13
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Marshallian Demand for Cobb-Douglas
Since $a + b=1$ the equations are exactly the same. Substituting in for $a+b$ with $1$ in the third and fourth equations gives the first and second equations.
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votes
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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 ...
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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 ...
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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 ...
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7
votes
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Why couldn't the Karush-Kuhn-Tucker optimization find the solution?
As @user32416 pointed out the first order stationarity conditions are not enough. Specifically it seems that you violate Slater's condition, which states that "the feasible region must have an ...
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Dynamic Optimization: What if the second order condition does not hold?
There is not a single answer, it will depend on the particulars of each problem. Let's look at a standard example.
Consider the benchmark intertemporal optimization problem for the Ramsey model
$$\...
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Marshallian Demand for Cobb-Douglas
This is how you get from your first equation to your second.
your utility function is $u(x_1, x_2)=x_1^a x_2^b$
since $a+b=1$ I'll change it slightly to a and (1-a)
In order to optimise these two ...
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Applications of Optimal Transport in Economics
Optimal transport methods are very much still in use in economics. The show up in two-sided matching with side-payments, contract theory, hedonic pricing, partial identification in econometrics, and a ...
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When can one drop time subscripts? Example from Angrist and Kugler (2003)
Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the ...
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Regression Optimization problem under constraints
The situation is given in the following picture
The black line is the true conditional mean $E(y|x)$. If we truncate the data, all observations above the truncation $Y^A$ are not observed.
For low ...
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Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Solving a non-linear programming (with inequality constraints) is about trial and error. You don't know a priori if a constraint is active. You consider all the possible cases satisfying your ...
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Is there a way to link Berge's theorem of maximum to Envelope theorem?
They are related and usually fall into the same discussion, but as @Alecos mentions in the comments, the two theorems show different things.
I suppose the connection that you're after is the fact ...
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Why couldn't the Karush-Kuhn-Tucker optimization find the solution?
This is an ill-posed question. Even without going through KKT, your constraint $(x + y - 2)^2 \le 0$, since the left-hand side is a square, means that the only solution that is feasible is the one ...
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Complementary slackness conditions (Kuhn-Tucker)
It is possible to have
$$g(x^*) = b\; {\rm and}\; \lambda^* = 0$$.
When the multiplier is zero and the constraint is equal to zero, then
a) The constraint does not really "bind"
b) That's ...
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Visualizing the expenditure minimization problem
I am not sure what you mean - the visualization is essentially the same, only the roles of the goal function and the constraint are switched. Given the appropriate utility and income levels the ...
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votes
Why couldn't the Karush-Kuhn-Tucker optimization find the solution?
Some confusion and incorrect statements in the answers already given, including the "accepted" answer. (E.g. obvious distinctions between necessity and sufficiency of different conditions for KKT are ...
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Leontief preferences
You are missing a $\min$ operator just before the bracket. The utility maximization problem is as follows, $$\max \ \min [\alpha x_1, ..., \gamma x_3] \\ \ \ \text{such that} \ \ \lambda_1x_1 + ... + \...
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votes
Reverse auction formula
A first price standard and reverse auction are formally equivalent to each other, and the same method can be used to solve both:
First Price Auction
In a first price auction, $n$ bidders choose ...
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5
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Derivative of CARA utility
All you need for this particular question is the following. Let $\mathbf{X}$ be a $T \times K$ matrix, $\mathbf{w}$ a K-dimensional vector and $\mathbf{y}$ a T-dimensional vector, then
$$
\begin{...
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Estimating the second derivative of function from optimizers
The first order condition of the maximization problem is
\begin{equation}
f'(x)-s=0\iff f'(x)=s
\end{equation}
We can then replace $x$ by $x(s)$ because this is the optimal value given $s$. Since ...
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Monetary policy optimization
You are unfortunately mistaken. DSGE models are at the heart of monetary policy and the most widely used class of models in this field. To work in monetary there is no real way around learning DSGE.
...
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How do you formulate a distance constraint and a budget constraint?
Because you are talking about constraints, it appears you do not consider the case of inserting such "access costs" (because this is what they are) in the utility function. It implies that ...
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Perfect substitutes and Lagrange
Your Lagrangian would be
$$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{\...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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} = \...
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