Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
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Remainder term in Linear Approximations going to 0
A number of proofs in optimisation use the idea that the remainder term in either the differential or the Taylor Approximation go to zero. For example:
Some envelope theorem proofs:.
Necessity and ...
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Notation for ∆y used in Necessary Second Order Condition Proof
Edit: I have updated the link so that it works!
I was watching a lecture for a proof that if $x^*$ is a local maximiser of $f$ then necessarily the hessian is negative semi definite.
However i've got ...
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Quasiconvex Constraints in Maximisation
Why do we have to have Quasi-convex Constraints for constrained maximisation? I think i'm missing something pretty simple as this feels like a basic question:
My current Logic: If both the objective ...
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2
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Derivation of Labor Function
I want to know if I'm taking the right steps to derive a labor equation from a utility function. Suppose $U(x,L)=x^{0.5}+l^{0.5}$ where $L$ is labor, $x$ is our one good of interest, and $l$ is ...
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How to use Leibniz Rule of integration to find interest rate in Expanding Variety model
I'm studying growth theory from Barro/Sala-i-Martin and I stumbled upon a problem where some more advanced level of calculus is required in chapter 6 (Models with Expanding Variety: p. 294 eq. 6.18).
...
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Can anyone help with these calculations?
This is from the this paper in section $3$ about the two period example.
Suppose that we have the following two period, $t=1,2$, sender(S) - receiver(R) model.
For an action path $a=(a_1,a_2)$ and a ...
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Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Complementary slackness condition (CSC) state that
$\lambda_j[g_j(x) − c_j] = 0 \hspace{5pt} \text{for} \hspace{5pt} j = 1, ..., m.$ Therefore, every constraint either needs to be an equality ...
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Source for maximising profit of a company
I was wondering if anyone could point me in the direction of a source, like a case study, where profit was maximised for a real (simple) company. I'm learning calculus for economics and a real life ...
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How to model the payoff (or utility function) of the information provider?
After a thorough look in the literature of information design like Bergemann and Morris and Kamenica and Gentzkow I am still not so sure how the utility gain or payoff of the information provider/...
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Budget-feasible set in a portfolio choice problem
I am going through Duffie's Dynamic Asset Pricing book, and already ran into something that confused me on the third page. First, some definitions.
Let $\{1, \cdots, S\}$ be a finite set of states, $D$...
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Deriving Demand for Exponent Term for Quasilinear Utilities
Consider a quasilinear utility function: $u(x, y) = x+4 y^{.5} \quad \text{s.t.} \, I=P_{x}x+P_{y}y$. I know how to calculate the demand for good $x$, beginning with
\begin{align*}
&\cfrac{\...
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How to interpretate the difference between elasticity and the price effect?
Suppose I have two different models to predict the sold quantity (Q) of apples. For simplicity, price (P) is the only explainatory variable in the model.
I assume that the following holds true for all ...
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How to determine convexity or concavity of an indifference curve?
I am at my wits end. Maybe one of you can explain it to me.
A utility function is given: $U(x,y)=\sqrt{x^2−y^2}$ and we should determine whether the indifference curve is convex.
From the lecture ...
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When the choice space is X=N*[-1,1], how do you determine whether the following lexicographic preference can be represented by a utility function
Here is the lexicographic preference be given,
And here is the choice space,
I'm really struggled with determining this proof, can someone help to solve the problem or give some hint? Thanks so much!...
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Are there any ``sophisticated'' mathematical modelling where they solve for the utility function?
Are there any references in literature of any ``sophisticated'' mathematical modelling where they solve for the utility function under specific conditions using differential equations theory?
In such ...
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What is the risk aversion domain and how this could change in a dynamic market game?
Most of the market microstructure theory models assume a risk aversion coefficient, say $\gamma$ that is indexed with $i$ since any individual $i$ has her own $\gamma_i$ coefficient. Also, the inverse ...
