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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2answers
73 views

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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49 views

Trying to find an optimization method

I'm working on a research about fertility choice and economic growth and I was trying to solve an Ahituv Moav paper where they develop a model of economic growth. My main problem is when I'm trying to ...
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2answers
89 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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1answer
85 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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1answer
36 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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2answers
584 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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2answers
185 views

Dynamic programming, optimal consumption-savings (finite horizon) problem

Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$, the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ ...
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1answer
55 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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36 views

Show that none of these critical points identifies a solution of the profit- maximization problem. Can you explain why this is the case?

Question: a firm produces a single output $y$ using three inputs $x_1,x_2,x_3$ in non-negative quantities through the relationship: $y=g(x_1,x_2,x_3)=x_1(x_2+x_3)$ The unit price of $y$ is $p_y>0$...
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31 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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26 views

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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1answer
43 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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4answers
819 views

Why couldn't the Karush-Kuhn-Tucker optimization find the solution?

I have the following utility maximization problem: $$\max (xy)$$ $$(x+y-2)^2 \leq 0$$ Conditions: $$y-2\lambda (x+y-2) =0$$ $$x-2\lambda (x+y-2) =0$$ $$\lambda(x+y-2)^2=0$$ When I set $\lambda>0$, ...
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1answer
85 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
53 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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1answer
68 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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1answer
183 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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1answer
152 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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1answer
33 views

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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4answers
935 views

Method of Lagrange multipliers with random variables

I'll illustrate the issue I'm having with a simple problem. Let $c_1, c_2 \in \mathbb{R}$, and $Z$ a real-valued random variable. Let $u:\mathbb{R} \rightarrow \mathbb{R} $ be a differentiable ...
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2answers
50 views

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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1answer
29 views

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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1answer
40 views

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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0answers
20 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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8 views

'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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35 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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28 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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29 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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34 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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1answer
57 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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0answers
25 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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1answer
221 views

Karush-Kuhn-Tucker in infinite dimension

Does the Karush-Kuhn-Tucker theorem on sufficient conditions for optimality of a convex program apply in countable dimension? For precisions, see Definition 4.1.1 and Theorem 4.1.4 of this course. ...
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42 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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28 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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1answer
70 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 ...
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2answers
211 views

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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1answer
150 views

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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0answers
99 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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0answers
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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1answer
127 views

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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0answers
17 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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0answers
49 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
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; ...
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0answers
12 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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1answer
38 views

Can I Upload my Preprint on Arxiv Before Submitting it to JPE

I wrote a paper relating the optimal deterrence strategy for crime to concepts on statistical physics, and I am considering submitting the paper to the Journal of Political Economy. I'm wondering if ...
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2answers
207 views

Cobb Douglas, Budget Line, Demand function question

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 ...
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2answers
120 views

Optimal point and MRS

I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that ...
3
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1answer
105 views

Maximising a partly concave and partly convex function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable and strictly increasing function. Suppose that we are searching for the numbers $x_1$, ..., $x_n$ that maximise $$\sum_{i=0}^{n}{f(...
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1answer
50 views

Have I found the correct Emission Price

Let's say that there is a hotel owner $(H)$ and a woodworker $(W)$ working in close proximity to one another. The woodworker produces $x$ units to sell at market at $p_{x}=6,5$. From the woodworking ...
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1answer
40 views

Typical Growth Model Social Planner Problem

Consider the following social planner's problem, hand-waving the usual assumptions on the preference, technology, endowment, and inelastic supply of labor: $V(k_o^*)= max_{\{c_t,k_{t+1}\}_{t=0}^{\...