Questions tagged [mathematical-economics]

The application of mathematical methods to represent theories and analyze problems in economics.

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22
votes
2answers
30k views

How can I obtain Leontief and Cobb-Douglas production function from CES function?

In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function, $$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$ (where the ...
67
votes
13answers
20k views

Fundamental equations in economics

For the other sciences it´s easy to point to the most important equations that ground the discipline. If I want to explain Economics to a physicist say, what are considered to be the most important ...
4
votes
1answer
548 views

Optimization problem with Kuhn-Tucker conditions

Consider a game with two players where each player $i=1,2$ has preferences $u_i=s_i^a c_i^{1−a}$, where $c_i$ is consumption and $s_i$ is social interaction. $s_i$ is given by $s_i=t_i+t_{ij}\times t_{...
12
votes
4answers
10k views

Is complex analysis used in economics?

It's frequently useful in physics and engineering applications; are there any applications in theoretical economics? (If not, were there any attempts at incorporating CA that just never caught on?) ...
3
votes
1answer
850 views

Showing that production technology exhibits decreasing returns to Scale

The Question Suppose a firm has a production function given by $$y=F(L,K)=L^{1/4}K^{1/4}$$ where L and K denote inputs used in the production of y units of output. (a) Determine whether marginal ...
16
votes
3answers
878 views

Complete Markets in Continuous Time

In the standard discrete time economies with a finite number of states, $n$, a complete markets economy is simply an economy with $n$ independent assets (Think Ljunqvist and Sargent Chapter 8). This ...
13
votes
3answers
947 views

Consumer optimum in an economy with a continuum of commodities

Consider an economy with a continuum of commodities, with one commodity for each point in $[0,1]$. Suppose a consumer wants to maximise $$U = \int_0^1 c_i^\theta\,di\qquad 0<\theta<1$$ subject ...
6
votes
2answers
3k views

Local Non-Satiation Proof

I have been having trouble with how to go forward with a proof for about three days now. I know the basic structure of the proof, but can't seem to construct it. Basically, I am trying to do a proof ...
9
votes
2answers
1k views

When can one safely talk about decreasing marginal utility?

One thing I hear a lot is talk of decreasing marginal utility—the idea being that additional units of a good become progressively less attractive the more units of that good one has already. However, ...
5
votes
1answer
5k views

What is a homothetic function?

Suppose we have the following equations for the MRS of a utility function. $$U(x, y)$$ Which of the following corresponds to a homothetic utility function? $$MRS (x, y) = \frac{x^2+y^2}{xy}$$...
2
votes
1answer
211 views

Kuhn-Tucker and optimization (continue)

This is a question related to the question: Kuhn-Tucker optimization problem and game theory .The question is: Some cultures emphasize more social interaction more than others. Is there a role for ...
1
vote
0answers
102 views

Revenue maximization problem

There are $N>0$ Households in an economy. The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones. ...
0
votes
1answer
58 views

Calculating elasticity between terms in a regression equation

Given the following regression: $ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$ I am asked to calculate the elasticity of wages with respect to age. Is the ...
16
votes
5answers
4k views

Topological concepts in economic theory

QUESTION: What are the major or systematic applications of post-1960s mathematics to microeconomics? For example, in the late 19th century, Fisher first used the mathematical ideas of Gibbs to ...
14
votes
4answers
2k views

Use of mathematics and imprecise definition of terms

As a postgraduate student of economics I've been trying to expand my mathematical "toolset". While doing so I've talked to engineers, physicists and mathematicians, many of which have ...
13
votes
2answers
418 views

Uses of convex analysis in Economics

I'm taking kind of a crash-course in convex analysis to complement my mathematical skills and was wondering if anyone knew about nice ways in which this kind of tools were used in Economics. To be ...
14
votes
5answers
6k views

Applications of Trig functions in Economics?

Are there any applications of trig functions (ie $\sin(x)$, $\cos(x)$,$\tan(x)$) in economics?
13
votes
1answer
1k views

Solving the Hamilton-Jacobi-Bellman equation; necessary and sufficient for optimality?

