# Questions tagged [optimization]

Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.

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### Is anyone familiar with the following basic resource sharing model?

Here is a resource sharing model, I do not remember where I came across it, I am wondering if this is well known in econometrics. Let $T > 0$ be the total quantity of resources. For example, ad ...
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### Symmetric Cournot equilibrium: suffciency without second order conditon

Let $q_i \in Q = \mathbb R_+$ denote the quantity produced by firm $i \in \{1,2\}$. Further let $\pi_i(q_1,q_2) = (1-q_1-q_2)q_i$ denote the profits of $i$. A Nash equilibrium $(q_1^*,q_2^*) \in Q^2$ ...
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### Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]

A consumer with a utility function given by $u(x_1, x_2) = \sqrt{x_1} + x_1x_2$ has an income of $100$. The unit prices of $x_1$ and $x_2$ are $4$ and $5$, respectively. (a) Compute the utility-...
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### Calculate optimal discount for product bundling

So recently I made some rules with my transaction data. Based on it I can determine which products are profitable to bundle it together. But even though I know e.g. product A→ product B, are there ...
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### How can this be proved? (Convex optimization)

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1$ and $\Omega_2$...
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### Generalizing demand for perfect substitutes utility function

I have the utility function: $U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero) $\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
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### When the global optimal is outside of the constraint set, what will be the demand?

$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex. $a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that ...
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### A maximization problem with multiple goods and integrated markets

Update: I will try to clarify the question: Let us say that the total harvest of the fish population at time t is $H_t$. Every harvest produce three types of fish: salmon ($f_1$), which is valuable ...
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### Simplex Lp interpretation of dual problem´s solution

I am wondering whether my interpretation of my simplex dual problem result is correct. The primal problem is: ...
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### Concavity of Cobb-Douglass Utility Function on Non-Open set

My textbook argues that the Cobb-Douglass utility function $u=(x1)^a(x2)^b$ with $a,b>0$ and $a+b<1$ is concave on $R2+$ by computing the Hessian and showing it to be negative semidefinite for ...
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### Taking the partial derivative of the demand function

Define the demand function which maximizes x -> U(x) as: $\sum_{i=1}^n$$p_i$$\zeta_i$(p, I) = I According to my textbook if I differentiate this with respect to $p_j$ I will obtain, $\zeta_j$(p, ...
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### 'Constrained optimisation' for mutually exclusive goods?

Taking the standard approach to constrained optimisation, where we maximise utility subject to a budget constraint with some allocation on the consumption of two goods, does it make apply the same ...
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### Optimization problem of a Cobb-Douglas function with 3 inputs

A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function: $$Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3}$$ ...
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### Is optimizing revenue and expense objectives simultaneously better than optimizing profit as composite objective?

In the profit maximization problem, I am curious if co-optimizing revenue and expense objectives simultaneously are better than optimizing profit (revenue - expense) as a single composite objective? I ...
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### On Demand Functions and Engel Curves

A consumer has utility function $U(x,y)=(x−2)y$, where $x≥2$ and $y≥0$. The price of $x$ is $P_x$, the price of $y$ is $P_y$ and the consumer's income is $I>2P_x$. ($x$ and $y$ do not have to be ...
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### Kuhn Tucker Maximization

I have to maximize following expected utility function using Kuhn tucker conditions - Since expected utility function are increasing $C_{1,t}$ and $C_{2,t}$ so constraints (i) and (ii) will hold with ...
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### Corner solution of the maximization problem

Answer Hello, I upload the actual question with my 8-pages answer. Please can you check it. Is there a corner dissolution for $c=\gamma$. Please share your ideas. Thanks.
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### A profit maximization problem (whole problem has been solved, I just have question about interpretation)

I would like to discuss with you about the following production function. $$y=f(t_m, t_l)=\rho t_m^m(n+t_l)$$ where $0<m<1$ and $n>0$ are fixed parameters. $t_m$ is manager time. $t_l$ ...
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### Revenue maximization problem

There are $N>0$ Households in an economy. The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones. ...
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### The centralized shift from barter to currency economy

Suppose some ancient king of small bronze age city-state wants to introduce universal currency instead of barter that is currently in overwhelming practice in his kingdom. In order to smooth the shift,...
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### Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)

I have question on central bank loss function. We know that the central bank loss function is $$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$ And we know that fisher equation is $$i=r+\pi^e$$...
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### Numerical Solution Using Excel about optimal consumption of households

I'm not sure how to solve this problem. I'm given the discount factor, interest rate, probability of high income shock, and various income shock sizes that I need to use to compute optimal consumption....
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### derive value function from utility function

We have the utility function. $$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$ And I am trying to find the value function. $U$ is utility function. $c_t$ is consumption at ...
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### Can be the duality theorem applied to not locally non-satiated utility functions?

I have the following not locally non-satiated utility function: $$U(x,y)=-(x-1)^2-(y-2)^2$$ where $U(x,y): \, \!R^n_+ \rightarrow \!R$ The 3D plot of this function is an infinite paraboloid; ...
use the general form of the Cobb Douglas utility function $U(x,y)= (x^a)(y^b)$ and the budget constraint in the form $B=p_{x}X + p_{y}Y$ to find the demand functions for good x and good y. Is this ...