Questions tagged [lagrangian]
The lagrangian tag has no usage guidance.
44 questions
2
votes
1
answer
76
views
How to Apply the Bellman Equation with Two Control Variables in an Optimal Growth Model
I'm currently studying an optimal growth problem involving a representative consumer, and I'm having trouble using the Bellman equation when there are two control variables involved.
Specifically, I ...
- 371
6
votes
1
answer
288
views
Lagrange with expectation: Why can we "ignore" the expectation operator?
I am considering a maximization problem:
\begin{align}
\max_{p(\theta),K} \mathbb E[S(p(\theta),\theta) - c D(p(\theta),\theta)]
\end{align}
subject to the constraint
\begin{align}
D(p(\theta),\theta)\...
- 3,840
2
votes
1
answer
61
views
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 $\frac{1}{2}$.
$$C_1 + S_1 = ...
- 223
3
votes
1
answer
67
views
Why can the Lagrangian Multiplier be dropped in the inverse demand function?
I'm deriving the Antras and Helpman (2004) paper. The model assumes a nested CES utility function
$$
U = x_0 + \frac{1}{\mu} \sum_{j=1}^{J} X_j^\mu
$$
where $X_j = \left[ \int x_j(i)^\alpha \,di \...
4
votes
2
answers
184
views
Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization
It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign.
If the above is ...
- 53
1
vote
0
answers
39
views
Missing Solutions in KKT Optimisation Problem
In the attached inequality, constrained, optimisation, problem. Looking at the specific case where $\lambda_1 = 0, \lambda_2 > 0$ that I am trying to solve, you can see that I have managed to find ...
- 1,011
1
vote
0
answers
27
views
Linear Dependence of Binding Constraint Qualifications in Karush-Kuhn-Tucker
When checking whether the CQ are satisfied in KKT, i.e. checking for Linear Independence amongst all combinations of the constraints.
Is it true to say we only need to check combinations that could be ...
- 1,011
5
votes
1
answer
111
views
Optimisation problem and KKT conditions (unsatisfied?)
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a ...
- 153
0
votes
1
answer
70
views
How should I add my period by period constraint in lagrangian?
This question just suddenly comes to my mind, and I'm not sure what i'm thinking is correct:
suppose I want to maximize my cumulative expected utility $E\left[ \sum_{t=0}^{T}\beta^tU(c_t) \right]$ by ...
- 11
2
votes
1
answer
199
views
CES function: Lagrangian or Kuhn-Tucker
Let's have the following problem:
$$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 ...
- 834
1
vote
2
answers
46
views
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}...
- 33
0
votes
1
answer
188
views
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?
1
vote
1
answer
87
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 ...
- 125
2
votes
1
answer
93
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 ...
- 155
1
vote
0
answers
21
views
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 ...
- 1,011
5
votes
1
answer
228
views
Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They give 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$
&...
- 1,011
2
votes
1
answer
369
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 ...
- 1,011
1
vote
1
answer
103
views
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 ...
- 1,011
2
votes
1
answer
60
views
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 ...
- 1,011
3
votes
4
answers
464
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 ...
- 1,011
1
vote
1
answer
441
views
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(...
- 1,011
3
votes
1
answer
88
views
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 ...
- 31
1
vote
1
answer
129
views
Supply function of a price-taking firm with a quadratic production function
For a firm with the production function
$$Q = 40L-L^2$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.
Following ...
- 13
2
votes
0
answers
39
views
Does local non-satiation hold for this problem?
I am getting some confusing results solving this problem:
$max_{c_0\geq0, c_1\geq0} \bigg\{EU = R(1-c_0) [p t_1 + (1-p) c_1^{-2} t_2] \bigg\} ~ s.t. ~c_0+c_1 \leq 1$
where $p$ is the probability of $...
- 33
3
votes
1
answer
97
views
Determine for which prices and income the constraint is binding
Under what conditions constraints start to bind and how to find it
I was trying the following optimization problem:
$$ \mathscr{L} = x_1 x_2 + x_2 + \lambda(M-P_1 x_1-P_2 x_2) + \mu x_1$$
The thing is,...
- 834
1
vote
1
answer
195
views
Compensating Variation - Interpreting the formulae
Assume $U(x,y) = x^{1/2}y^{1/2}$ s.t. $P_xx + P_yy = m$
And a price increase from $P_x$ to $P'_x$:
$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$
Compensation variation formulae is: $\frac{M + ∆M}{2(P_x'P_y)^{1/2}...
- 1,011
2
votes
2
answers
93
views
What is the Lagrangian a function of?
I understand the role of Lagrangian in constrained optimisation, and that we could conceptualise it as for example, a penalty function.
What I don’t understand is the notation, and perhaps any deeper ...
- 1,011
3
votes
1
answer
156
views
How can I solve a Utility Maximization problem using the Lagrangian method where the Utility formula has an exogenous constant $a$?
The utility function is given by: $$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*}\...
- 31
3
votes
2
answers
281
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{...
5
votes
1
answer
354
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 ...
- 834
1
vote
1
answer
52
views
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 ...
- 77
1
vote
1
answer
325
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 ...
- 141
0
votes
1
answer
135
views
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,...
- 105
2
votes
0
answers
27
views
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 ...
- 243
4
votes
1
answer
692
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 ...
- 43
1
vote
1
answer
99
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 (...
- 244
3
votes
1
answer
173
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 ...
- 8,847
1
vote
0
answers
31
views
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 ...
- 11
1
vote
1
answer
68
views
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 ...
- 21
1
vote
1
answer
211
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 $[...
- 342
2
votes
0
answers
49
views
Solving for the efficient subsidy amount with an externality
I am dealing with a problem that is set up as follows: Actors A and B get utility from consumption ($c_i$) and disutility from safety measures ($s_i$), however their chance of getting sick is reduced ...
- 21
2
votes
1
answer
534
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 ...
- 121
0
votes
1
answer
655
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 ...
- 13
0
votes
0
answers
42
views
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{*} ...
- 1,935