Questions tagged [mathematical-economics]

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

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A derivative with respect to Euclidean distance?

I have a utility function $u(x,z)$ from $\mathbb{R}_+$ to $\mathbb{R_+}$, where $x,z \in \mathbb{R}_+$. I would like to turn the following statement into math: "the utility function $u$ is ...
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Solve Joint Distribution From An Equation

In Postel-Vinay & Robin 2002 they show an equation: $$\left\{\delta+\mu+\lambda_{1} \bar{F}(p)\right\} \ell(\varepsilon, p)=\left\{(\delta+\mu) h(\varepsilon)+\lambda_{1} \int_{p_{\min }}^{p} \ell(...
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Right-to-manage wage bargaining (simple algebraic steps)

I am following (and trying to understand) a paper where the wage of unskilled workers is determined as the outcome of wage bargaining between a single union and a single firm in a right-to-manage ...
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Nash equilibrium with mutually exclusive actions

consider a zero-sum game where action space for both players are the same. I want to compute the equilibrium under the constraint that players do not use the same action. For example, if player 1's ...
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1 answer
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Convexity of preferences (dissimilar definitions)

Varian's Intermediate Microeconomics describes convexity as $$\text{Given } x, y \in X: x \sim y \implies \forall t \in [0,1], tx + (1-t)y \succeq x,y$$ The other definition I read everywhere is: $$\...
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Solve Value Function Analytically

The context is the Schumpeterian growth model with incumbent innovation (see Aghion, Akcigit, Howitt 2014 Handbook of economic growth): the value function is $$ r V_{t}(n)-\dot{V}_{t}(n)=\max _{z_{i} \...
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Average Discounted Payoffs in Continuous Time

While reading some theory papers that model a continuous time problem, I have noticed the way they represent the expression for average discounted payoff is of this form, where $r> 0 $ is the ...
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How do I calculate ROI if a cash flow is given?

In the image in the link below you see a cashflow. I wanted to calculate the ROI manually from this cashflow, but I don't seem to get it right. The program which also generates this cashflow gives me ...
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Do practicing economists actually use complex mathematics, or is it just a thinking tool?

I'm currently undertaking postgraduate theory units in microeconomics and macroeconomics for the first time and it seems like I am supposed to turn into an applied mathematician, not an economist. I ...
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Fundamental knowledge about invest, stock and economics for self learning [closed]

I am not expertised in the field of economics and stock so I do not know anything about invest. Would anyone please recommend any book/textbook for the initial leaner to get understanding of all these ...
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What would be some of the most basic and universal economic principles (independent of the school of economic thought)?

Comparing the principles of the neoclassical, Keynesian, neo-Keynesian, post-Keynesian and other schools, it is difficult to see a consensus on certain universal principles. There are ideas and ...
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Can someone please explain this housing prices equation?

Can someone please explain to me the following formula from this paper ? It's on the end of page 16 and the beginning of page 17. I don't know if it is asking too much, but I am looking at these ...
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Conventions for reading mathematically rigorous academic articles in economics

How would an informed reader typically go about reading a mathematically rigorous academic article in economics, with lots of notations, assumptions and postulations (e.g, a typical article published ...
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Risk aversion using multinomial choice model

Recently, I have stumbled upon this paper by R. Zhang. It introduces binary choice model to compute risk aversion equations for the natural selection process. My idea is to do something similar but ...
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Transversality Condition Requires Constant to be Zero

I'm a little confused. In Barro and Sala-i-Martin (2004, p. 208), it is mentioned that the transversality condition requires the constant to be zero, i.e. \begin{equation} \lim_{t \rightarrow \infty}\...
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Is this a function or vector space?

I want to make a function that states consumer $i's$ consumption $C_i$ in a shared bundle $X^n$ $\subset$ $\mathbb R^n$ depends on their preferences. That is, something like $C_i(\succeq_i)$. Since $\...
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Where to start studying the maths in economics?

I'm about to start my undergraduate in a few months and wanted to study some economics in depth. I already have some vague idea because of my background. However, most books recommended at that level ...
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1 answer
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How to mathematically denote that a consumer behaves according to their preference structure at every point in time?

