Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
482
questions
0
votes
0answers
15 views
Total factor productivity persistence
Would it make a difference whether a decrease in total factor productivity z on the labor, goods, and money market is persistent? (That is, whether decreased productivity today also is predictive of ...
1
vote
1answer
58 views
Linear Utility?
Consider a preference relation $\succeq$ on $X\subseteq\mathbb R^2$. If $\succeq$ satisifies:
$$
\begin{align}
&1.\mbox{ }(a_1,a_2)\succeq (b_1,b_2)\implies(a_1+t,a_2+s)\succeq (b_1+t,b_2+s),\...
1
vote
0answers
51 views
Dynamic programming, optimal consumption-savings (finite horizon) problem
Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$,
the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ ...
1
vote
0answers
62 views
What is Economic Interpretation of three nonlinear equations?
I have the following nonlinear system
$w$ is wealth
$c$ is consumption
$r(w)$ is gross return on wealth
$a,b, d$ are parameters which are strictly positive and fixed.
$$\dot{w} =r(w)w-c$$
$$\...
0
votes
1answer
30 views
The assumptions of Rational Expectations Models
What are the assumptions between rational expectations models and how restricted are there for the following results of economic theory? Where can I find them all gathered in some textbook or in the ...
1
vote
1answer
37 views
Regarding the Expenditure Function Underlying a Bliss Point
I've been looking at expenditure systems and have been really interested in the behaviour of the demand system that underlies bliss points:
Consider the bliss point utility function of the following ...
2
votes
0answers
25 views
Software used for solving demands of different functions
I'm currently doing some research on demand systems and have been experimenting with different underlying utility functions which will generate different systems of demands. However I've been doing ...
1
vote
0answers
29 views
Two different definitions for a Complete Relation
Many sources show this definition for completeness of a relation
$$
\forall a, b \in A, a \neq b, (aRb) \text{ or } (bRa) \tag{1}
$$
Others show only
$$
\forall a, b \in A, (aRb) \text{ or } (bRa) \...
4
votes
5answers
640 views
Is it possible to have a preference relation that is complete but not transitive?
I've been doing my own reading on non-rational preference relations.
Im currently under the impression that transitivity follows as a direct result of completeness of preferences. However my (much ...
3
votes
0answers
48 views
Existence of symmetric trembling hand perfect equilibria
Consider symmetric and finite game. By Nash (1950), the game must have at least one symmetric equilibrium (proof). Also, it must have at least one trembling hand perfect equilibrium (proof).
...
2
votes
2answers
102 views
Optimal Production Input in Relation to Cost Minimization Problem
I was doing my homework and got really confused about how to approach the optimal levels of inputs when there are three variables. My current understanding is that the problem is to solve the ...
2
votes
1answer
56 views
Regular annual pension
I am supposed to solve the problem:
A 24-year-old man decides to invest 200,000 euros at a 7% annual interest rate to bring him a regular annual pension from 31 to 50 years inclusive. What will be ...
1
vote
1answer
40 views
Short cuts to solve Cobb Douglas Utility function (minimization)
Say a Cobb Douglas like:
$$\max_{X,Y\: s.t. X \cdot P_x+ Y \cdot P_y=I} U=X^\alpha Y^\beta$$
When it comes to maximization I would do the following way (for the fastest result):
x: $\alpha/(\alpha +...
1
vote
0answers
75 views
Revenue maximization problem
There are $N>0$ Households in an economy.
The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones.
...
1
vote
0answers
28 views
The centralized shift from barter to currency economy
Suppose some ancient king of small bronze age city-state wants to introduce universal currency instead of barter that is currently in overwhelming practice in his kingdom. In order to smooth the shift,...
0
votes
0answers
9 views
Overlapping Generations Model Pension System Question
Part 1 Pension System
OLG Model with pension system:
Each individual lives up to two periods.
The surviving probability at period 2 is p.
At period 1, the young household consumes c1, saves s1, and ...
0
votes
2answers
110 views
Algebraic approach towards convexity
I have a function: $ u(x) = x_{1} + x_{2} + \min\{x_{1}, x_{2}\}$. How do we algebraically show if it's convex or not? Also, what would be the general way to show if any given function is convex.
0
votes
1answer
70 views
Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)
I have question on central bank loss function.
We know that the central bank loss function is
$$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$
And we know that fisher equation is $$i=r+\pi^e$$...
1
vote
0answers
41 views
derive value function from utility function
We have the utility function.
$$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$
And I am trying to find the value function.
$U$ is utility function. $c_t$ is consumption at ...
1
vote
0answers
41 views
Elasticity of substitution
So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:f(x,y)
this function have the elasticity of substitution(EOS): 1/(1-beta). a,...
1
vote
0answers
23 views
What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy
In my research I am trying to find minimal conditions to guarantee a quasi-equilibrium must always be a typical Arrow-Debreu equilibria in a rather specific production setting.
This may be rather ...
0
votes
1answer
54 views
Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed
Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed.
Saw this statement in the textbook but I'm not sure how this is the case when we don't have any restrictions on $u(x)$ such as ...
3
votes
1answer
45 views
Adverse selection problem
The classic literature refers to the problem where information asymmetry exists between an informed and an uninformed counterpart as the adverse selection problem, but how can we verify what kind of ...
1
vote
0answers
39 views
Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?
Quasilinear preference is defined to be:
$x \sim y \Rightarrow x+ae_1 \sim y+ae_1$ and $x + ae_1 \succ x$ with $e_1 = (1,0,0,...)$,
Given a quasilinear preference, if f $x \succeq y - ae_1$, does ...
1
vote
1answer
38 views
Supporting Hyperplane Theorem and quasiconcave utility function
My notes says that if $u(.)$ is strictly quasiconcave and differentiable, by the supporting hyperplane theorem, there exists $p >>0$ and $w \geq 0$ such that $ x = x(p,w)$ $\forall x$. I am ...
