Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
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Maximization of CD production function
I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem:
$$\max_{L_{t}}=\left\{ p_{t}\right\}L_{t}^{\alpha}K_{t}^{...
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Question where sectoral shares and their annual growth rate is given. How will the sector share change with each year? (Exact question attached below) [closed]
In an economy, the agriculture, industrial and services sectors have initial shares of 50, 20 and 30 percent respectively in the total GDP. They also subsequently grow at the following constant annual ...
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Hardcore elasticity of substitution (bad results)
I have a following function and would like to find the elasticity of substitution between pairs:
$$U = \left( x_1^\delta + x_2^\delta + x_3^\gamma + x_4^\gamma \right)^{\frac{1}{\delta + \gamma}}$$
...
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Convex curves and quasi-concave [closed]
For two goods, like U(x,y) = k, where k is a natural number, we define the Marginal Rate of Substitution (MRS) as the ratio of the partial derivatives of U with respect to x and y. If we take the ...
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What is the difference among information acquisition, information providing and information design?
Taking a look at the literature of information economics I see three different terminologies about information that seem to have some intersection, though my understanding is not so good. These terms ...
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Elasticity of substitution between capital and effective labour
While going through the derivation of elasticity of substitution between capital and effective labour in economic materials for a Slow growth model, I found the following step there:
$\frac{\partial ...
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stochastic optimal control/derive the consumption process
(the paper link: p.32, Equation (26), (27))The output process: $d\log y_t = \sigma dZ_t$. The problem is:
$$\max_{C_t, B_t} \mathbb E_0 \int_0^\infty e^{-\rho t} \left(\frac{C_t^{1-v}}{1-v} + (y_t)^{...
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Different elasticities of substitution
I have been reading into generalizations of the concept of elasticity of substitution for more goods/inputs and three main possibilities emerged:
Hicksian EOS
Allen-Uzawa EOS
Morishima EOS
HICKS
As ...
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stochastic optimal control/FOC/Reis(2021)
Reis (2021) `The constraint on public debt when $r < g$ but $g < m$' :
HJB:
$$\rho V(a, q) = \max_{c/a, k/a} [\log c + V'(a,q)[r + (mq-r)\frac{k}{a} - \frac{c}{a}] a + \frac{V''(a,q)}{2} (k/a)^2 ...
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Complete markets and convenience yields
I have been reading some papers on the safety/liquidity of US government debt and got a bit perplexed by the assumptions made in some of those papers. For example, this paper by Mehrotra and Sergeyev ...
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$N(A) \oplus R(A) = V \; \forall A$?
If $A$ is a $m•n$ matrix.
Question: Is $N(A) \oplus R(A) = V \; \forall A$ ?
Update: I now think this question makes sense only for square matrices, as noted below.
Terminology
$R(A)$ By this I ...
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Does local non-satiation hold for this problem?
I am getting some confusing results solving this problem:
$max_{c_0\geq0, c_1\geq0} \bigg\{EU = R(1-c_0) [p t_1 + (1-p) c_1^{-2} t_2] \bigg\} ~ s.t. ~c_0+c_1 \leq 1$
where $p$ is the probability of $...
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Determine for which prices and income the constraint is binding
Under what conditions constraints start to bind and how to find it
I was trying the following optimization problem:
$$ \mathscr{L} = x_1 x_2 + x_2 + \lambda(M-P_1 x_1-P_2 x_2) + \mu x_1$$
The thing is,...
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Questions regarding Solving Melitz (2003) model
I am editing my question regarding solving the model in order to be more specific.
Regarding the demand side, In the beginning of the model, we have a CES utility function over a continuum of goods
...
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Rochet-Tirole price level formula
I have to show that this equation (price level) holds for any pair of monopolist profit-maximizing prices ($f^M, m^M$), where f is the rider's fee and m the driver's fee:
$$\frac{(f^M + m^M - c)}{f^M} ...
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1
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Pareto optimal allocations with uncountably many agents
Consider an economy with some $n$ agents with continuous utility functions $u_1,\ldots,u_n$. It is easy to prove that a Pareto-optimal allocation exists: define the welfare of an allocation $x$ as: $W(...
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Every Submodular Valuation Can Be Represented as a Maximum of Additive Valuations
According to this paper, "every submodular function can be represented as a maximum of additive valuations." It gives an algebraic description as well, but I am having trouble internalizing ...
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How to create a composite good?
Let's say I would like to create some composite score for multiple of goods...
EDIT: More concise version based on @BrsG comments... I would come up with the following scenario. I have a consumer with ...
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How do I rebase this NDP per capita data?
So i got two different data sets that I want to put together, one is from 2004-05 to 2014-15 with the base year 2004-05 and the other one is 2011-12 to 2021-22 with base year 2011-12. I have also ...
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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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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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Maximising social welfare in time inconsistency
I have solved a number of issues that have a loss function which I need to minimize subject to the Philips curve.
I was wondering if anyone knew the method for solving this model generally, to acquire ...
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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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Total Retention Rate Calculated from Categories
I am calculating retention for 3 categories and then total, and I am trying to double check my total, but my check formula isn't working.
I am comparing the last 14 days (let's call it Period 1) to ...
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4
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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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2
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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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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 ...
3
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0
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80
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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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1
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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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