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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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13
votes
1answer
1k views

First Order Condition for Profit Maximization in Gambling Industry

I am working on a model of optimal payout percentages in the gambling industry. Because the nominal price of a \$1 ticket is always \$1, we use an effective price strategy where Q = \$1 in won ...
11
votes
6answers
983 views

References to learn continuous-time dynamic programming

Does anyone know of good references to learn continuous-time dynamic programming? The references don't have to be books. They could be links to online resources as well. Links to clear, concise ...
10
votes
2answers
26k views

Marshallian Demand for Cobb-Douglas

When trying maximize the utility having a cobb-douglas utility function $u=x_1^ax_2^b$, with $a+b = 1$, I found the following formulas (Wikipedia: Marshallian Demand): $x_1 = \frac{am}{p_1}\\ x_2 = \...
9
votes
1answer
654 views

Dynamic Optimization: What if the second order condition does not hold?

Consider the following dynamic optimization problem \begin{align} &\max_u \int^T_0{F(x,u)dt}\\ \text{s.t.}~& \dot{x} = f(x,u) \end{align} FOCs The Hamiltonian is given by \begin{align} H(x,u,...
8
votes
2answers
595 views

Is there a way to link Berge's theorem of maximum to Envelope theorem?

Berge's theorem states Let $X \in \mathbb R^m, \Theta \in \mathbb R^n $, $f : X \times \Theta \to \mathbb R$ be a jointly continuous function, $C : \Theta \rightrightarrows X$ be a continuous(both ...
8
votes
1answer
5k views

Leontief preferences

I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really ...
7
votes
3answers
736 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 $\lambda>0$, ...
7
votes
2answers
111 views

Identification of switching costs from price shocks

What, if anything, can we learn about customer switching costs by looking at price, revenue, profit, and quantity responses of producers to cost shocks? For example, we can define the profit equation ...
7
votes
1answer
198 views

Optimization: Dynamic Programming vs Kuhn-Tucker

Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to ...
7
votes
1answer
98 views

Mythbusters - Determine optimal boarding strategy based on time and satisfaction score

Most airlines board passengers starting from the back of the plane and then working their way towards the front (after boarding priority classes and passengers). In an episode of Mythbusters, Adam ...
6
votes
1answer
59 views

Overlaping jurisdictions Model: Proof of Lemma 1; The Size of Nations

I've been reading the book 'The Size of Nations' by Alberto Alesina and Enrico Spolaore (can be found on the net if you know where to look) and I'm having trouble following their "proof" of the first ...
6
votes
1answer
130 views

Simple Derivation of Maximum Principle

Consider the simplest problem of optimal control \begin{align} &\max_u\int^T_0{F(y,u)dt}\\ \text{s.t.} \quad&\dot y = f(y,u)\\ & y(0) = y_0\\ & y(T)~~\text{free} \end{align} ...
6
votes
1answer
231 views

Optimization of Households' utility in “ Rule-of-Thumb Consumers and the Design of Interest Rate Rules ” (Gali et al., 2004)

I can't figure out how the calculation of first order conditions was carried out. I can't figure out where the stochastic discount factor came from.
5
votes
4answers
678 views

Examples of non-differentiable problems in economics

As a research project, we're investigating various algorithms developed for non-differentiable, convex (or concave, if you're into economics) optimization. I'd like to find some good examples of real ...
5
votes
3answers
455 views

Kuhn Tucker Conditions with fewer non-negativity constraints than number of variables

I have a following type of problem: $Maximize\,\, F(s,x,y,z)$ $s,x,y,z$ s.t. (i) $g(x,y,z) \le I$ (ii) $x \ge 0$ (iii) $y \ge 0$ (iv) $s > 0$ That is there is no non negativity constraint on ...
5
votes
1answer
223 views

Estimating the second derivative of function from optimizers

Consider the following optimization $$x^*(s) = \max_{x\in X} \big(\,f(x)-sx\,\big)$$ where $f$ is assumed to be a strictly concave function and $X$ is an interval constraint, e.g $X = [0,b]$. We do ...
5
votes
1answer
743 views

Does the Marshallian demand function always include prices and income?

I have the following utility function: $$U(x_i)=x_1x_2+x_3$$ with budget constraint: $$p_1x_1+p_2x_2+p_3x_3\leq I$$ I use the Kuhn-Tucker method to find the optimal choices of the Utility ...
5
votes
1answer
94 views

Finding a maximal growth portfolio

I have the following problem that asks me to solve for the "maximal growth portfolio." Suppose that the equilibrium stochastic discount factor evolves as $$ \log S_{t+1} - \log S_t = \kappa_s(X_t,...
4
votes
2answers
2k views

Transformation Function

In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
4
votes
1answer
138 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. ...
4
votes
1answer
233 views

Calculating mean variance portfolio with risk aversion parameter

I want to calculate the classic mean variance portfolio (Markowitz) with a risk aversion parameter $\gamma$. I have the following problem where I want to maximize: $max(x_t) \ \ x_t^T\mu_t - \frac{...
4
votes
1answer
785 views

static/dynamic optimization

The interesting paper Calvo and Obstfeld (1988) uses two-stage optimization on an OLG model which then reduces to a standard representative agent framework. First stage optimization consists on a ...
4
votes
1answer
165 views