2
votes
0answers
46 views
Neoclassical Economic Growth Model Shadow Price for Discrete vs Continuous Time
I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional ...
0
votes
0answers
21 views
What is the the return to scale of per capita production function?
If $Y=f(AL,K)$ is CRS, $a_k+a_L=1$ by the Euler Theorem.
However, I wanted to know the return to scale of $y=f(1,k)$ (i.e. $Y$ divided by $AL$).
I tried $z=p/AL$ ,
then $py=f(p,pk)$, differentiate w....
0
votes
0answers
16 views
Questions regarding 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015
This question is with reference to the paper 'Efficient Trade with Interdependent Values' by Marek Pycia and Peng Wang,2015(See here).
At page no. 4 of the paper, the authors describe $v_i$ as the ...
0
votes
0answers
26 views
Modeling risk aversion. Expected utility function
I want to model an Expected Utility Function for risk aversion but my problem is uncertainty in itself.
I want a function(a special case) $$f(x, y) =\left\{\begin{matrix} h(bx^{1-C}+ay^{1-C}),C\neq 1\...
4
votes
2answers
82 views
Set Theory Properties of the Budget Constraint
In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is
Bounded
Closed
Convex
Non-empty
The 1. 2. and 4. are very straight forward and the benefits in terms of ...
0
votes
1answer
36 views
Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition
I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the ...
0
votes
1answer
25 views
Derive the growth rate of an equation
I have the following equation:
$$\mu =\left [s_{\pi }-v(s_{\pi }-s_{W})+\zeta \right ]$$
And I have to derive its growth rate, which is:
$$\dot \mu =-\frac{v}{\mu } (s_{\pi }-s_{W})\dot v$$
Do ...
2
votes
1answer
25 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
65 views
Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)
A specialized question for those, who excel in monopolistic competition and modern trade theories. I am interested in a derivation of an aggregate demand function for a model of monopolistic ...
0
votes
0answers
30 views
In this price model, employment grows with higher wages. Makes no sense
Using a Sraffian price model we have that, $p=(1+r)(pA+wl)$ where w (nominal wage rate), r (average profit rate) are scalars; p (market prices), l (labor input coefficients) are vectors; A (producers' ...
0
votes
0answers
20 views
Understanding lognormal distribution of stock prices
I am trying to understand how to correctly use the lognormal distribution. Let's say I have a list of Adjusted Price of a stock, I know that its daily returns is ...
2
votes
1answer
50 views
Abraham (1987) Simple Job Market Matching Model
I have a question about a derivation in Abraham (1987)'s simple job market matching model (equations 3 through 7):
She begins by writing down tautologies:
J - V = L - U = E
where J is the number of ...
1
vote
1answer
78 views
How do I derive Hessian of this function and check for concavity?
The function is $f(K,L)= AK^{a}L^{b}$ on the set of points $(K,L)$ with $K\geq 0$ and $L\geq 0$, assuming $A>0$
How do I find the Hessian and check for concavity?
0
votes
0answers
32 views
Moral hazard, bubbles, and economies of scale in limited companies and organizations
http://eprints.lse.ac.uk/100058/1/Goodhart_CEPR_DP13494.pdf
What would be the disadvantage of limiting the liability to the product of the proportion of the company and the debt(a person with 2% of a ...
1
vote
1answer
25 views
The notation cl co (A)
I stumbled upon this notation, (cl co(A+C), while reading "Set Optimization—A Rather Short Introduction" by Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff and Carola Schrage.
Is it ...
0
votes
2answers
91 views
Log linearising (Gali textbook)
In Gali (2015)'s textbook Monetary Policy, Inflation, and the Business Cycle, variables in levels are denoted with capital letters, logged variables with lowercase letters.
However, when a log-...
0
votes
1answer
54 views
Mathematical framework for modelling the relationship between price and sales of a product
In my job as a data scientist, I am required to model the relationship between the price of a product and the sales or number of unit sold. I am trying to build a simplistic model, the assumptions of ...
3
votes
1answer
61 views
Direct revelation mechanism's sets of strategies and types
Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows:
The employed notation is the following:
$θ_i$: Player i's type
$Θ_i$: Set of types for ...
0
votes
0answers
11 views
Calculating weighted average of operating margin (ROS) in benchmarking analysis
I'm looking for the formula for calculating the weighted average of operation margin data.
From all the sources I've found so far it calculated as following: the sum of all EBIT data divided by the ...
1
vote
1answer
16 views
Difference between social choice function and mechanism outcome function
Mas-Colell, Whinston and Green's Microeconomic Theory (3rd edition) defines the social choice function as the following:
Later, the mechanism outcome function is also defined:
The relationship ...
0
votes
1answer
65 views
Bolzano-Weierstrass Theorem and Pareto Efficient Allocation
Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.'
However, I couldn't ...
2
votes
1answer
34 views
Meaning of assumptions regarding continuum of agents in economics models
I am reading an economics paper which contains a model with households and banks. In the model, banks are owned by households. The authors make this assumption when they discuss households (section 2....
0
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0answers
15 views
Meaning or Interpretation of the denominator in the implicit function theorem
Consider $p=g(p,y)$. Rewrite this equation as $p-g(p,y)=F(p,y)=F(P(y),y)=0$, where $P(y)$ is the implicit function of $p$.
By the implicit function theorem, we obtain $\partial P(y)/\partial y=-F_y/...
3
votes
1answer
100 views
Maximising a partly concave and partly convex function
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable and strictly increasing function. Suppose that we are searching for the numbers $x_1$, ..., $x_n$ that maximise
$$\sum_{i=0}^{n}{f(...
