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Questions tagged [mathematical-economics]

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

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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 ...
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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),\...
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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 $w_{t+1}$ ...
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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. $$\dot{w} =r(w)w-c$$ $$\...
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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 ...
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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 ...
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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 ...
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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) \...
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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 ...
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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). ...
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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
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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 ...
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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 +...
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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. ...
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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,...
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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 ...
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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.
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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 $$i=r+\pi^e$$...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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' ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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/...
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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(...