Consider the following differential equation \begin{align} \dot x(t)=f(x(t),u(t)) \end{align} where $x$ is the state and $u$ the control variable. The solution is given by \begin{align} x(t)=x_0 + \...
5
votes
2answers
249 views

Foundational equations or concepts of Finance

For the other sciences it´s easy to point to the most important equations, inequalities, propositions or concepts that ground the discipline. If I want to explain Finance to a physicist say, what are ...
6
votes
4answers
16k views

Calculating rate of growth of per capita income

Given this question: National income is increasing by 1.5% a year and population by 2.5% a year. What is the rate of growth of per capita income? Attempt: Since per capita income is GDP/ ...
6
votes
1answer
160 views

Splittet Value Function and Hamilton-Jacobi-Bellman equation

General Problem Let $k\in\mathbb{R}_+$ be the state variable, $k=k^*$ a fixed point (saddle) and $v(k)$ a value function. The problem is, that the value function has two distinct functional forms, ...
3
votes
3answers
5k views

Alpha interpretation in Solow growth model

Consider the Solow model (without technology): $Y = F(K, L) = K^\alpha L^{(1-\alpha)}$ What's the economic interpretation of $\alpha$? Prove and argue the result. I see it as a share that goes into ...
11
votes
4answers
2k views

Why is the Cobb-Douglas production function so popular?

As relatively novice quantitative analyst/ Cost analyst, Ive been asked to estimate the level of a given organizations productivity more than once, and then forecast for the next couple of periods. ...
7
votes
1answer
296 views

Compute evolution of a distribution over time

We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the ...
6
votes
1answer
278 views

Envelope theorem for discrete choice sets?

If we have a function $$f(x)=\max_yg(x,y)$$ Then we can find the derivative $d/dx \ f(x)$ by realizing that $$(1): \quad \frac {\partial }{\partial y}g(x,y^*)=0$$ because of the first order ...
6
votes
1answer
283 views

Present value of a payment

Suppose I've just won 1'000'000 dollars in a game show. At the end of the program they tell me that they will pay me the prize as following: they will deposit in my bank 50'000 dollars every year for ...
4
votes
1answer
4k views

Mixed strategy Nash equilibrium in 3x3 game

What is the MSNE for the following game? I think you can eliminate strategies $A$ for player 1 and $C$ for player $2$, as these will are weakly dominated by all other strategies. Then, the game ...
4
votes
2answers
736 views

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 ...
4
votes
1answer
419 views

Aren't all cost functions step functions?

Long Question: Overview: While I understand that over certain intervals (i.e. less than the width of the step), they may appear not to be because the step is outside of consideration, and over other ...
3
votes
3answers
446 views

How to solve a linear forward-looking equation $x_t = \beta E_t[x_{t+1}] + k$ where $\lim_{t \to \infty} x_t = 0$ and $0<\beta < 1$?

How does one solve a linear forward-looking equation $x_t = \beta E_t[x_{t+1}] + k$ where $\lim_{t \to \infty} x_t = 0$ and constant $k,\beta \in \mathbb{R}$, $0<\beta <1$?
2
votes
1answer
110 views

How to relate real rate of return on capital to bond interest rate: Lagrangian

Suppose that household resource constrain equation is as follows: $$P_tC_t + Q_tB_t+ P_tI_t \leq W_tL_t+R_tK_t+B_{t-1}+D_t$$ where $P_t$ is price at time $t$, $Q_t$ is the price of one bond quantity, $...
0
votes
1answer
26 views

Hep with total differentiation of an AD function [closed]

Is there anyone who can help me with a total differentiation exercize. I am starting with the following formula for AD: $$x=\mu ^{-1}(g+i+e)$$ Where $\mu$ is the Keynesian multiplier. And have to ...
-4
votes
1answer
491 views

Question about budget constraint and utility maximization [closed]

I have also following budget set $$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$ where y is income. Assume that there are two stories. The agent can shop in both of them. The first store ...
6
votes
1answer
392 views

Does the envelope theorem hold at a corner solution?

Suppose we have the following production function: $$F(L,K)=\max_{L_K}H(L,L_K,K)=\max_{L_K}\left[(L-L_K+1)^\alpha(L_K+K)^{1-\alpha}\right]=(L-L_K^*+1)^\alpha(L_K^*+K)^{1-\alpha}$$ With the ...
6
votes
2answers
194 views

Multiple equilibria: which one to select?

