Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
500
questions
0
votes
1answer
17 views
Interpretation of dummy variable : Random Effects Model
I am running a RE regression and I have export similarity index between two countries as the dependent variable and have a dummy variable such as share a border(=1 if countries share border, 0 ...
0
votes
0answers
27 views
Why decreasing co-state variable (over time) is Compulsory?
I have a short question.
In the continuous time infinite horizon optimal control maximization problem Like
$$\max \int^{\infty}_0 u(c_t)e^{-pt}dt$$
I use current hamiltonian function and I obtain ...
0
votes
0answers
38 views
What is the economic interpretation of the solution of this optimal control problem?
I have the following optimal control problem
$$\max_{c_t} \int^{\infty}_0 e^{-p_it}\ln(c_t(i))dt$$
subject to $$\dot{w_t}(i)=rw_t(i) -n_ic_t(i)$$
$$w_0(i)=w_0>0$$
I have some wealthy and ...
1
vote
1answer
39 views
How to find mixed optimal strategies in this zero-sum game?
I'm trying to solve this problem from last year final exam in game theory:
Consider the zero-sum game $G=(X, Y, g)$ where $X=Y=[0,1]$, and $$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(...
0
votes
1answer
18 views
Is there some definition about risk sharing?
I was searching for a definition of risk sharing and I have found the following:
$\underline{Definition:}$ Risk Sharing — also known as "risk distribution," risk sharing means that the premiums and ...
-1
votes
0answers
12 views
New equilibrium when ad-valorem tax is imposed on supplier [closed]
In a question in a past final from my intro Microeconomics courses we're given the following demand and supply equations:
$P_s=50-\frac Q2$
$P_d=\frac Q2$
Which give an initial equilibrium quantity ...
-3
votes
2answers
32 views
What is wrong with the idea that resource depletion causes recessions? [closed]
How can you refute this theory, such that it never comes up again.
1
vote
1answer
29 views
Dimensional analysis of GDP
The GDP is sometimes given by
$$
GDP=P\cdot h\cdot e \cdot F
$$
Where $P$ is the Productivity, $h$ is the number of hours worked, $e$ is the employment rate and $F$ is the size of the labor force.
...
3
votes
1answer
45 views
Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$
I am interested in an economic interpretation for the ratio of partial derivatives of a demand function $Q_1(p_1, p_2)$, which is
\begin{equation}
t=\frac{\frac{\partial}{\partial p_1}Q_1(p_1, p_2)}{\...
1
vote
0answers
57 views
Writing constraint
A firm accumulates useful knowledge $k$ by investing in R&D activities. Specifically, if the firm invests $r > 0$ dollars into R&D, the stock of useful knowledge grows by about $2\sqrt{r}$ ...
3
votes
1answer
53 views
Can we have a Non-Reflexive Preference Relation?
I've been thinking about preferences alot recently and have been specifically thinking about the reflexivity requirement.
That is:
$$x \succsim x$$
Though this is apparent and obvious, I have been ...
2
votes
0answers
44 views
Finding optimal path in continuous time
I have the following optimal control problem
$$\max_{c_t,l_t} \int_0^{\infty} [ln(c_t)+\theta ln(1-l_t)]e^{-pt}dt$$
st. $$\dot{k_t}=k_t^{1/2}l_t^{1/2}-c_t-\beta l_t$$ $$k_0>0$$
I do big part of ...
1
vote
2answers
36 views
How to use an instrumental variable to estimate the parameter?
I have the following linear model of log wages (w) explained using years of schooling (S), years of experience and its square $(E,E^2)$ and 3 dummy variables indicating whether the individual was ...
3
votes
0answers
69 views
Bliss point hamiltonian function
I have the following utility function
$$u(c_t)=g(c_t-b)^2$$ for constants $g,b>0$
Edited: in order to be bliss utility function, It must be g is negative. But in the question g is given to be ...
1
vote
1answer
34 views
Linear Probability Model Instead of Logit in Fixed Effects Regression
In our panel data analysis we estimated a fixed effects linear probability model (LPM) instead of a fixed effects logit regression because our sample size was quite small (600 individuals) and the ...
2
votes
1answer
73 views
Present value heuristics problem
No idea if this is the right stackexchange for this, feel free to point me elsewhere!
I'm teaching business calculus and one of the problems the students have is to figure out the present value of a ...
0
votes
1answer
24 views
Consumption Set in Arrow-Debreu
I'm very inexperienced in mathematical economics, so when I came across the idea of the consumption set in Arrow-Debreu, I was a little confused. So for each element x in consumption set X, what does ...
1
vote
1answer
33 views
How to show that the estimator is consistent?
$Y_i=\beta_0+\beta_1X_i+U_i$ is my regression model for an I.I.D. sample with N=1000 observations. Suppose $U_i\sim I.I.D.(0,\sigma^2)$ and Xi are also I.I.D for i=1,2,3......1000. Xi is independent ...
2
votes
1answer
41 views
How to test if the effect of one regressor entirely comes from other regressors?
I have a regression model that includes IQ test scores as the dependent variable; my own education, my father's education and my mother's education as independent variables. Suppose I want to know ...
0
votes
0answers
16 views
Total factor productivity persistence [closed]
Would it make a difference whether a decrease in total factor productivity z on the labor, goods, and money market is persistent? (That is, whether decreased productivity today also is predictive of ...
1
vote
1answer
79 views
Linear Utility?
Consider a preference relation $\succeq$ on $X\subseteq\mathbb R^2$. If $\succeq$ satisifies:
$$
\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),\...
2
votes
1answer
91 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
77 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.
$$\dot{w} =r(w)w-c$$
$$\...
0
votes
1answer
34 views
The assumptions of Rational Expectations Models
What are the assumptions between rational expectations models and how restricted are there for the following results of economic theory? Where can I find them all gathered in some textbook or in the ...
1
vote
1answer
40 views
Regarding the Expenditure Function Underlying a Bliss Point
I've been looking at expenditure systems and have been really interested in the behaviour of the demand system that underlies bliss points:
Consider the bliss point utility function of the following ...
2
votes
0answers
25 views
Software used for solving demands of different functions
I'm currently doing some research on demand systems and have been experimenting with different underlying utility functions which will generate different systems of demands. However I've been doing ...
1
vote
0answers
31 views
Two different definitions for a Complete Relation
Many sources show this definition for completeness of a relation
$$
\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
648 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
52 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
106 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
44 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
79 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
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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
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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
111 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
74 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
46 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
1answer
36 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
55 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
48 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
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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
69 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
50 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
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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
87 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
38 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 ...
