Questions tagged [lagrangian]

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Why do we optimize utility from X1 according to itself?

In lecturer's notes we have a utility function $U_i = X_i^A * (1-L_i)^{1-a},$ $ 0< a< 1$, $ i = 1,2 $ $MP_1 = w_1 $ $MP_2 = w_2$ $X_1 + X_2 = w_1 * L_1 + w_2 * L_2$ And we need to form a ...
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Lagrangian when ICs are tangent to the budget line

Suppose the graph below shows three Indifference Curves such that $t > s > r$, and the budget line $p_1x + p_2y = I$. I was wondering if we set the Lagrangian as $\mathcal{L}= U(x,y) - \lambda (...
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Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?

This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts. To preface suppose we have the ...
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Optimal Policy under Commitment

Hi I'm trying to wrap my head around an equation in Jordi Galís book "Monetary Policy, Inflation and the Business Cycle" Under commitment the Central Bank has the following optimization ...
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Lagrangians of a negative minimum function [duplicate]

if I rewrote min -f(x) s.t. g(x)<k as -max f(x) s.t. g(x)<k, what would my Lagrangian look like?
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minimisation problem as a maximisation problem for lagrangians?

if I have a problem min(-f) s.t. g<0, I can rewrite it as -max(f) s.t. g<0. In this case, if I take Lagrangians, would my lagrangian be L=f- lambda(g-0) or would I have to have a negative in ...
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What is the meaning of lagrange multiplier (especially in ramsey problem)

Consider lagrange function for ramsey problem: $L=E_0 \sum_{t=0}^{\infty} \beta^t \{u(c,l)+\gamma_t (s^t)[E_0 \sum_{j=0}^{\infty} \beta^j u_c (s^{t+j}) z(s^{t+j}) -u_c (s^t) b_t(s^{t+1})] \}$ where $[...
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Solving for the efficient subsidy amount with an externality

I am dealing with a problem that is set up as follows: Actors A and B get utility from consumption ($c_i$) and disutility from safety measures ($s_i$), however their chance of getting sick is reduced ...
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How does this imply that a Pareto optimum maximizes a weighted average of utility functions?

I'm reading a passage from Asset Pricing and Portfolio Choice Theory by Kerry Back, and I don't understand some of it. I would appreciate any help anyone could provide me. In the passage, Back is ...
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Calculate hicksian demand with utility function (with restriction)

$U(x_1, x_2) = 1/2 * x_1 $ I am trying to calculate the Hicksian demand when when $U(x_1, x_2) = 2$ and the value of the minimum expenditure when $p_1 = 9$ and $p_2 = 16$ For the hicksian demand I ...
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FOC for stochastic Ak Model

(Note: the other posts do not cover this part of the derivation) I have tried to compute the FOC of $k_{t+1}(s^t)$. I get that $0 = -\lambda_t(s^t)$; I can't see why the sigma remains for the FOC of $...
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How to derive consumer expenditures in EMEA 14.2.5

I am working on a problem 14.2.5 from EMEA by Sydsaeter, Hammond and Strom. Consider the consumer demand problem: $$ \max_{x,y} U(x,y) = \alpha \ln(x-a) + \beta \ln(y-b) \text{ s.t. } px+qy=m \tag{*} ...
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