9
votes
Accepted
Karush-Kuhn-Tucker for series
Yes, Bachir et al. (2021) extend the Karush-Kuhn-Tucker theorem under mild hypotheses, for an infinite number of variables (their Corollary 4.1).
I give hereafter a weaker version of the ...
- 191
9
votes
Real Life Examples of Optimization in Economics
Optimization problems
Some of the problems you mention do not seem that simple to me, e.g., "farmers choosing between different crops to grow based on expected harvest and market price" can ...
- 29.6k
9
votes
Minimisation problem turned into Maximisation
The Lagrangian is not really symmetric; something that is easier to see if you formulate it without the calculus implementation. First-order conditions for maxima and minima might look similar, but ...
- 13.5k
9
votes
Accepted
Cost Minimization and Karush-Kuhn-Tucker
The term $\lambda_2(x_1-1)$ in your Lagrangian is incorrect; it treats the second constraint as an equality rather than an inequality. To allow for the constraint being an inequality you can include a ...
- 8,584
8
votes
Accepted
Two-Stage Utility Maximization Problem
Consider the Wikipedia definition (take from Hal Varian's book) of a quasi-linear utility function
$$u(w,x_1,...,x_n) = w + h(x_1,...,x_n),$$
where $h$ is strictly concave.
The example in this ...
- 3,798
8
votes
Accepted
Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Solving a non-linear programming (with inequality constraints) is about trial and error. You don't know a priori if a constraint is active. You consider all the possible cases satisfying your ...
- 1,262
7
votes
Applications of Optimal Transport in Economics
Optimal transport methods are very much still in use in economics. The show up in two-sided matching with side-payments, contract theory, hedonic pricing, partial identification in econometrics, and a ...
- 13.5k
7
votes
Accepted
When can one drop time subscripts? Example from Angrist and Kugler (2003)
Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the ...
- 2,506
7
votes
Accepted
Regression Optimization problem under constraints
The situation is given in the following picture
The black line is the true conditional mean $E(y|x)$. If we truncate the data, all observations above the truncation $Y^A$ are not observed.
For low ...
- 12.5k
6
votes
Complementary slackness conditions (Kuhn-Tucker)
It is possible to have
$$g(x^*) = b\; {\rm and}\; \lambda^* = 0$$.
When the multiplier is zero and the constraint is equal to zero, then
a) The constraint does not really "bind"
b) That's ...
- 33.9k
6
votes
Accepted
Perfect substitutes and Lagrange
Your Lagrangian would be
$$L = (ax+by)+\lambda (I−p_x x−p_y y) +\mu_x(x−0)+\mu_y(y-0),$$
where the final two terms represent the restriction that $x,y\geq0$.
You then arrive at conditions
$$\frac{\...
- 5,291
6
votes
Visualizing the expenditure minimization problem
I am not sure what you mean - the visualization is essentially the same, only the roles of the goal function and the constraint are switched. Given the appropriate utility and income levels the ...
- 29.6k
6
votes
Accepted
Lagrange Multiplier Dual Meaning?
The Lagrange multiplier is both of the above, both statements are equivalent.
In the classic consumer problem
$\max U(x,y)$
s.t. $p_x x + p_y y = I$
whose Lagrangian is
$\mathcal{L}(x,y,\lambda) = U(x,...
- 2,259
6
votes
Cost Minimization and Karush-Kuhn-Tucker
Here is the cost minimisation problem that we need to solve:
\begin{eqnarray*} \min_{x_1,x_2} & w_1x_1+w_2x_2 \\ \text{s.t. } & \sqrt{x_1x_2}=\overline{y} \\ \text{and } & x_1\geq 1, x_2\...
- 9,421
5
votes
Does the Marshallian demand function always include prices and income?
Given that $u(x_1,x_2, x_3)=x_1x_2+x_3$, demand is the solution to the following problem:
\begin{eqnarray*} \max_{x_1,x_2,x_3} & x_1x_2+x_3 \\ \text{s.t. } & p_1x_1+p_2x_2+p_3x_3 \leq I \\ \...
