Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
333 questions
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Estimating the second derivative of function from optimizers
Consider the following optimization
$$x^*(s) = \max_{x\in X} \big(\,f(x)-sx\,\big)$$
where $f$ is assumed to be a strictly concave function and $X$ is an interval constraint, e.g $X = [0,b]$. We do ...
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1
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What is the interpretation of the output matrix of pivoting?
I have the following matrix:
$$A= \begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & 4
\end{bmatrix}$$
After pivoting, I got this matrix:
$$B= \begin{bmatrix}
1 & 2 & 3 \\
...
- 1,044
1
vote
1
answer
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Duality of cost minimization and profit maximization
The firm tries to maximize profits $\Pi$
\begin{align}
\max_{K,L}\{\Pi(K,L) = F(K,L) - RK - wL\}
\end{align}
where $F$ is the linear homogeneous production function, $R$ the rental rate of capital $K$...
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4
answers
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Why couldn't the Karush-Kuhn-Tucker optimization find the solution?
I have the following utility maximization problem:
$$\max (xy)$$
$$(x+y-2)^2 \leq 0$$
Conditions:
$$y-2\lambda (x+y-2) =0$$
$$x-2\lambda (x+y-2) =0$$
$$\lambda(x+y-2)^2=0$$
When I set $\lambda>0$, ...
- 1,044
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votes
1
answer
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Simple Derivation of Maximum Principle
Consider the simplest problem of optimal control
\begin{align}
&\max_u\int^T_0{F(y,u)dt}\\
\text{s.t.} \quad&\dot y = f(y,u)\\
& y(0) = y_0\\
& y(T)~~\text{free}
\end{align}
...
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3
votes
1
answer
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Can be the duality theorem applied to not locally non-satiated utility functions?
I have the following not locally non-satiated utility function:
$$U(x,y)=-(x-1)^2-(y-2)^2$$
where $U(x,y): \, \!R^n_+ \rightarrow \!R $
The 3D plot of this function is an infinite paraboloid; ...
- 1,044
4
votes
1
answer
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Calculating mean variance portfolio with risk aversion parameter
I want to calculate the classic mean variance portfolio (Markowitz) with a risk aversion parameter $\gamma$.
I have the following problem where I want to maximize:
$max(x_t) \ \ x_t^T\mu_t - \frac{...
- 41
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votes
3
answers
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Does the Marshallian demand function always include prices and income?
I have the following utility function:
$$U(x_i)=x_1x_2+x_3$$
with budget constraint:
$$p_1x_1+p_2x_2+p_3x_3\leq I$$
I use the Kuhn-Tucker method to find the optimal choices of the Utility ...
- 1,044
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votes
2
answers
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Intermediate macroeconomics: optimal bundle for quasilinear utility?
How would I go about solving this question:
Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. ...
- 133
3
votes
0
answers
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Optimizing Cobb-Douglas like functions
Economics isn't my home field, but I'm looking for references for a paper I'm working on and I'm hoping one of you can help. Are there many good references for optimizing a Cobb-Douglas like utility ...
- 31
3
votes
2
answers
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Why is instantaneous utility of current period discounted?
Consider a two period model of consumption.
I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the ...
- 31
4
votes
1
answer
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static/dynamic optimization
The interesting paper Calvo and Obstfeld (1988) uses two-stage optimization on an OLG model which then reduces to a standard representative agent framework.
First stage optimization consists on a ...
- 2,155
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votes
0
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Appropriate economic/econometric tools to analyze segmented promotion optimization problem
I'm trying to determine which micro-economic/econometrics concepts, models, and/or tools are appropriate for an analysis of promotions.
Below I
Describe the problem in general terms
Give ...
- 380
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0
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Converging Trajectories and Sufficiency for Optimality
(The question is loosely relatet to this thread.)
In the paper "Feedback Equilibria for a class of non-linear Differential Games" by Mäler et al. it is stated (p. 14)
In fact sufficiency is ...
- 1,579
12
votes
1
answer
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Dynamic Optimization: What if the second order condition does not hold?
Consider the following dynamic optimization problem
\begin{align}
&\max_u \int^T_0{F(x,u)dt}\\
\text{s.t.}~& \dot{x} = f(x,u)
\end{align}
FOCs
The Hamiltonian is given by
\begin{align}
H(x,u,...
