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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How can you interpret one of the parameters of optimal consumption at 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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37 views

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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70 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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1answer
57 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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24 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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19 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
64 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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165 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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97 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
95 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
61 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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41 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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40 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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30 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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103 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
38 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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2answers
57 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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37 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
55 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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26 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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9 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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52 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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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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96 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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35 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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57 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
74 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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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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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
75 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
194 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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239 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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102 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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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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1answer
160 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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1answer
152 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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61 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
39 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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295 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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196 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 ...
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1answer
107 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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44 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}^{\...
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1answer
60 views

Stokey, Lucas (1989) p 11 FOC

My Lagrangian is: $L=\sum\limits_{t=0}^T \beta^tU(f(k_t)-k_{t+1})+\sum\limits_{t=0}^T\lambda_t(f(k_t)-k_{t+1}).$ My FOC for $[k_{t+1}]$ is: $\beta^tU'(f(k_t)-k_{t_1}^*)(-1)-\lambda_t^*+\beta^{t+1}U'...
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81 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
445 views

Maximization problem FOC and Euler equation

Can someone please help me with the Lagragian and the derivation of the following objective function ? Beneath I provide the objective function, the constraint and the Euler equation that results from ...
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1answer
53 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
95 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
79 views

Pareto efficiency (optimality conditions) in simple New Keynesian model

I am looking for the pareto-optimal equilibrium for a central planner's problem in a simple New Keynesian model. The planner's problem is to choose $\{ C_{t}, H_{t}, Y_{t}, \pi_{t}, \{h_{t}(j)_{j=0}^{\...