Questions tagged [optimization]

Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.

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• 11
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Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)

Could you please help me evaluate this logic? I've been told that "if preferences are strongly monotonic, duality holds." In the case of utility u(x,y) = x^2 + y^2, we will get a corner ...
• 53
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Modeling Approach to Adjust linear Elasticity Effect in Pricing Optimization

I am working on a pricing optimization model for a product where the price depends on the competition as well as our costs. The current formulation of the model is: ...
• 11
1 vote
38 views

Missing Solutions in KKT Optimisation Problem

In the attached inequality, constrained, optimisation, problem. Looking at the specific case where $\lambda_1 = 0, \lambda_2 > 0$ that I am trying to solve, you can see that I have managed to find ...
• 991
1 vote
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Linear Dependence of Binding Constraint Qualifications in Karush-Kuhn-Tucker

When checking whether the CQ are satisfied in KKT, i.e. checking for Linear Independence amongst all combinations of the constraints. Is it true to say we only need to check combinations that could be ...
• 991
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Optimisation problem and KKT conditions (unsatisfied?)

I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$. Now, I solved the problem quite easily in a ...
• 153
1 vote
183 views

Why this optimisation problem cannot be solved with "usual" KT conditions?

I have this optimisation problem: $$f(x, y, z) = 2xy + yz \qquad \text{subject to} \qquad \begin{cases} x+y+2z \leq 1 \\ x \geq 0, y \geq 0, z\geq 0 \end{cases}$$ I solved it with "a certain ...
• 111
1 vote
62 views

Why do multiple investment funds exist?

Say that you are a head of an investment fund. Your goal is to maximise the return on money entrusted to you by investing in various enterprises. You look to your left, and see another investment fund....
53 views

When is a value function twice differentiable?

Consider the optimization problem $$V(\gamma) \ \equiv \ \max_x \ f(x,\gamma) \quad \text{s.t.} \quad g(x,\gamma) \leq 0$$ Roughly, the Benveniste-Scheinkman theorem implies that if $f$ and $g$ are ...
• 191
50 views

How should I add my period by period constraint in lagrangian?

This question just suddenly comes to my mind, and I'm not sure what i'm thinking is correct: suppose I want to maximize my cumulative expected utility $E\left[ \sum_{t=0}^{T}\beta^tU(c_t) \right]$ by ...
• 11
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Suppose there is one asset that you can buy and sell, and that you know what was the selling price and buying price at all times between times t0 and t1, both in the past. If you would have had an ...
• 121
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Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?

Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable. ...
• 23
1 vote
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Cost function from a weighted CES production function

I want to find the cost function given the CES production function: $$Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho}$$ with $0<\rho<1$. So far I have set up the ...
• 11
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Assumption of interior solution in the Lagrangian method

Why do we need to assume an interior solution before using Lagrangian method for utility maximization problems?
65 views

Relation between second-order derivatives and corner solutions

Imagine I have the simple maximisation programme: $$\max_{x,y}U(x,y)$$ $$\text{subject to: } B=x+y$$ $U(\cdot)$ satisfies the usual properties: it is increasing and concave in both arguments. This ...
1 vote
14 views

Normalization for model comparisons

I have a time series applying the Markov Switching model, which is estimated in about 15 different versions. One or two of the time series had to be normalized in order to converge. That is 1-2 out of ...
• 21
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Profit maximization; how to derive

Can someone expalin me the math process step by step behind this? I haven't been able to figure it out and I cannot find anything in google
1 vote
125 views

Conditions for an interior solution to the UMP

I was wondering under what set of conditions one is allowed to assume an interior solution to the Utility Maximisation Problem. In most of my classes and lecture notes, interior solutions are assumed ...
1 vote
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• 315
1 vote
175 views

• 11
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Habit formation ala Constantinides (1990)

Consider the following problem, from Constantinides (1990). \begin{align} V(W_0, x_0) \equiv \max_{c, \alpha} \mathrm{E}_0 \int_0^\infty e^{-\rho s}\gamma^{-1}[c(s) - x(s)]^\gamma \mathrm{d}s, \end{...
1 vote
274 views

Question About Non-Degenerated Constraint Qualification (NDCQ)

I am studying constrained optimization using Mathematics for Economists by Simon and Blume, and I have some difficulties understanding the Non-Degenerated Constraint Qualification (NDCQ). I would like ...
• 505
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Leontief function nested in a cobb-douglas function for a computable general equilibrium

I am currently trying to build a CGE model, and I'm stuck with the specification of the agriculture sector. I'm trying to understand how to do nested production functions and also how to solve them. I ...
• 31
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Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32

I am studying Implicit Function Theorem and its application on comparative statics using Mathematics for Economists by Simon and Blume. Here is the question: Consider a pure exchange economy with two ...
• 505
196 views

Intertemporal Utility Optimization For Multiple Goods

I'm building an economic simulation game and I'm trying to solve for the values that a person will spend on each good and the amount they will save in the current period, taking into account all ...
39 views

Is there a labor vs leisure model with work experience?

I find the labor-leisure model with utility functions interesting, but I find it lacks the factor of work experience, which is very important in the real life labor market. This is a reason people why ...
• 2,259
1 vote
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Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7

I am working on Mathematics for Economists by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that ...
• 505
1 vote
38 views

Signalling a relation

I have a very basic, trivial question to ask. (I apologise if it is not a worthy problem) I have been trying to formulate an international Economics model where a larger country is trying to signal a ...