Questions tagged [optimization]

Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.

Filter by
Sorted by
Tagged with
3
votes
1answer
292 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 ...
7
votes
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 ...
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 (...
-1
votes
1answer
237 views

Price Optimization from Data

How can I find the optimal price that maximizes profits, given past sales data? I thought I could do this, but I've been running into problems. Data: ...
1
vote
1answer
192 views

Find Indifference curve/s and Marginal Rate/s of Substitution given only one point

Arka likes fries. She wants to consume as much as possible. She consumes either regular (1 oz) or large sizes (5 oz). Draw her indifference curve through $(x_R, x_L) = (10,0)$ and her indifference ...
2
votes
0answers
64 views

Does this conditional increase in income affect a budget line in the same way as an unconditional increase in income would?

It's been awhile since I've taken introductory microeconomics. I remember increases in income move budget line outward. What if the increases have some condition? The problem: Jill has $I$ to ...
0
votes
1answer
282 views

Can budget lines be discontinuous?

It's been awhile since I've taken introductory microeconomics. I read budget lines can be discontinuous awhile ago but am suspicious as to such a concept being discussed in intro micro classes ...
1
vote
0answers
96 views

Dynamic programming: verification principle

Consider the Gale cake problem with $u(c) = log(c)$, so that the problem becomes:$$\max_{x}\sum_{t=0}^{\infty} \beta^{t}log(x_t-x_{t+1})\: sub\: 0<x_{t+1}\leq x_t,\: x_0>0\ given$$ It is ...
2
votes
1answer
250 views

Constraints on a symmetric pareto allocation under uncertainty

I've been trying to figure out how the author came up with the constraints for this liquidity model in a textbook I'm reading. details: http://imgur.com/TpVjg4w $U = \pi_1u(C_1) + \pi_2u(C_2)$ where ...
3
votes
1answer
124 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 &...
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 $Q$?
1
vote
0answers
55 views

Separate optimization problem versus one big Lagrangean

I am a bit confused about when to consider a separate optimization problem and when to combine different problems into a single optimization problem. Consider an individual who derives utility from ...
1
vote
2answers
157 views

Profit Maximisation if MC is still falling after intersecting MR

really basic one here, I am just in the process of re-covering old ground. I understand that for any profit maximising firm FOC: MC=MR SOC: MR'< MC' But suppose MC has a minimum beyond the ...
3
votes
0answers
315 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 ...
5
votes
1answer
273 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 ...
0
votes
1answer
64 views

What is the interpretation of the output matrix of pivoting?

I have the following matrix: $$A= \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}$$ After pivoting, I got this matrix: $$B= \begin{bmatrix} 1 & 2 & 3 \\ ...
1
vote
1answer
1k views

Duality of cost minimization and profit maximization

The firm tries to maximize profits $\Pi$ \begin{align} \max_{K,L}\{\Pi(K,L) = F(K,L) - RK - wL\} \end{align} where $F$ is the linear homogeneous production function, $R$ the rental rate of capital $K$...
8
votes
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 $\lambda>0$, ...
6
votes
1answer
144 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} ...
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; ...
4
votes
1answer
280 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{...
5
votes
1answer
848 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 ...
2
votes
2answers
1k views

Intermediate macroeconomics: optimal bundle for quasilinear utility?

How would I go about solving this question: Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. ...
2
votes
0answers
246 views

Optimizing Cobb-Douglas like functions

Economics isn't my home field, but I'm looking for references for a paper I'm working on and I'm hoping one of you can help. Are there many good references for optimizing a Cobb-Douglas like utility ...
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 ...
4
votes
1answer
850 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 ...
3
votes
0answers
125 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 ...
10
votes
1answer
784 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,...
5
votes
4answers
724 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 ...
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 ...
3
votes
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 ...
2
votes
0answers
1k views

First and Second Order Lagrangian-Multiplier Conditions for Optimization

The Statement of the Problem: Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier ...
1
vote
0answers
88 views

Simple Neoclassical Growth Model with Elastic Labor and Non-Standard Capital Adjustment Costs

I have the following social planner problem to maximize $\{c_t, k_t, n_t \}$ $\begin{gather*}E_0 \sum_{t=0}^\infty \beta^t U(c_t, 1 - n_t), 0 < \beta < 1\end{gather*}$ subject to $\begin{...
2
votes
2answers
974 views

Editing formula for finding Marshallian Demand with Cobb-Douglas utility function

Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$: $x_1 = \frac{am}{p_1}\\ x_2 = \frac{bm}{p_2}$ But what if the utility function looks like $u=...
10
votes
2answers
30k 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 = \...
1
vote
1answer
158 views

Derivative of CARA utility

Can someone help explain the passage here? I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much ...
2
votes
0answers
262 views

Calculating the optimal portfolio for an investor with quadratic utility

The problem is from Asset Pricing and Portfolio Theory by Back and can be found here. The relevant info from section 2.5 can be found here. Given that we have the Expected value and the variance of ...
4
votes
1answer
182 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 ...
8
votes
2answers
640 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 ...
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 ...
4
votes
1answer
837 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 ...
6
votes
1answer
243 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
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,...
11
votes
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 ...
7
votes
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 ...
13
votes
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 ...

1 2
3