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
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1answer
2k 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
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6answers
1k 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 ...
11
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2answers
31k 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 = \...
10
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1answer
819 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
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4answers
832 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 $\lambda>0$, ...
8
votes
2answers
655 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
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1answer
243 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. ...
7
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2answers
114 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
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1answer
73 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 ...
7
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1answer
148 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} ...
7
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1answer
215 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
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1answer
123 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
247 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
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4answers
748 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
532 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
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1answer
291 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
899 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
98 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
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4answers
977 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 ...
4
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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
287 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
865 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
185 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
257 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
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 $Q$?
4
votes
1answer
866 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 ...
4
votes
0answers
190 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
640 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
1answer
117 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
2answers
268 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
2k 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
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 ...
3
votes
1answer
326 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
2answers
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}...
3
votes
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(...
3
votes
1answer
229 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
494 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
1answer
157 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 ...
3
votes
0answers
185 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 ...
3
votes
1answer
126 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
324 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
126 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
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 ...
3
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0answers
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 ...
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
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2answers
962 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
664 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 ...