- 1,579
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votes
4
answers
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Examples of non-differentiable problems in economics
As a research project, we're investigating various algorithms developed for non-differentiable, convex (or concave, if you're into economics) optimization. I'd like to find some good examples of real ...
8
votes
1
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Leontief preferences
I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really ...
- 81
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votes
0
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Optimisation of bonds
I'm doing an optimisation problem but don't understand what the terms mean.
Suppose someone wants to invest $110,000.
They have 4 choices as to what they invest their money into:
municipal bond ...
- 131
2
votes
0
answers
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First and Second Order Lagrangian-Multiplier Conditions for Optimization
The Statement of the Problem:
Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier ...
2
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0
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Simple Neoclassical Growth Model with Elastic Labor and Non-Standard Capital Adjustment Costs
I have the following social planner problem to maximize $\{c_t, k_t, n_t \}$
$\begin{gather*}E_0 \sum_{t=0}^\infty \beta^t U(c_t, 1 - n_t), 0 < \beta < 1\end{gather*}$
subject to
$\begin{...
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votes
2
answers
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Editing formula for finding Marshallian Demand with Cobb-Douglas utility function
Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$:
$x_1 = \frac{am}{p_1}\\
x_2 = \frac{bm}{p_2}$
But what if the utility function looks like $u=...
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votes
2
answers
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Marshallian Demand for Cobb-Douglas
When trying maximize the utility having a cobb-douglas utility function $u=x_1^ax_2^b$, with $a+b = 1$, I found the following formulas (Wikipedia: Marshallian Demand):
$x_1 = \frac{am}{p_1}\\
x_2 = \...
- 245
1
vote
1
answer
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Derivative of CARA utility
Can someone help explain the passage here? I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much ...
- 237
2
votes
0
answers
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Calculating the optimal portfolio for an investor with quadratic utility
The problem is from Asset Pricing and Portfolio Theory by Back and can be found here.
The relevant info from section 2.5 can be found here. Given that we have the Expected value and the variance of ...
- 237
4
votes
1
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Markov decision processes, contractions and value iteration
I am reviewing Markov decision processes (MDP) and there is something I am missing with respect to the contraction argument. I am pretty sure it is a silly mistake somewhere (maybe computational), but ...
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2
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Is there a way to link Berge's theorem of maximum to Envelope theorem?
Berge's theorem states
Let $X \in \mathbb R^m, \Theta \in \mathbb R^n $, $f : X \times \Theta \to \mathbb R$ be a jointly continuous function, $C : \Theta \rightrightarrows X$ be a continuous(both ...
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3
votes
1
answer
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Reverse auction formula
I am studing a little bit of auction theory. I found the optimal bid value in the Milgrom paper for the first price auction that is
$$
P=v \frac{n-1}{n}
$$
where $P$ is the optimal bid, $v$ is the ...
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4
votes
1
answer
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Portfolio choice problem of a CARA investor with n risky assets
Ok, I am working on a problem that consists of the following:
I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
- 237
6
votes
1
answer
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Optimization of Households' utility in " Rule-of-Thumb Consumers and the Design of Interest Rate Rules " (Gali et al., 2004)
I can't figure out how the calculation of first order conditions was carried out.
I can't figure out where the stochastic discount factor came from.
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votes
1
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Finding a maximal growth portfolio
I have the following problem that asks me to solve for the "maximal growth portfolio."
Suppose that the equilibrium stochastic discount factor evolves as
$$
\log S_{t+1} - \log S_t = \kappa_s(X_t,...
- 9,385
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votes
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References to learn continuous-time dynamic programming
Does anyone know of good references to learn continuous-time dynamic programming? The references don't have to be books. They could be links to online resources as well. Links to clear, concise ...
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2
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Identification of switching costs from price shocks
What, if anything, can we learn about customer switching costs by looking at price, revenue, profit, and quantity responses of producers to cost shocks?
For example, we can define the profit equation ...
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votes
1
answer
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First Order Condition for Profit Maximization in Gambling Industry
I am working on a model of optimal payout percentages in the gambling industry.
Because the nominal price of a \$1 ticket is always \$1, we use an effective price strategy where Q = \$1 in won ...
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