Questions tagged [mathematical-economics]

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

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What is the math behind the coefficient of Absolute Risk aversion? [duplicate]

I have a good grasp of Calculus but I have never used the ratio of second and first derivatives. So, I am having a hard time understanding what it does and how?
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Find payoff matrix of B from payoff matrix of A [closed]

Here, I need to confirm that the answer of question (A) can be answer just by a(i,j) = - A(i,j) Where a(i,j) and A(i,j) are the elements of i-th row and j-th column of payoff matrix of B and A ...
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Violation of the transitivity axiom [duplicate]

I'm struggling to think of a way to violate the transitivity axiom. I was thinking about the following scenario: Let's say that you're voting on three different choices, A, B and C. If you pit A vs. B,...
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Derive Marshallian and Hicksian demand

How do I find the Marshallian and Hiksian demand functions with the utility of the following kind? Can I still use the Lagrangian? And how do I consider boundary cases?
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Question about an interpretation of the MRS

Given the marginal rate of substitution of $x$ for $y$ : $\frac{u'(x)}{u'(y)} $ I know one can interpret this as the amount of $y$ one is willing to give up for an additional unit of $x$, or the ...
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2answers
52 views

Returns to Scale Microeconomics

Are there any production function $f(x_1,\ldots,x_n)$ that is having decreasing returns to scale, given that the marginal product in every input $i$ in the function $f$ is constant?
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1answer
35 views

What is state space representation for DSGE modeling

I'm beginning with DSGE modeling, and a mathematical representation (perhaps trivial for most of the people that are more with this topic) is the space-state state representation of a dynamical model, ...
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23 views

What does it mean satisfy budget equality? [closed]

given a demand function x:Rn++ → R+ → Rn+ what does it mean x satisfies the budget equality? (The answer: it just means a consumer buys things until they've spent all the money that they have)
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Proving that Marshallian demand is of the form: $x_i^*(p,I) = \hat{x}_i^*(p)I$ with certain conditions

Can I please have some feedback/help proving the following. My proof is below but I am quite uncertain as to whether my solution is efficient. Thank you. If $u(x)$ is a homothetic utility, then show ...
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Optimization in discrete time

I have made optimizations in continuous time that belong to the control theory, for example one case: $\max(\min)V[u(t)]=\int_0^Tf(t,x(t),u(t))dt$ constraint to: $\dot x=g(t,x(t),u(t))$ Where: $x(t)$: ...
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Mankiw's version of the Cagan model - need help interpreting it

To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is $$m_t - p_t = -\gamma(p_{t+1}-p_{t})...
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Conditional distributions in model with continuum of agents

Many economic models consider a continuum of agents, $i \in [0,1]$. Suppose these agents have characteristics $(x_i, y_i)$, which are independently distributed. Are all possible values of $(x_i,y_i) \...
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Continuous time optimization with two laws of motion (the Hamiltonian with two laws of motion)

How would we deal with a continuous time optimal control problem with two laws of motion? Suppose we have the following RCK like environment with human capital investment. $$\max_{c(t),k(t),h(t)}\int_{...
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57 views

Continuous logit models - random utility with uncountable choice set

This question is about the mathematical foundations of the continuous logit model, as derived in McFadden (1976) (https://eml.berkeley.edu/reprints/mcfadden/math_theory.pdf) and Ben-Akiva et al (1985) ...
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61 views

Current Value Hamiltonian VS Present Value Hamiltonian in Economics

I've been looking at a number of optimal control problems and have been wondering under what conditions one should use the current value Hamiltonian over the present value Hamiltonian. Does it depend ...
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Topological intuition to continuous preference relation

For a Microeconomics Course, we are going through MWG, and in the lecture we discussed the notion of a continuous preference relation. A preference relation $\succsim$ on a set $X$ is called ...
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Some questions about Kyle's model in Continuous Auctions and Insider Trading (1985)

I was trying to understand Kyle'e Theorem 1 in page $1319$ in Continuous Auctions and Insider Trading in 1985. As we can see by the proof, this factor $\beta=\frac{1}{2\lambda}$ refers to the ...
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1answer
542 views

Is there an economic interpretation of the envelope theorem?

I wonder if envelope theorem has also some hidden economic interpretation. For example, Lagrangian multiplier in economics can have interpretation of 'shadow price' which is useful economic concept. ...
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2answers
252 views

Has anybody seen anything this expression before?

Suppose that $z(\cdot)$ is the demand function of an individual (consumer/investor) and $p$ is the price of the commodity/asset demanded. Does anybody know what is the intuition behind the following ...
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Are trig functions not used in economics? [duplicate]

I noticed that many mathematics textbooks designed specifically for economics students like for example Essential Mathematics for Economic Analysis - Knut Sydsaeter / Peter Hammond / Arne Strom / ...
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1answer
136 views

To what extent does economics use Mathematical Logic?

My economics program had a class that transitions or introduces to proofs with books like Bridge to Abstract Mathematics , Reading, Writing, and Proving: A Closer Look at Mathematics or A Transition ...
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1answer
51 views

Are there some examples of theoretical models that discuss pandemics?

I am looking for some papers/references that showcase theoretical economic models that incorporate pandemics. Most work seem to be purely empirical. Does anyone know of some papers that try to model ...
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1answer
50 views

Informed traders do know the cross section of the privately known signal between each other

I am having the following setup of privately known signals and I am trying to understand an assumption. Here, I quote the setup. Consider two agents idexed by $i=\{1,2\}$ and each one observes some ...
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105 views

How can I build a fixed point theorem argument in pure strategies?