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Nonlinear budget constraints (for quantity discounts)
I was thinking about quantity discounts and if there is a possibility to model them not as bundles (as is typical for second price discrimination) but rather as prices being some continous functions ...
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In the Diamond-Dybvig model, why does the long-term investment not offer a liquidity risk premium?
In the basic version of this model (see link below), both the short-term asset (deposit) and the long-term investment (“technology”) are considered to be risk-free (that is, there is only one possible ...
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Solow model continuous time - definitions
Given that the per-capita capital $ k = \frac{K}{L}\ $ (total capita divided by labor force), and we want to find $ \dot{k}\ $, it seems that
$ \frac{\dot{k}}{k}\ = \frac{\dot{K}}{K}\ - \frac{\dot{...
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What is the intuition behind the different information structures from the static to the continuous time ones?
It is a difficult and challenging problem at the same time, to model the information structure in theoretical models of economics and finance. The information structure in most of the literature is ...
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What does it mean when I say that CDF is bounded away from 1?
Suppose $\theta \in [\underline\theta, \bar\theta]$ is distributed with CDF F(.). What does it mean when I say that this F is bounded away from 1? Does it mean that F can never take the value 1 in ...
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Geometric Interpretation of the Potential Function of a Game
One geometric interpretation of (at least one term of) the potential function I've come across is as the Riemann-approximated area under an individual player's cost as a function of the number of ...
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estimation of certainty equivalent without given utility function
The body of question is:
Assume the decision maker is risk averse, $u(40)=\frac{1}{2}(u(0)+u(100))$, $u(m)=\frac{1}{2}(u(0)+u(180))$, try to estimate the range of m.
It is easy to get the infimum of m:...
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Understanding the terminal condition in the Taylor rule
I am currently reading this summary of various theories on how central banks control inflation. However, I got quite confused in section 3.2.4. Here is the context.
Suppose we have a log-linearized ...
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1
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85
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Rationalizable strategies/ Nash equilibrium
For the question below, how can we solve it generally for every value of θ? As the θ is not discrete, I am not sure how to apply iterated elimination of dominated strategies in this question. And is ...
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How to convert unchained nominal US personal income values to 2012 chained dollars
I have 1980-2021 data for US personal income (nominal, unchained) and I need to convert all those years to chained 2012 dollars, and then convert it from nominal to real.
I looked up the formulas and ...
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CALCULATING REVENUES OF EACH PRODUCT IN RSTUDIO [closed]
Hello everyone,
I need to work with the data presented above. This question involves a little bit of knowledge of Rstudio, but I assume some of us work with this programming language throughout our ...
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How to prove that these level curves intersect
I have two functions:
$$F(x,y) = U_1'(c_1)(a_1-b_1)+U_2'(c_2) = 0$$
$$G(x,y) = U_1'(c_1)+U_2'(c_2)(a_2-b_2) = 0$$
where $c_i=a_i(x-y)+yb_i, i=1,2.$ the utility functions $U_1,U_2$ are twice ...
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Why isn't this state a price equilibrium with transfer?
This is probably a silly question, but I am misunderstanding something about the definition of a price equilibrium with transfers. Consider an Edgeworth Box (2 goods, 2 agents) consumer economy with ...
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Derivation of index decomposition analysis
I’m currently reading a paper on index decomposition. The paper is here for reference : https://www.sciencedirect.com/science/article/pii/S0140988315001772
The paper is setting out how it has gone ...
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Monopoly markets
A monopoly can produce any output level at a constant marginal (and average) cost of θ per unit. Assume the monopoly sells its goods in two markets separated by some distance.
The demand curve in the ...
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How to prove Kelly criterion maximizes wealth compared to other betting strategies, and how to generalize such a proof
The Kelly criterion approach to a betting situation basically involves maximizing the expected logarithm of one's wealth. The assumption is that maximizing the logarithm appears to inadvertently ...
