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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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Why is incidence not included in social welfare maximization?

I am very confused on why incidence is not included in social welfare maximization of one good. Typically, I see the optimization over price done something like this: $C$ ~ production cost function $...
EngineerinEcon's user avatar
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Help with Deriving Hicksian Demand in the Monocentric City Model?

I have a fairly standard Alonso-Muth-Mills model, but struggling to derive the Hicksian demand. Starting with the basic utility function: And this Budget Constraint: Housing Floor-space is ...
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Do standard consumer theory axioms rule out corner solutions?

By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves. If these axioms are not sufficient ...
Santiago Valdivieso's user avatar
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A question about Lagrangian, KKT theorem, consumer's problem

Suppose we want to maximise a expected utility function: $$E_1(u(C_1,C_2,C_3)) $$ subject to following constraints. There are two possible situations each with probability $\frac{1}{2}$. $$C_1 + S_1 = ...
Nonenicht's user avatar
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Derivation of Euler Equation in presence of the Dixit-Stiglitz Aggregator

I reading the working paper of Sebastian Banz (2012). I have an issue with the derivation of the Euler equation. The author models the demand side of the economy as follows The representative consumer ...
Maximilian's user avatar
9 votes
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Cost Minimization and Karush-Kuhn-Tucker

A firm produces an output $y$ using two inputs $x_1$ and $x_2$, where the production function is given by $y = \sqrt{x_1 x_2}$ for any $(x_1, x_2) \in \mathbb{R}^2_+$. Union agreements obligate the ...
bruno's user avatar
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Proving quasi-concavity for a utility function

I have a utility function, and I want to prove that it is a quasi-concave function: $$ u(x_1,x_2)= 2x_1x_2+x_1+2x_2 $$ I do this by showing that the set of points where the utility is larger than or ...
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Using lagrange on a quasi-concave utility function

A consumer has the following utility function $$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$ I have maximized his utility function, and found its demand functions, for $x_1$ and $x_2$, using Lagrange. However, is it ...
Noah's user avatar
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Solving utility maximization, and finding demand function

A consumer has the following utility function $$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$ I want to maximize his utility function. $$max: 2x_1x_2+x_1+2x_2. uc:p_1x_1+p_2x_2=y_A$$ Using Lagrange, I get $$L(x_1,...
Noah's user avatar
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When is argmax increasing in some multiplier of the objective function?

Let $f : R_+ \to [0,1]$ be continuously differentiable and strictly increasing with $f(0)=0, \lim_{x\to\infty}f(x)=1$ and let $c : R_+ \to R_+$ be continuously differentiable, strictly increasing, ...
raving-bandit's user avatar
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Irrelevance of Heterogeneous Agent Modelling

This is a question from a previous year PhD entrance exam. I have outlined how I have tried to tackle the problem as well: N.B. 1 This exam is of 100 points and this particular problem is of 25 points....
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2 answers
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Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization

It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign. If the above is ...
hillard28's user avatar
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question about production optimization

the question is, if Q = AK^a(HL)^b and the parameters are: (A =100) (K = 10000$) (H = 1) (L = 100 person) (a = 0.5) (b = 0.5) P = 5 per unit, R = interest rate of 3 percent per year , W = 3 per ...
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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 ...
Martin's user avatar
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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: ...
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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 ...
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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 ...
CormJack's user avatar
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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 ...
Henry's user avatar
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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 ...
Heidegger's user avatar
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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....
Iron filings's user avatar
3 votes
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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 ...
John Sturm's user avatar
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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 ...
Eileen's user avatar
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Finding the optimal trading strategy

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 ...
user171780's user avatar
2 votes
1 answer
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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. ...
opre's user avatar
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1 answer
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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 ...
fabs's user avatar
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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?
Ranu Jain's user avatar
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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 ...
ju_pi_car's user avatar
1 vote
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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 ...
David K's user avatar
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1 answer
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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
matias's user avatar
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1 answer
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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 ...
TomUcl2003's user avatar
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Durable goods in a (two sector) necolassical growth model

i want to add a firm to a neoclassical growth model that produces a durable good which it rents out in each period to the consumers. Right now i'm using the following approach: The firm maximizes: $\...
mfba's user avatar
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1 answer
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FOCs for profit maximization using a transformation function

I'm (still) reading the microeconomics textbook of Mas-Colell et al. On p. 135, the profit maximization problem (PMP) for producers is introduced; characterizing the technology as $Y = \{ y \in \...
chsk's user avatar
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1 answer
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How do I show that the minimization problem has a solution?

Consider an inner product space $X$ with the induced metric $d$ (induced by the inner product). Suppose that the induced metric space $(X,d)$ is complete. Moreover, for all $x,y,z\in X$, $$[d(x,y)]^2+[...
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why is the MRS same for everyone?

If the consumers are optimizing and at interior solutions and facing the same prices, then the MRS=p1/p2 will be the same for everyone no matter the preferences and income. but why? I don't understand ...
tessa's user avatar
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help with nested integrals in common agency public goods paper

I'm trying to derive the example function used in a paper I am reading and am stuck. Please help. Below is the equation. For now, all I am requesting is someone to help me solve for p. More details ...
KatLeigh11's user avatar
2 votes
1 answer
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Help with a proof for an quite intuitive Utility optimization problem

Assume $U(x,y,a,c )= - c x + B(x,y,a)$, with $\frac{\partial B(x,y,a)}{\partial c }=0$, and with $a$ and $c\geq 0$ being parameters, and with $x$ and $y$ being variables. Further, $B(x,y,a)$ is ...
Paul's user avatar
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Is there a name for this generalized assignment problem variant with an additional indicator constraint

I have an optimization problem that's closely related to the generalized assignment problem with an additional constraint. Each knapsack $i$ has capacity $b^i$. There is a number of tasks $j$, each ...
retVI23's user avatar
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1 answer
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Effect of a change in prices when the consumer starts from a bundle of goods rather than wealth

I'm struggling to make progress with the following problem: Assume that instead of income, the agent starts with an endowment of goods c (not necessarily her optimal bundle) & can buy and sell ...
TomUcl2003's user avatar
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1 answer
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Inequality and effect on pricing

Suppose there’s an island that for whatever reason has a finite amount of US Dollars; 1000 residents live on the island and use this fixed amount of currency to exchange goods and services. Suddenly, ...
jbuddy_13's user avatar
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1 answer
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Problem with Optimizing Profit in Log-Linear Demand Model

I've trained a linear model to predict log(volume) to capture the elasticity of demand with respect to price difference between my product and a competitor's: $$log(volume)= constant+elasticityCoef×(...
MarcM's user avatar
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0 answers
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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{...
Wittgenstein's Poker's user avatar
1 vote
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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 ...
Beerus's user avatar
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2 votes
0 answers
83 views

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 ...
Meg's user avatar
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2 votes
2 answers
262 views

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 ...
Beerus's user avatar
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3 votes
1 answer
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 ...
Aidan Loten's user avatar
3 votes
0 answers
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 ...
Nicolas Torres's user avatar
1 vote
2 answers
116 views

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 ...
Beerus's user avatar
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0 answers
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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 ...
Elina Gilbert's user avatar
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24 views

Can a business's search for profits be considered as movement in state space?

When businesses make decisions to increase profits, they have to adjust several "parameters" of the business such as quantity to output, pricing, choosing a production mix, brand positioning,...
Joebevo's user avatar
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0 answers
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Expected value in budget constraint

I have been reading this paper by Yang Liu, Lukas Schmid and Amir Yaron, which contains a very elegant mechanism that generates an endogenous liquidity premium for US government debt. However, I got ...
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