To begin with, I am recalling the Banach Fixed Point Theorem. Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T:X\to X$. Then $T$ admits a unique fixed-point $x^*$ in $X$ ...
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62 views

Find a perfect Bayesian equilibrium

Each of two sellers, $1$ and $2$ owns one indivisible object that a buyer would like to buy. The two objects are identical. The buyer´s valuation depends on the number of objects he gets. The ...
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1answer
28 views

What is the probability of an unemployed worker receiving no job offer during a time period?

we are currently covering one sided search models and I had a question for you all. I kind of understand the raw calculus behind finding the probability of a job offer over a time interval h, but what ...
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Simplex Lp interpretation of dual problem´s solution

I am wondering whether my interpretation of my simplex dual problem result is correct. The primal problem is: ...
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18 views

Solving a HJB with additional constraints on control and state variables

I am trying to solve a Hamiltonian-Jacobi-Bellman equation with additional constraints on the state and control variables, but I am a bit confused on how to do that. In Intrilligator 2002, it is ...
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Numerical Backward Induction Optimal portfolio choice

I am currently considering a simple life-cycle problem. We consider a market with equity risk only, which follows a geometric Brownian motion. We seek to maximize the terminal wealth of a CRRA utility ...
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1answer
77 views

Game theory and differential equations

Does anybody know, any (advanced/classic) textbook that combines game theory with differential equations in mathematical economics. Specifically, I an interested in asymmetric information problems.
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60 views

Generalizing demand for perfect substitutes utility function

I have the utility function: $U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero) $\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
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50 views

How to optimize this dynamical system? Needing guidelines

I'm trying to solve a growth model, where the author indicates is a dynamical system. I want to ask if someone would help me with some guidelines of how to optimize this, I've been trying to solve it ...
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1answer
36 views

How to determine a corner solution? Growth Model

I'm working on a paper called "Fertility clubs and economic growth" of Ahituv and MOav (linked below) and arrived at this point: The paper shows three optimal conditions respect to ...
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164 views

Solving a HJB with a probability to transit to a new state

I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time. The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}...
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how to calculate Leontief demand functions from first order conditions of a CES function when sigma tends to 0?

This question is NOT about how to approximate a CES function to a leontief function. Knowing that: $i= good (\begin{array}{*{20}{c}} {1}&{or}&{2} \end{array})$ $j= firm (\begin{array}{*{20}{c}}...
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4answers
111 views

What are good advanced textbooks to learn mathematics for economist?

I am looking for books or other sources that focus on math that is above bachelor level (i.e. above just simple calculus). I am not looking for a specialized text for some field but just general ...
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25 views

How to choose the autocorrelation type ? MA(q)^2 or AR(1)

I have heteroskedastic and serrialy correlated (autocorrelated) panel data. I want to test it with both dynamic and static models. For the dynamic test, I use GMM and the results of GMM is parallel to ...
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1answer
64 views

Interpretation of $x c '(x)$

Consider a cost function that is continuous, differentiable and (possibly) convex: $c(x):\mathbb{R}^+\to \mathbb{R}$. I was wondering if there is a "common" way to interpret the expression: $...
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1answer
22 views

Expenditure function. Prove that this set is bounded

I need to prove that the following set is bounded in order to derive the expenditure function: $e(p,v)=min_x px$ ST $\{x \in R^n_+$ such that $U(x)\geq v\}$. Knowing that $U(x):R^n \longrightarrow R$ ...
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2answers
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Prove that budget constraint is Lower Hemi Continuos (LHC)

I need to prove that the following constraint is LHC. $B=\{x \in R^n : px\leqslant pw)$ But Im not capable of finding and sequence $\{x_n\}$ such that $x_n \in B(p_n,w_n) \forall n$ and that $x_n\...
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1answer
59 views

Can the weierstrass and the Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?

Question: An agent who consumes three commodities has a utility function given by: $u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$ Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’...
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Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form $y=f(L)$, for $L≥...
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46 views

Private signals [closed]

Following the literature of information asymmetry (see in Kyle among others), we have seen that many papers introduce a private signal, that is $$S=\tilde{v}+\tilde{e}$$ where, $(\tilde{v},\tilde{e})$...
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200 views

Why do we need the specific utility assumption?

In many microeconomic models, we read the assumption that agents have CARA normal utility preferences. We need the assumption of the utility to do pricing, but should someone do differently? Namely, ...
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28 views

Auction Makrets and delearship markets

The largest difference is that the NYSE is an auction market and the Nasdaq is a dealer market. In the former, the highest bid for a stock is matched with the lowest asking price. In the latter, ...
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1answer
76 views

Certainty Equivalent

The certainty equivalent is a guaranteed return that someone would accept now, rather than taking a chance on a higher, but uncertain, return in the future. Otherwise, some definitions say that the ...
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1answer
83 views

Help with weierstrass’ theorem

Question: Use the Weierstrass Theorem to show that a solution exists to the expenditure minimization problem of subsection 2.3.2, as long as the utility function II is continuous on $\mathbb{R}$ and ...
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1answer
38 views

Neutral technical progress

Recently I learnt that Cobb-Douglas production function has elasticity of substitution equal to 1, therefore it has neutral technical progress. Then Leontief production function has zero elasticity of ...
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19 views

Lerner Index Interpretation?

How do you interpret the Lerner Index? I ask because at Uni. we have been told that, for monopoly, when the producer max. his profit, he sets the price such as the demand is elastic and the lerner ...
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1answer
52 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 ...

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