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Reference request: book on international trade theory will full mathematical derivations
I am doing some self-study on international trade theory, and have been using the book by Feenstra Advanced International Trade Theory as my textbook. The book is good and relatively rigorous, but my ...
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Differentiating over multiple time horizons to get FOCs
First of all, I'd like to say sorry if I couldn't be more specific in the title, I really tried to synthesize the core of my doubt.
I was reading The Econometric Analysis of Calibrated Macroeconomic ...
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Looking for a term I'm pretty sure exists
Let me describe the situation:
Company is selling a product; they buy it at x, sell it at some % over for profit. Taken on a monthly scale, you can see the profit of that particular object by ...
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1
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452
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Why is isoelastic utility function so prevalent?
An isoelastic utility function is used in both simple and advanced models. I understand that it is fairly convenient to work with mathematically and that there are empirical estimations of its ...
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How to find the share price with respect to the type of share?
As someone coming from a mathematical background who has started reading into some basic finance, there are a few concepts that I am struggling to understand and would be grateful if I could check to ...
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Demonstrate order and rank conditions for identification with instrumental variables
For an equation in a simultaneous system to be identified two conditions must hold: i) the order condition, and ii) the rank condition.
b_IV=(Z'X)^(-1) Z'Y
How to demonstrate in matrix form that both ...
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2
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Can data be created using Monte Carlo Simulation
I am aware that Monte-Carlo Simulation is used for making accurate assumptions by introducing randomness. But can it be used to synthesize or create a dataset? If yes, can someone share an example?
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Why is the demand function homogenous of degree $0$ in all prices and income?
Why is the demand function homogenous of degree $0$ in all prices and income?
If I know that the expenditure function is homogenous of degree $1$ in prices and income, how do I show, using the lemma ...
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Macroeconomics / Differential Equations
Could someone help me to solve the following differential equation:
$\dot L(t) = nL(t) + b$ with $n>0$, $b>0$, $L(O)∈R$
$\dot L(t)=$ the time derivative of $L(t)$
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Long-run equilibrium number of firms is indeterminate when all firms in the industry share the same constant technology and factor prices are same
Why is the long-run equilibrium number of firms indeterminate when all firms in the industry
share the same constant returns-to-scale technology and face the same factor prices? How to show it ...
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VAR model and simulation scenario
I have a VAR model to forecast key economic variables like gdp and inflation, but I want to be able to do some sort of simulation to study a particular scenario.
I am running inflation expectations ...
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1
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Write equations for $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$
I am working with a VAR and trying to understand the dynamics of it for forecasting.
Currently, I am trying to generate conditional forecasts by expressing the equations in the form of conditional ...
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0
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Elasticity of substitution for 3 and more goods (interpretation)
Elasticity of substitution for 3 and more goods (interpretation)
Hello everyone,
I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function ...
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0
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28
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Modelling common incentives and strategic manipulation
There are many situations in the markets when small or larger portions of traders collude and make a strategy manipulation through communication, even they have heterogeneous endowments and ...
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log differencing and VAR model
I am working on a VAR model to forecast inflation using variables like CPI prices, oil prices, unemployment rates, PMI, inflation expectations, policy rates, and GDP.
To use the VAR model for my ...
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0
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538
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Utility function distinguishing between complements/substitutes
Distinguish between complements/substitutes in utility function or production function
Hello everyone,
I would like to know if there exists some utility function $U(x)$ for $n$ goods that is able to ...
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I'm trying to understand a proof for First order stochastic dominance
Here's the theorem, consisting of 2 statements:
The equivalence is proven with the aid of this:
There are 2 things I don't understand about it. Firstly, why $ U(x).H(x)|\infty, 0 = 0 $. And secondly,...
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how and why Roy's equation works?
i know that the roy's identity is: ${\displaystyle x_{i}^{m}(p,w)=-{\frac {\frac {\partial v}{\partial p_{i}}}{\frac {\partial v}{\partial w}}}}$
but i can't understand why it works.
why the fraction ...
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