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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26 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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37 views

Lagrangian multiplier and optimal bundle

I would like to know where I am wrong (if I am) and why I am wrong here please: If a consumer has an income of 600 euros to spend for good x (Px = 10 euros) and good y (Py = 5 euros). What is the ...
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It is argued that in optimization the first part of the second-order condition appears opposite to their interpretation? [closed]

It is argued that in optimization the first part of the second-order condition appears opposite to their interpretation
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58 views

Contradictory FOC and maximizing solution

I have to maximize the following function - $\max_{x \in (0,1)} (((p_1x)^{2r} + (p_2(1-x))^{2r})/2)^{1/r}$ where, $p_1$ and $p_2$ are drawn from uniform distribution [0,1] and are considered to be ...
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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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72 views

Find Pareto optimal allocations and the core for the following economies

Find Pareto optimal allocations and the core for the following economies. There are two consumers and two goods. Utility functions are $u_1(x_1,y_1)= 10x_1-(y_1-2)^2$ and $u_2(x_2,y_2) = 10y_2 − (x_2 −...
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52 views

Essential goods: How does one restrict the utility function?

I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure ...
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25 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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Illustrating Karesh Kuhn Tucker with two non-nonnegativy constraints binding

I'm teaching Karesh-Kuhn-Tucker, and looking for papers, ideally in the fields of development, agricultural or environmental economics, and ideally in good journals, that I can use to illustrate the ...
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142 views

What does binding mean?

I am curious how to solve the utility maximization problem if the representative agent has borrowing constraint.
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62 views

setting of Lagrangian function

Consider a simple consumer's problem: Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$ $w$ is initial endowment. We can set the Lagrangian function to solve this problem. $L=u(X)+\lambda ( \...
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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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39 views

Is a binding ZLB a binding constraint?

Usually, in an optimisation problem, a binding constraint is one at which the optimal solution holds at the constraint with equality, i.e. it's a boundary solution. However, in many articles, for ...
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68 views

Perfect substitutes and Lagrange

How does one solve utility maximization of perfect substitutes using Lagrangian function? Consider the problem $$\max_{x,y} ax +by $$ subject to the constraint that $$px + qy \leq I$$ where $a,b,p,q,...
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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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1answer
18 views

The optimal price for a demand curve with a steep slope

Given the demand function, $$D(p)=A-ap$$ I've found the optimal price, $$p=\frac{A+ac}{2a}$$ Where $c$ is cost and $A,a >0$. My question is how is the optimal price is dependent of $a$ (1) - what ...
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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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93 views

Applications of Optimal Transport in Economics

The 1975 Nobel Prize winner in Economics was Kantorovich who reformulated the optimal transportation theory of Monge and applied it to optimal resource allocation. The Wasserstein distance is central ...
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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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39 views

Practice question on Correspondences and maximization

We're learning about Theory of the Maximum. I tend to struggle with correspondences in this context, so I'm trying to work through some practice questions. I will start with some general notation of a ...
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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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54 views

How do you formulate a distance constraint and a budget constraint?

Everybody knows about budget constraints and how they are represented: but what if I want to represent a distance constrain from the shop you buy the goods? How can I build that?
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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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71 views

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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58 views

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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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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25 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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103 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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174 views

Solving a HJB with a probability to transit to a new state

I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time. The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}...
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105 views

In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?

In lecture, my professor defined the following 2-period consumption model: $c_i = $ consumption in period $i$. $y =$ endowed income in period 1. $r = $ interest rate in perfect credit markets. $h = $ ...
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110 views

Solving Constrained Optimization Problem with Two-Period Model of Human Capital

I'm trying to solve a constrained optimization problem in human capital model. The objective function is $\max_{c_1,c_2,\nu} U = u(c_1) + \beta u(c_2)$, subjected to $c_1 = w +(1-\nu)\theta_1 h_1^a$ ...
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71 views

Can the weierstrass and the Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?

Question: An agent who consumes three commodities has a utility function given by: $u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$ Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’...
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45 views

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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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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Using ML to estimate demand function

Say, I am looking to estimate the demand curve for rental of a real estate property. The demand varies depending on time of the year, location, economic and demographic variables. I'd like to ...
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159 views

How to find the Utility Possibility Frontier when there are Perfect Substitutes?

I am trying to derive the Utility Possibility Frontier (UPF) when both utility functions display perfect substitutes (in an Edgeworth economy with to consumers and two goods). The specific problem: $...
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Expectational stability: adaptive learning of RE equilibria in dynamic systems

There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand. Step 1. What does this formula below mean, ...
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Economics of Justifying N95 masks and Mass COVID testing [closed]

The US has shutdown a significant fraction of its economy because of COVID-19. Eventually we will all migrate in a pre-COVID direction. Obviously, too fast would be a medical disaster, too slow ...
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Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line?

This a person's budget line with various points, and their consumption, C*, and their endowment e, which is worth $5000 (unimportant). Also shows is their initial indifference curve. The difference ...
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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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Complementary slackness conditions (Kuhn-Tucker)

Consider the problem of maximising a smooth function subject to the inequality constraint that $g(x) \leq b$. The complementary slackness condition says that $$ \lambda[g(x) - b] = 0$$ It is often ...
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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 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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116 views

Budget Constraint in Utility Maximisation Problem with Lagrange Multipliers

Lets say we have a utility function $U: \mathbb{R}^{2} \to \mathbb{R}$ given by $U(x,y)$ and a binding budget constraint $p_{x} x + p_{y} y = m$, where $p_{x}, p_{y}$ are prices of goods $x,y$ and $m$ ...
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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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78 views

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 ...