Questions tagged [optimization]

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

42 questions with no upvoted or accepted answers
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177 views

Dynamic demand model in many good competitive markets and price optimization

This is a question about demand models, price optimization, dynamic pricing, big data, online learning, so I will cross-post in other communities. $\mathbf{Background}$ I am interested in dynamic ...
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171 views

Solution to Dynamic Programming (Bellman Equation) Problem

Could someone please provide pointers on how to solve the below? If any theoretical approximations are possible, that would be very helpful. If numerical solutions are the right approach, could you ...
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118 views

Appropriate economic/econometric tools to analyze segmented promotion optimization problem

I'm trying to determine which micro-economic/econometrics concepts, models, and/or tools are appropriate for an analysis of promotions. Below I Describe the problem in general terms Give ...
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66 views

Converging Trajectories and Sufficiency for Optimality

(The question is loosely relatet to this thread.) In the paper "Feedback Equilibria for a class of non-linear Differential Games" by Mäler et al. it is stated (p. 14) In fact sufficiency is ...
3
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61 views

Optimisation of bonds

I'm doing an optimisation problem but don't understand what the terms mean. Suppose someone wants to invest $110,000. They have 4 choices as to what they invest their money into: municipal bond ...
3
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1answer
101 views

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; ...
2
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1answer
130 views

Can $u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ be solved by Kuhn–Tucker conditions?

Consider $\max_{x_1, x_2, x_3, x_4} u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ s.t. $\; p_1x_1 + p_2x_2 + p_3x_3 + p_4x_4 \le w$ I know we can solve the max problem through separately considering ...
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40 views

Finding savings in an Overlapping Generations model

I have not seen this question asked anywhere, so I'm posing it here in case anybody else (hopefully) can help me get to the answer. In a nutshell, my question is: how do we arrive at the saving ...
2
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395 views

Second order condition for symmetric game

Denote by $i \in \{1, \ldots, n\}$ an economic agent. Let $\mathbf x \in \mathbb R^n$ denote a vector of actions and $x_i \in \mathbf x$ a typical element. Let further $f_i : \mathbb R^n \to \mathbb ...
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63 views

Does this conditional increase in income affect a budget line in the same way as an unconditional increase in income would?

It's been awhile since I've taken introductory microeconomics. I remember increases in income move budget line outward. What if the increases have some condition? The problem: Jill has $I$ to ...
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240 views

Optimizing Cobb-Douglas like functions

Economics isn't my home field, but I'm looking for references for a paper I'm working on and I'm hoping one of you can help. Are there many good references for optimizing a Cobb-Douglas like utility ...
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1k views

First and Second Order Lagrangian-Multiplier Conditions for Optimization

The Statement of the Problem: Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier ...
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241 views

Calculating the optimal portfolio for an investor with quadratic utility

The problem is from Asset Pricing and Portfolio Theory by Back and can be found here. The relevant info from section 2.5 can be found here. Given that we have the Expected value and the variance of ...
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15 views

Kuhn-Tucker conditions in linear cost minimization

Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by $$ f(x,y) = ax + by $$ and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} ...
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22 views

What does the elasticity say about the fraction of total cost used on input 1?

A firm have the following production function $$ y=x_{1}^{\alpha} x_{2}^{1-\alpha}, \quad 0< \alpha < 1 $$ $w_1>0$ is the cost of input 1 and $w_2 > 0$ is the cost of input 2. (1.1) ...
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86 views

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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29 views

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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43 views

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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1answer
40 views

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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1answer
76 views

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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1answer
141 views

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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137 views

Natural borrowing/debt limit and other borrowing constraints

When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading) $$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
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30 views

Optimal population allocation layout over the Earth

Does exist some model of optimal people occupancy over whole our planet? Something accounting for climate, resources (with and without existing settlements) availability, progress and grows ...
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21 views

Perfect complement outputs with each output being composed of substitutable inputs

How does one solve the following maximization problem? $\underset{K_1, K_2, L_1, L_2}{\text{maximize }} min\{K_1 + L_1,K_2 + L_2\}$ subject to $c(K_1 + \mu K_2) + \beta c(L_1 + \mu L_2)$ where $c(...
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43 views

Local maximum when Hessian is negative semi-definite?