As the title says, I would like to mathematically denote that a consumer behaves according to their preference structure at every time $t$, $t+1$, $t+2$ and so on in a finite consumption set $X$, by ...
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Nash Equilibrium with Constraints on Decision Variables

I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this: $$\max_{x_1} f(x_1, x_2)$$ $$\max_{x_2} g(x_1, x_2)$$ subject to $$x_1 + ...
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For the cost function question (equation in image), I have found TVC as AVC*P. However the MC curve derived is giving imaginary number solution

For the cost function question (equation in image), I have found TVC as AVC*P. However MC curve is showing an imaginary number as solution.
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1 vote
1 answer
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Confusing on the CRS Property of CES Function

Say a CES function is that $$Y = A\left[\alpha K^{\rho}+ \beta L^{\rho}\right]^{\frac{1}{\rho}}$$. Clearly this function is constant return to scale whatever the values of $\alpha$ and $\beta$ take. ...
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Preference notation by agent $i$

I want to say that the preferences of agent 1 on a consumption set $X$ are the same as agent 2's and the inverse of agent 3's (hence 2's is also the inverse of 3's). How would I do this using the ≽ ...
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1 answer
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Constrained optimization to find utility maximizing allocation

I am trying to find the allocation of goods X and Y in order to maximize utility between two consumers. The two utility functions are: $$ U1 = xy^5 $$ $$ U2 = 10xy $$ There are 8 of good X and 8 of ...
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Understanding the Formula For the Accumulation/Distribution

My understanding of the essence of the Accumulation/Distribution Index is that it tracks the closing price of a security during each period relative to its price range for that period, something like ...
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3 votes
1 answer
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Solow residual with cost minimization, calculus (Roeger, 1995)

I am trying to get a good understanding of the steps involved in solving the dual of a maximization problem, namely cost minimization. At some point (last two steps), the author ends up with the ...
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2 votes
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Definition of TFP in Acemoglu & Restrepo Ecta Forthcoming

The production structure in the paper: $$y=\left(\frac{1}{M} \int_{\mathcal{T}}(M \cdot y(x))^{\frac{\lambda-1}{\lambda}} \cdot d x\right)^{\frac{\lambda}{\lambda-1}}$$, where $$y(x)=A_{k} \cdot \psi_{...
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How to index the same variable by two different groups

I think this is a simple concept - I will try to be as clear as possible given that I do not have a full understanding and am currently self-learning. Suppose I have the amount of good $i$ represented ...
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High Kurtosis with normal Distribution

I need help. Here is my problem, I am testing normal distribution on covid, 2008 crisis, and beetween these. So I have 3 timezone and 7 indexes. When I use SPSS to test the normal distribution with ...
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which papers consider entry of buyers or consumers in a Salop (1979) circle?

I am considering a platform shaped like a Salop circle where sellers and buyers interact. I want to study free entry of buyers which amounts to an increase of the length of the circle or an increase ...
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How do we have no–envy for the Final Target Score here in assessing the consequences of individual changes on game dynamics and appeal?

Assume that a new Chess-variant designed to increase the level of competition throughout the game, provide additional excitement at the finish: "A Final Target Point will be set." The Final ...
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1 vote
1 answer
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How can I choose how much to invest in each stock of my portfolio?

I'm trying to find a mathematical way to decide what percentage of my capital I should invest in each stock of my portfolio to maximize my profit. Here's my attempt to the solution: Let's say my ...
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2 votes
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Is the mixed strategy $\sigma_i^*:S_i\times\Theta\to\Delta(A_i)$ chosen by player $i$ linear?

Suppose that we have a Bayesian game, where the number of players is $I$ and we refer to the generic player with $i$ $\Theta$ stands for the state of the world, where $\theta\in\Theta$ is the typical ...
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Loss function to State space

I have a quadratic equation and a loss function and need to write then in a state space format according to a model below. My equation is the following below, where T is the degree of a custom index ...
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I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?