- 9,421
5
votes
Accepted
Transformation Function
I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was ...
- 76
5
votes
Why couldn't the Karush-Kuhn-Tucker optimization find the solution?
Some confusion and incorrect statements in the answers already given, including the "accepted" answer. (E.g. obvious distinctions between necessity and sufficiency of different conditions for KKT are ...
- 2,619
5
votes
Accepted
How do you formulate a distance constraint and a budget constraint?
Because you are talking about constraints, it appears you do not consider the case of inserting such "access costs" (because this is what they are) in the utility function. It implies that ...
- 33.9k
5
votes
Accepted
setting of Lagrangian function
It's a matter of choice how one writes the Lagrangian in the context of Lagrange/KKT. Depending on how it's written, the gradients of the objective and constraint functions are either parallel or anti-...
- 2,619
5
votes
Accepted
Contradictory FOC and maximizing solution
As alluded to by Bertrand in his +1 comments this is because FOCs do not tell you where maximum or minimum occurs. This is common misconception among some students but it simply does not hold.
FOCs ...
- 57.7k
5
votes
Accepted
How do I find the socially optimum and equilibrium value?
You need to think about what the total costs are and what the marginal costs are. The social optimum is where marginal costs are equal to the outside option which is riding the bus.
The story goes ...
- 3,798
5
votes
Real Life Examples of Optimization in Economics
One could use simple games of resource management, this way the scope is fairly finite. I do research in digital games and economics, which students tend to be interested in. Some relevant papers are ...
- 3,477
5
votes
Real Life Examples of Optimization in Economics
So one thing to keep in mind is that the "simple" examples you're looking at are very much real-world. Any particular optimization decision needs to be simple enough for the people making ...
- 151
5
votes
Real Life Examples of Optimization in Economics
The NFL uses optimization when scheduling.
In 2013, the NFL began using Gurobi’s mathematical optimization solver to tackle this incredibly complex scheduling problem. With mathematical optimization, ...
- 151
5
votes
Accepted
Calculating cost-minimizing inputs with 3 inputs in production function
The production function
$$F(x) = x_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3},$$
has the derivative with respect to $x_j$ where for sake of example I choose $j=1$
$$\frac{\partial F(x)}{\partial x_1} = \...
- 3,798
5
votes
Accepted
Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?
If by $\lambda_i$ you mean the multiplier belonging to constraint ($i$), then $\lambda_i$ and $\lambda_j$ being positive do mean that these constraints are active/effective/realized as equalities.
Now ...
- 29.6k
5
votes
Accepted
How can I show convexity of this value function?
Suppose that $u(C,l)=\sqrt{C}-l^2$ and $f(l,A)=\big(l+g(A)\big)^2$, where $g$ is any function of $A$ that is not convex.
Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$
The optimal labor supply is given by $...
- 13.5k
5
votes
Accepted
Concave utility functions solution example
This is the problem we want to solve:
\begin{eqnarray*} \max_{x_1, x_2} & x_1 +\ln x_2 \\ \text{s.t.} & \ p_1 x_1 + p_2x_2 \leq w \\ \text{and} & \ x_1\geq 0, x_2>0 \end{eqnarray*}
Here ...
- 9,421
5
votes
Accepted
What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?
On pages 53-55 of the Stokey, Lucas, with Prescott (1989) book they discuss the Contraction Mapping Theorem. This theorem guarantees existence and uniqueness of the solution (one fixed point).
The ...
- 574
5
votes
Accepted
Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
Proof by contradiction. Suppose $(x_1^*, x_2^*)$ solves the problem and $x_1^* \neq x_2^*$ is true. By symmetry of the utility and equal prices, $(x_2^*, x_1^*)$ which is not the same bundle as $(x_1^*...
- 9,421
Only top scored, non community-wiki answers of a minimum length are eligible