Questions tagged [lagrangian]
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35
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Technology Parameter In Converted Minimisation Problem
Question:
I want to understand what's going on with respect to the technology parameter $A$ when i convert this minimisation problem into a maximisation problem. The issue is only revealed when i use ...
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Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They given the following reason as outlined in the problem below.
$Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$
&...
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Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
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Logarithmic Utility function Algebra
Question:
I'm told the following (by an exam mark scheme):
Using $a + b =1$
$a[ln(\frac{am}{p_1})] + b[ln(\frac{bm}{p_2})] = ln(m) - aln(p_1) - bln(p_2)$
I can't get this to hold without the ...
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Envelope Theorem and Factor Mix Intuition
Brief Summary of question:
I'm frustrated with the intuition of the envelope theorem that when input costs change, our envelope theorem tells us we do not need to re-optimise demand levels.
Context:
I ...
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1
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Inappropriate use of Calculus in estimating ΔCost?
I have the following model, and i solve for my optimised conditional factor demands, and minimised cost functions $C$. (Note: I have turned a minimisation problem into a maximisation problem). Let's ...
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4
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Lagrange Multiplier Dual Meaning?
Is the Lagrange multiplier:
The marginal cost of the constraint?
The marginal benefit of relaxing the constraint?
Through duality, both interpretations imply the other?
If anyone were so kind, I ...
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Non-Negativity Constraints KKT
When we take our Lagrangian and we include non-negativity constraints. If a variable $x = 0$ do we take FOC first or set $x=0$ first?
E.g.
$Max \; L(x, y, λ) = f(x,y) - λ_1(g(x,y) - k) - λ_x(-x) - λ_y(...
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Finding the constraints for a profit maximization program in Acemoglu et al, 2016
I have been reading the paper of Acemoglu et al., 2016 [Networks and the macroeconomy : an empirical exploration][1], and I have been struggling with a maximization ...
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Max and Min with $\leq$ and $=$ constraints. General questions
I wrote this question on Maths.stackexchange but perhaps this community suits better (?)
I need to ask you for this question, which is a rather general one, in order to understand how to behave when ...
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Supply function of a price-taking firm with a quadratic production function
For a firm with the production function
$$Q = 40L-L^2$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.
Following ...
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37
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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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Compensating Variation - Interpreting the formulae
Assume $U(x,y) = x^{1/2}y^{1/2}$ s.t. $P_xx + P_yy = m$
And a price increase from $P_x$ to $P'_x$:
$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$
Compensation variation formulae is: $\frac{M + ∆M}{2(P_x'P_y)^{1/2}...
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Compensating Variation for $U = (xy)^{1/2}$ and $U = xy$
I want to check my calculations for these compensating variations regarding an increase in $P_x$ to $P'_x$.
Below I have used $-∆M$, this is how my course first laid it out but I appreciate that it's ...
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What is the Lagrangian a function of?
I understand the role of Lagrangian in constrained optimisation, and that we could conceptualise it as for example, a penalty function.
What I don’t understand is the notation, and perhaps any deeper ...
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How can I solve a Utility Maximization problem using the Lagrangian method where the Utility formula has an exogenous constant $a$?
The utility function is given by: $$u(x, y) = 2x^{\frac{1}{2}} + 2ay^{\frac{1}{2}}.$$
The optimal bundle should be expressed as a function of $a$. Other variables are given by:
$$\begin{eqnarray*}\...
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How to handle multiple lagrange multipliers in a maximization problem?
Let's assume a standard household maximization problem of the form:
\begin{align}
\underset{C_t}{max} \sum_{t=0}^{\infty} \beta^t U(C_t)
\end{align}
subject to a standard Budget constraint:
\begin{...
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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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Differentiating over multiple time horizons to get FOCs
First of all, I'd like to say sorry if I couldn't be more specific in the title, I really tried to synthesize the core of my doubt.
I was reading The Econometric Analysis of Calibrated Macroeconomic ...
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how to derive marshallian demand functions from leontief preferences?
For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a ...
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Questions about Lagrangian and consumer's problem
Background: Suppose a consumer has the following utility function $u(x,y)=\sqrt{xy}$, then the Lagrangian equation is $\mathcal{L}(x,y)=\sqrt{xy}+\lambda(I-xp_x-yp_y)$. Then the optimal bundle is $(x,...
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Relation between KKT necessary conditions
I am trying to understand the relationships in the KKT theorem between being a maximizer, satisfying the first order conditions (FOCs) and complementary slackness (CSC), and the linearly independent ...
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Concave utility functions solution example
In the following post an example is given of the corner solution for a concave utility function. I tried solving it but got stuck. I have no idea how these types of problems are solved so if you could ...
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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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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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