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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Solve the Ben-Porath Model (Optimal Control Problem)

Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
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194 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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199 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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49 views

Optimization problem of the firm

I have been reading an Economics working paper and trying to derive the first-order conditions of a seemingly complicated optimization problem. The optimization problem with choice variables $P_{t}^{R}...
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108 views

Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?

I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving in Matlab and Python and could not find any) so that I cannot get an explicit expression ...
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123 views

What are the boundary value conditions for generic HJBs in economics?

Consider a routine continuous time optimization problem: $ V(t,a_{t}) := \max \int_{\tau=t}^{\tau = T} e^{-\rho (\tau -t)} u(c_{\tau})d\tau $ $\text{ s.t. }$ $\dot{a}_{t} = y + ra_{t} - c_{t}$, $a_{...
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71 views

How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?

Statement: Let the dynamics of wealth of the agent satisfy $$dX_{t} = \pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$ where $(\pi,c)$ is an ...
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135 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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67 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 ...
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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 ...
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1answer
117 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; ...
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45 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 ...
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436 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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64 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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270 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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2k 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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332 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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26 views

Total Derivative of a Max Function: Maximizing Social Welfare Function

I'm studying public economics but my question here is purely mathematical in nature. I have a function: $$ V(1-\tau, R) = \max_zu((1-\tau)z+R,z) $$ I need to take the total derivative of this, in my ...
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83 views

Conic optimization in economics

Are there any mainstream economic models that rely on conic optimization to solve for decision variables? Conic optimization is a type of convex optimization problem, different from linear and ...
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25 views

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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26 views

Solving a HJB with additional constraints on control and state variables

I am trying to solve a Hamiltonian-Jacobi-Bellman equation with additional constraints on the state and control variables, but I am a bit confused on how to do that. In Intrilligator 2002, it is ...
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1answer
172 views

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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44 views

Analytical approach to estimate equilibrium price for Real Estate Property

I am looking to calculate the equilibrium price, i.e an optimal price that I can set without affecting demand and maximize revenue. I've gathered historical data: occupancy rates, asking rents for ...
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109 views

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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50 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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114 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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26 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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111 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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30 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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91 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
373 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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218 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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22 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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56 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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103 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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58 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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93 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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1answer
57 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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31 views

Stochastic Optimization

I am working on a Data Panel project that is about macro volatility, taking into consideration some quality indicators. According to my model, indicators definitely determine macro variables. Data are ...
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39 views

Optimization with tax and deduction constraint

I want to find optimal demands for a utility function in the form: $$U_{1}(c_{1},c_{2})=c_{1}^{1-a}\cdot c_{2}^{a}$$ It is related to an altruistic behaviour, where $c_1$ is the consumption of the ...
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27 views

Solving for parameter value

I have the following maximization function - $\max_{x \in (0,1)} (((p_1e_1x^2)^{r} + (p_2e_2(1-x)^2)^{r})/2)^{1/r}$ where, $p_1$ and $p_2$ are drawn from uniform distribution [0,1] and are considered ...
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9 views

Algorithms/Models to solve minimal Matchings for consumer producer household pairs

I’m working on the following problem: Minimising the electricity price for household trading pairs. There’s producer and consumer households. Trades are just possible between producers and consumers. ...
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53 views

Cost-optimal p2p-trade in a community of households

I’m trying to solve the following problem and I’ve been working on it for a long time already: I want to optimize electricity-costs in a smart grid. There’s producer and consumer households in the ...
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53 views

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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30 views

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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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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97 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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2answers
719 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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1answer
248 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: ...