I consider the following communication game. Suppose that we have $I$ players and each one of them learns a private signal $s_i=(s_{i,1},s_{i,2},...,s_{i,k})$, where $k$ is finite and also, every ...
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2 votes
0 answers
40 views

Optimal Control Problem with Bang-Bang Solution

I am reading Acemoglu, Robinson, Verdier 2017 jpe and stuck at the derivation of the optimal control problem in section IV.D. The current-value Hamiltonian of the problem is $$H(m(t), u(t), \mu(t))=\...
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4 votes
1 answer
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Convex CES Aggregator

I just find it seems that a CES aggregator e.g. $\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}$ with $\sigma<0$ is called a general convex aggregator and its limit as ...
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1 vote
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Cost per men-hour not adding up

In a company with two departments, we know the actual & budget of working hours (WH) and the labor cost (LC) of both of them. I want to compute the cost per men-hour (CPMH) and the economic impact ...
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1 vote
1 answer
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Understanding a paragraph in Debreu's Theory of Value

I wish to understand the following paragraph (from section $2.7$ of Debreu's Theory of Value): Imagine that a certain good circulates as money at location $s$, at date $t$, and let $k$ be the index ...
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Solving this budget constraint tangent to indifference curve without a graphical approach

Question: Let $V(x, y) = (1-\overline{p}) U(x) + \overline{p} U(y) - \overline{p} U(F)$ where $U$ is a strictly concave function ($U'>0$ and $U''<0$) with $U(0)=0$ and $0<\overline{p}<1$ ...
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Line equation tangent to convex level curve

Suppose we are given a differentiable function $f(x,y)$ such that $\forall$ $t \in \mathbb{R}$, $f = t$ yields strictly convex level curves. If we are given a line equation $y = mx + c$ such that it ...
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How does elasticity of demand of a public good affect tax rate?

Consider a situation with 2 consumers with one private good, and one public good; where consumers are endowed with fixed amount of private good $M_i$, and give up portion of their private good to ...
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2 votes
1 answer
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Why is human capital per worker multiplied by wage per worker?

I am studying a paper entitled "The Past and Future of Knowledge-Based Growth" by Strulik et al (2013) where the budget constraint is written as: $$w_{t}h_{t}(1-\tau n_{t})=c_{t}^{1}+s_{t}+...
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Proving that Every preference relation can be represented by utility function

Im looking the prove the following: every preference relation ⪰ over finite/countable set X can be represented as a utility function u such that ...
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2 answers
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How do you get to the formula L(t) = ln[L(0)] + nt on the Solow Model?

I understand the idea that the growth rate of a variable equals the derivative of the natural log of that variable, even so, i can't figure out how to get the following equation: L(t) = ln[L(0)] + nt. ...
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1 vote
2 answers
117 views

Conditions to use the Lagrangian method

I have seen that the prices and $\text{MU}_{i}$ are assumed to be positive (or, the preferences monotonic). This is always mentioned when a utility maximization problem is being solved with the ...
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3 votes
1 answer
92 views

Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?

This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts. To preface suppose we have the ...
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1 vote
2 answers
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What it is a utility function that it is quasi-concave but not concave?

If we have complete, transitive, continuous, and strictly convex preferences, then all the utility functions that represent them are strictly concave. I know that strictly convex preferences are ...
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3 votes
1 answer
180 views

Steady-state savings rate

I'm having trouble with the steady-state savings rate type of problems. Here is the problem I'm stuck on: The production is $Y = 0.5*K^{1/3}(AN)^{2/3}$. If savings is $s$%, what are the steady-state ...
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1 answer
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Optimization problem - wage determination

Could someone please show how the author derives the the first order condition (14) of this optimization problem using the expressions shown here ? For context this is taken from a labour market model ...
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1 vote
1 answer
59 views

What is a simple demand function that allows for different price and income elasticities than 1 and -1?

Cobb-Douglas utility functions assume price elasticity of $-1$ and income elasticity of $1$. Are there any utility functions with two goods that lead to a demand function, where you have the choice of ...
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