Questions tagged [lagrangian]

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CES function: Lagrangian or Kuhn-Tucker

Let's have the following problem: $$U(\boldsymbol{x}) = \left(x_1^\rho + x_2^\rho \right)^{\frac{1}{\rho}} \qquad s.t. \qquad P_1 x_1 + P_2 x_2 \leq M$$ Is it optimal to solve this problem via pure ...
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Deriving FOC with non-substitable goods

I'm trying to derive the first-order condition of cost minimization when inputs are non-substitutable. The question is based on the problem raised by Rubens (AER 2021). Suppose firms $f$ produce $Q_{f}...
nini'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
1 vote
1 answer
76 views

Dealing with integrals in econ models

I am currently studying econ models, and trying to calculate the FOC for different kind of models. I use J.Gali "Monetary Policy, Inflation and the Business Cycle - An Introduction to the New ...
krauuuus's user avatar
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Lagrangian Problem (hopefully basic)

I have a 3 period (0, 1 and 2) problem with a savings asset and 2 consumption goods. The interest rate is given (SOE) as $R^*$ and I am trying to solve for consumption demand. I have 3 budget ...
econstudent's user avatar
2 votes
1 answer
83 views

Understanding Duality between Individual and Collective Maximization in Macroeconomic Models

I'm currently studying macroeconomic models, specifically from the book "Recursive Macroeconomic Theory." In Chapter Seven, it is mentioned that some economic models involving firms and ...
jrudd's user avatar
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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 ...
CormJack's user avatar
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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. $Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$ &...
CormJack's user avatar
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195 views

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 ...
CormJack's user avatar
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1 vote
1 answer
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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 ...
CormJack's user avatar
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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 ...
CormJack's user avatar
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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 ...
CormJack's user avatar
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3 votes
4 answers
325 views

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 ...
CormJack's user avatar
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1 answer
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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(...
CormJack's user avatar
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3 votes
1 answer
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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, and I have been struggling with a maximization programm in the early pages of the ...
PGCD's user avatar
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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 ...
Martin and Friends's user avatar
1 vote
1 answer
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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 ...
honkhonk's user avatar
2 votes
0 answers
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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 $...
L1234's user avatar
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3 votes
1 answer
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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,...
Athaeneus's user avatar
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1 answer
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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}...
CormJack's user avatar
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2 votes
2 answers
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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 ...
CormJack's user avatar
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3 votes
1 answer
127 views

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*}\...
mrtavsh's user avatar
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3 votes
2 answers
230 views

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{...
mindandfields's user avatar
5 votes
1 answer
277 views

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 ...
Athaeneus's user avatar
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1 vote
1 answer
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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 ...
PGabriel96's user avatar
1 vote
1 answer
233 views

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 ...
Tatanik501's user avatar
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1 answer
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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,...
Twilight's user avatar
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0 answers
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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 ...
Golden_Ratio's user avatar
4 votes
1 answer
516 views

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 ...
Victor Yerz's user avatar
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0 answers
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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 ...
Yekta Aktaş's user avatar
1 vote
1 answer
89 views

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 (...
not tdm's twin's user avatar
3 votes
1 answer
139 views

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 ...
EconJohn's user avatar
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1 vote
0 answers
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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 ...
Caba25's user avatar
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1 vote
1 answer
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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 ...
theshadowers's user avatar
1 vote
1 answer
156 views

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 $[...
guest's user avatar
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2 votes
0 answers
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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 ...
BDot35's user avatar
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2 votes
1 answer
391 views

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 ...
Ben Phronesis's user avatar
0 votes
1 answer
540 views

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 ...
Coco Garazzo's user avatar
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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 $...
JustBlaze's user avatar
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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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