There are two agents $i=1,2$. Consider the following programm \begin{align} &V_1(x_0) := \max_u \int^\infty_0 e^{-\rho t}F_1(x(t),u(t),v(t))dt\\ &V_2(x_0) := \max_v \int^\infty_0 e^{-\rho t}...
4
votes
0answers
61 views

How to econometrically identify perfect complements in production?

The production $$f(x_i,...,x_n)=\min\{x_i,...,x_n\}$$ is pretty straight forward and usually with smaller size data sets and can usually be picked up on rather quickly in an intuitive sense. ...
4
votes
2answers
564 views

Modifying Hotelling's lemma: Is this valid?

Hotelling's lemma is stated as: $$\frac{\partial \pi}{\partial p}=y$$ knowing however that on the more basic level, output $y$ is determined by the input(s) $x(p,w)$,let the profit function be ...
4
votes
1answer
424 views

How does the companies set up utility function for its own purposes?

How does the companies set up utility function for its own purposes? In another word, what is the types statistics data that is usually consider and statistical method being used to set up these ...
4
votes
4answers
1k views

list of math intense graduate level microeconomics books?

List of math intense graduate level microeconomics books? Except Reny's book, krep's books, varian's book and mas-collell's book books from subfields are acceptable, however by math intense i mean ...
3
votes
2answers
105 views

Properties of orders and preference relations

Suppose I have alternatives $A$, $B$, and $C$. If I have strict preferences, that means that for any $x,y \in \{A,B,C\}$ such that $x \ne y$, either $x \succ y$ or $y \succ x$. Assume transitivity, ...
3
votes
2answers
328 views

Correlated Equilibrium for Rock Paper Scissors

Consider the game of Rock, Paper, Scissors (RPS), with payoffs given as follows: Is there a correlated equilibrium in this game? Consider, for example the signal given to both players not to play ...
3
votes
1answer
2k views

Positive Monotonic Transformations and Nested Functions

Suppose there is an economic agent with the utility function $u(x,y)$. A second agent has the utility function $h(g(f(u(x,y))))$. Am I correct in thinking that if $f'(x)>0$, $g'(x)>0$, and $h'(...
2
votes
1answer
78 views

What is the difference between the demand schedule and the demand curve?

What is the difference between the demand schedule and the demand curve that a monopolist or monopsonist has? Could someone provide details for both cases or even some paradigm or a graph. I can not ...
2
votes
1answer
27 views

Rules of total differentials [closed]

What are the rules of total differentials? I. e., given an arbitrary expression of which to take the total differential, what rules can be applied to arrive at the desired result, and how does ...
1
vote
1answer
105 views

Short cuts to solve Cobb Douglas Utility function (minimization)

Say a Cobb Douglas like: $$\max_{X,Y\: s.t. X \cdot P_x+ Y \cdot P_y=I} U=X^\alpha Y^\beta$$ When it comes to maximization I would do the following way (for the fastest result): x: $\alpha/(\alpha +...
1
vote
1answer
327 views

Question on subgame perfect equilibrium

Consider a world of complete information with two agents X and Y and two time periods 1 and 2. Person X only lives in second period. Person Y lives in 1st and 2nd periods both. X and Y are each ...
1
vote
0answers
104 views

Kuhn Tucker optimization problem and game theory [duplicate]

Consider a game with two players, where each player i= 1 ,2 has preferences $u_i$= $s_i^a$$c_i^{1-a}$, where c_i is the consumption and $s_i$ is social interaction. $s_i$ is given by : $s_i$ = $t_i$ + ...
1
vote
1answer
315 views

What are some applications of Real Analysis in Graduate Economics?

I am interested as to what areas of masters/PhD coursework that learning the fundamentals of Real Analysis would be beneficial for? I am aware of its applications in Econometrics proofs and analysis, ...
1
vote
1answer
299 views

What is the concept of ordinal utility?

I have read in many books that since utility cannot be measured - so ordinal concept or comparison concept is used. If that is so, how can one define a mathematical function for utility which gives a ...
0
votes
2answers
127 views

Algebraic approach towards convexity

I have a function: $ u(x) = x_{1} + x_{2} + \min\{x_{1}, x_{2}\}$. How do we algebraically show if it's convex or not? Also, what would be the general way to show if any given function is convex.