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 ...
bixiou's user avatar
  • 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 ...
Giskard's user avatar
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8 votes
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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 ...
Jesper Hybel's user avatar
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8 votes
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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 ...
Tony's user avatar
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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 ...
Michael Greinecker's user avatar
7 votes
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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 ...
Alalalalaki's user avatar
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7 votes
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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 ...
tdm's user avatar
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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 ...
Alecos Papadopoulos's user avatar
6 votes
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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{\...
Bayesian's user avatar
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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 ...
Giskard's user avatar
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6 votes
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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,...
Nicolas Torres's user avatar
5 votes
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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 ...
Julia B's user avatar
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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 \\ \...
Amit's user avatar
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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 ...
Michael's user avatar
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5 votes

Interpretation of lagrange multiplier

As mentioned in the other answer, the Lagrange multiplier is the marginal effect on the value (optimized) function, when the constrained is "relaxed" marginally. In your case then it should be ...
Alecos Papadopoulos's user avatar
5 votes

Monetary policy optimization

You are unfortunately mistaken. DSGE models are at the heart of monetary policy and the most widely used class of models in this field. To work in monetary there is no real way around learning DSGE. ...
BB King's user avatar
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5 votes
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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 ...
Alecos Papadopoulos's user avatar
5 votes
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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-...
Michael's user avatar
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5 votes
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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 ...
1muflon1's user avatar
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5 votes
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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 ...
Jesper Hybel's user avatar
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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 ...
RegressForward's user avatar
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 ...
Perkins's user avatar
  • 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, ...
gomory-chvatal's user avatar
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} = \...
Jesper Hybel's user avatar
  • 3,366
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 ...
Giskard's user avatar
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5 votes
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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 $...
Michael Greinecker's user avatar
5 votes
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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 ...
Amit's user avatar
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5 votes
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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 ...
Fića's user avatar
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5 votes
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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^*...
Amit's user avatar
  • 8,476
4 votes
Accepted

Solving a maximization problem by substitution when the constraint is in implicit form

Here are two methods. First method: the substitution can be made by inverting $f$. Since $f$ is strictly increasing and continuous, $f^{-1}$ is well-defined. The constraint can therefore be written \...
Oliv's user avatar
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