Markov decision processes, contractions and value iteration

I am reviewing Markov decision processes (MDP) and there is something I am missing with respect to the contraction argument. I am pretty sure it is a silly mistake somewhere (maybe computational), but ...
4
votes
3answers
165 views

A question about Lagrange multiplier(when $\lambda=0$)

I need help in a maximization problem(finding the optimal investment portfolio). where $R_s$ and $\Phi$ are $n$ by $1$, with other variables being scalars. $C^s$ is consumption (or wealth) of an ...
4
votes
0answers
168 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 ...
3
votes
2answers
551 views

Solving a maximization problem by substitution when the constraint is in implicit form

I am trying to understand how the first order conditions for an interior solution of a maximization problem were derived using the substitution method. The problem is: $$\max\limits_{x\ge0,y\ge0}P(a-...
3
votes
3answers
810 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 ...
3
votes
2answers
200 views

Indirect changes in Marshallian Demand

Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$. It can be shown that the Marshallian demand for $x$ ...
3
votes
1answer
110 views

Monetary policy optimization

I was wondering if anyone could give me some advice / lectures / introduction to stochastic optimization that could be applied to monetary policy. I have heard of the Dynamic stochastic general ...
3
votes
1answer
1k views

Reverse auction formula

I am studing a little bit of auction theory. I found the optimal bid value in the Milgrom paper for the first price auction that is $$ P=v \frac{n-1}{n} $$ where $P$ is the optimal bid, $v$ is the ...
3
votes
1answer
99 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(...
3
votes
1answer
209 views

Constrainted optimization: merge two constraints into one

Consider the following problem \begin{align} &\max_u F(x,u)\\ \text{s.t. }& u \in [0,\bar u]. \end{align} Any idea how to merge the two constraints $u \geq 0$ and $\bar u - u \geq 0$ into one ...
3
votes
1answer
217 views

Topkis' Theorem

Suppose my optimization problem is stated as follows $\max\limits_x f(x,t)$ $s.t.$ $g(x,t) \leq 0$ I am interested in finding the direction $x^*$ changes with the parameter $t$. Can someone ...
3
votes
1answer
342 views

Externalities - First order conditions

I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell on my own. I'm reading the externalities chapter, and i found an interesting example: There are just two firms: ...
3
votes
2answers
1k views

Why is instantaneous utility of current period discounted?

Consider a two period model of consumption. I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the ...
3
votes
2answers
2k views

Optimize by MR = MC vs TR = TC

I know that I should optimize production by solving $MR = MC$ with respect to $Q$. But if $TR > TC$, I am making a profit. Why is not enough to just solve $TR = TC$ with respect to $Q$?
3
votes
1answer
701 views

Portfolio choice problem of a CARA investor with n risky assets

Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
3
votes
1answer
107 views

Dynamic optimization with assets as state variable: interpreting capital gains and losses

Given a hamiltonian of the form: \begin{equation} H_{t} = ln(c_{t}) \dot{} e ^{-\rho t} + \lambda_{t}(w+ra_{t}-c_{t}), \end{equation} with $c_{t}$ consumption at time t (the control variable), $\rho &...
3
votes
0answers
301 views

Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$ [duplicate]

Suppose preferences are represented by the following utility function \begin{equation} u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \} \end{equation} Write the ...
3
votes
0answers
112 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 ...
3
votes
0answers
65 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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0answers
60 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
votes
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
votes
2answers
620 views

Interpretation of lagrange multiplier

A student wishes to minimize the time required to gain a given expected average grade, 𝑚, in her end-of-semester examinations. Let $\displaystyle {t}_{i}$ be the time spent studying subject i ∈ {1,2}....
2
votes
2answers
2k views

Utility maximization question setting up.

Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$ Assume the agent's income is $y=5$. The price of one unit ...
2
votes
1answer
251 views

Weierstrass Theorem in Optimization

Weierstrass Theorem states that any bounded sequence has a convergent subsequence. I did that in my maths course and understood it completely. But when I was learning optimization techniques in ...
2
votes
1answer
39 views

Computing optimum efforts

Consider the following cost function: $$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$ The value function is: $$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$ How do I compute the optimum efforts $e_1$ and $...
2
votes
1answer
268 views

CV, EV for additive utility; confirm or deny

I'm currently a TA for a class and recently graded a midterm. I gave the answer key back to the teacher, after going over part of the exam in a study hall. I was going to go over the rest of it ...
2
votes
1answer
341 views

Is this Cost function concave or convex?

Given the following cost function, where t is the quantity of some product. $$C(t) = 1/3t^3 - 7t^2 +11t + 50$$ here is a graph between $t= 0$ and $t = 25$ We are asked if this function is convex or ...
2
votes
2answers
736 views

Show that First order conditions are necessary and sufficient for utility maximization

I have a budget set $$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$ where $y>0$ is income. Assuming the preferences are strictly monotonic and convex, I want to show that first order ...