If it possible to have a local maximum when the Hessian is only negative semi-definite (i.e., there is one zero eigenvalue and all other eigenvalues are negative). If not, what it the ultimate ...
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95 views

Dynamic programming: verification principle

Consider the Gale cake problem with $u(c) = log(c)$, so that the problem becomes:$$\max_{x}\sum_{t=0}^{\infty} \beta^{t}log(x_t-x_{t+1})\: sub\: 0<x_{t+1}\leq x_t,\: x_0>0\ given$$ It is ...
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55 views

Separate optimization problem versus one big Lagrangean

I am a bit confused about when to consider a separate optimization problem and when to combine different problems into a single optimization problem. Consider an individual who derives utility from ...
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88 views

Simple Neoclassical Growth Model with Elastic Labor and Non-Standard Capital Adjustment Costs

I have the following social planner problem to maximize $\{c_t, k_t, n_t \}$ $\begin{gather*}E_0 \sum_{t=0}^\infty \beta^t U(c_t, 1 - n_t), 0 < \beta < 1\end{gather*}$ subject to $\begin{...
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24 views

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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39 views

What is the economic interpretation of the solution of this optimal control problem?

I have the following optimal control problem $$\max_{c_t} \int^{\infty}_0 e^{-p_it}\ln(c_t(i))dt$$ subject to $$\dot{w_t}(i)=rw_t(i) -n_ic_t(i)$$ $$w_0(i)=w_0>0$$ I have some wealthy and ...
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27 views

Second-Order Conditions under Kuhn-Tucker Formulation

How should I address second-order conditions if I use the Kuhn-Tucker formulation of constrained optimization as opposed to the usual one? For instance, suppose an agent wishes to maximize $f(x_1, ...
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16 views

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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11 views

Designing the payment function in Mechanism Design problems

Suppose we have a network in which agents request access to its resources. Thus we have a resource allocation problem. Ideally, we want to incentivize agents to send social-welfare supporting ...
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27 views

optimization problem for two individuals

Two flat mates 1 and 2, rent a flat and play their own music on the only CD player owned by flat-owner. They both like their own music, but dislike the music played by the other. Given the timing ...
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27 views

Constrained Optimisation: Why is it that when I merge constraints, I get different results?

The problem that I am given is the following: $ \max \ln c_0 + \beta \mathbb{E} [\ln c_1 ] \\ \text{ s.t. } c_0 + x_g q_g + x_b q_b = y_0\\ c_g = y_g + x_g\\ c_b = y_b + x_b $ Where $y_0$, $ y_b$ ...
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51 views

Finding the optimal consumption bundle given the strictly concave utility function $v(x,y) = U(x) +y$?

I am also finding it difficult to understand what are the fundamental differences between analysing optimal bundles between concave and convex functions ? Does it also happen that the optimal bundle ...
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1answer
41 views

Neoclassical model with proportional taxes

In a certain economy, time is discrete with periods $t=0,1,2,...$. The economy is populated by many households and identical firms. The utility of a household is: $\displaystyle\sum^{\infty}_{t=0}\...
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16 views

Monetary Policy under commitment. How to solve the optimization problem?

Under commitment the CB might follow this problem as Monetary Policy strategy: $$ \min_{\pi_t,x_t}=E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( \pi_t^2 + \alpha x_t^2 )\right) $$ $$ \text{s.t. }\...
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2answers
537 views

Price optimization with demand forecast

I have one year sales data of a retail company and lets say I am forecasting the next month sales for the product. I have got the sales using time series in R. Now I want to forecast the price as well....
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42 views

Differentiability vs Continuous Differentiability

Why do optimization textbooks, as well as appendices for MWG JehleReny etc. simply state conditions in terms of continuous differentiability, even when the weaker condition of differentiability is ...
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1answer
226 views

Price Optimization from Data

How can I find the optimal price that maximizes profits, given past sales data? I thought I could do this, but I've been running into problems. Data: ...
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11 views

Crusoe Economy maximization

u(c,l)=c^a(1-l)^a-1 0<a<1 Y=Al^b Find the optimum c and l.