B, y_B)=\max\{x_B, y_B\}$
Endowments are $w_A(1,\alpha)$ and $w_B(1,\alpha)$ ...
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Reflexivity of the preference relation
Advanced Microeconomic Theory by Jhele and Reny states the axioms of consumer choice: completeness, transitivity, continuity, local-non satiation, strict monotonicity, convexity and strict convexity.
...
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Stock price change, quantitatively
People say that when you sell a stock its price goes down, and when you buy it the price goes up.
This is clear, qualitatively, but I never really understood how much the price changes.
Could you ...
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Meaning of Mathematical Statements in Kelly Criterion
The Wikipedia article on Kelly Criterion states, "the Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric ...
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1
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Solving for the total derivative
In my Microeconomics course, we derive the theorem of Sufficiency of Consumer's First Order Conditions. We derive the following expression, where u(x,y) and the t just comes from the quasiconcavity of ...
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All-pay auction question
Two players take part in the following auction for a £1000 prize. The two players submit bids simultaneously, and the higher bid wins the prize (if bids are identical each gets £500). Both the winner ...
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1
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Current value of option
risk free rate=$r$
volatility of stock price=$\sigma$
continuous dividend rate=$q$
$a>0,K>0$
If your stock price S becomes below $K$ at maturity T,
the option A pays you $aS_T$.
Otherwise, this ...
- 159
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31
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Supermodularity of the product of two functions
Function $V(s,x)$ is supermodular in $s$ and has increasing differences with respect to (s,x). $x$ lives in a finite set $X$.
Function and $g(x,a)$ is the conditional distribution of $x$ given $a \in ...
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How to derive this optimality condition from Current value hamiltonian
I am reading a paper from https://academic.oup.com/ej/article-abstract/131/634/658/5876342
and don't know how to derive the optimality condition (7) from the current value hamiltonian. (p.663 in the ...
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Proving that $\exp(-x^2-y^2)$ is a strictly quasi-concave function
Prove that
$$f(x,y)=\exp(-x^2-y^2)$$ is a strictly quasi-concave function, that is,$\forall (x_i,y_i)\in \Re^2, (x_i,y_i)\neq (x_j,y_j), \lambda\in(0,1)$,
$$f(\lambda x_1+(1-\lambda) x_2,\lambda y_1+(...
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1
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Rebalancing a stock portfolio with a given amount of extra cash
Suppose I have an investment portfolio made up of $n$ stocks each worth $p_i$, with the count of each being $c_i$, where $i \in [1, n]$.
Let's say I've found that the optimal/target split of each ...
- 1,062
3
votes
1
answer
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Writing down Bellman equation in recursive macroeconomics world.
Assume an infinite horizon representative agent economy with the following consumer preferences $u(c_t)$
The production technology of this economy uses capital and land, which is fixed amount in ...
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1
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Partial derivative of distribution function
I'd recently read an article that said:
Let $F(q|p)$ be the distribution function of demand, where $p$ is the retail price. It's natural to assume demand decreases stochastically in price, i.e., $\...
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optimal control: What are the conditions imposed on nonlinear state evolution for a continuous and discrete optimal control problem
I have been looking at some texts on optimal control and its applications in economics. In particular, I am looking at a really nice book by Sydsaeter and Seirstad Further Mathematics for Economic ...
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14
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Coefficient of determination using direct, indirect, and net effect?
I've been reading a paper on regression modeling and there is a paragraph that confuses me. it states (Lipovetsky, 2001):
Many different techniques are used in applied regression analysis to
evaluate ...
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1
answer
33
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Conjecture: maximization of linear combination
For $x\in \mathbb{R}$ I have two continuous and differentiable real-valued functions $f(x)$, $g(x)$, which are single-peaked and maximized at $x^f$ and $x^g$ respectively, with $x^g<x^f$. Let $\...
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Textbooks with rigorous treatment of microeconomics without using local nonsatiation but strong monotonicity instead as main underlying assumptions
In the class I am taking, to simplify the proofs, the professor mainly uses strong monotonicity of preference( x > y for vector x,y is defined as $\forall i [x_i \geq y_i] \land \exists j[ x_j ...
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Hungarian algorithm for multiple items auction with unit demand; How to get market clearing prices?
The well known "Hungarian Algorithm" is capable of finding a matching in a weighted bipartite graph with minimal (respectively maximal) weight, and therefore can be used to find an optimal ...
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0
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32
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Shapley value of a parliament with infinitely many parties
Question:
A parliament has $n$ parties. The two main parties (we call them $A$ and $B$) have $\frac{1}{3}$ of the total seats each. The remaining $\frac{1}{3}$ of the seats are equally divided among ...
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Conditional maximization of consumer utility
I'm trying to solve the following consumer problem:
Consumers: The economy is populated by an infinity of homogeneous individuals who inelastically supply an amount L of work. The individual has ...
