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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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0answers
21 views

Maximum utility of 3 commodities using Lagrange Multiplier?

Given a consumer utility function consuming commodity x, y, and z as U(x,y,z) = √x + √y + √z and the price of x, y and z are $2, $3 , & $4 respectively. What should be the consumer's purchase of ...
1
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1answer
21 views

What is the “bequest condition” in a finite-horizon discrete optimization problem?

For a finite-horizon discrete time optimization problem, my textbook provides a condition called the "bequest condition", which I'm not familiar with. Specifically, where the state at time $t$ is ...
1
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2answers
135 views

Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$?

Will all the solutions be in the corner or will the cusp in the middle give us any interior solution? This is by the intersection of the budget line. I am getting this type of a shape: But I am not ...
2
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2answers
76 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
55 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
72 views

Utility Function Implies Consumption of Not All Goods

Suppose we have a utility function with three inputs, $j, k,$ and $s$ described by $$u(j,k,s) = A\ln(k^\alpha + \beta j^\alpha) + B\ln(s).$$ The price of $j, k,s$ are $p_j, p_k, p_s$, respectively, ...
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1answer
189 views

Concave utility functions corner solution explanation

I seem to not be getting this. Could someone explain me the mathematical way to show a concave utility function [like (ax^2+by^2)] subject to a budget constraint has a corner solution. I get the ...
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1answer
121 views

Investor's optimization problem with risk aversion

Consider an investor with initial wealth $w$ and has to decide how to invest it. There is a riskless asset with rate of return $r$. The risky asset has return $x_i$ with probability $\pi_i$ for $i=1,2,...
4
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3answers
99 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 ...
2
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1answer
155 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
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1answer
43 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 ...
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1answer
49 views

General Equilibrium with Linear Production

I don't think I understand how optimization problems with a linear function work as of now. If you have a production economy with two agents, two goods and Cobb-Douglas utility representation, and you ...
2
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0answers
23 views

Finding savings in an Overlapping Generations model

I have not seen this question asked anywhere, so I'm posing it here in case anybody else (hopefully) can help me get to the answer. In a nutshell, my question is: how do we arrive at the saving ...
1
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2answers
62 views

Difficulty in an economics' optimization problem using Kuhn-Tucker conditions (interpretation difficulty)

I am having troubles in solving correctly the following problem: A company wants to minimize its total costs, on the condition that the income obtained from the sale of the quantities $x_1, x_2$ of ...
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0answers
54 views

Natural borrowing/debt limit and other borrowing constraints

When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading) $$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
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2answers
61 views

Economies of scale: when is it disadvantageous?

So, I watched a video on economies of scale. It makes sense to me but I'm wondering, is there a point where say doubling the production rate makes the product even more expensive? How can I figure out ...
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1answer
216 views

Question about budget constraint and utility maximization [closed]

I have also following budget set $$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$ where y is income. Assume that there are two stories. The agent can shop in both of them. The first store ...
2
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2answers
324 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 ...
2
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1answer
34 views

Linear programming, shadow price range

I'm not sure how to determine the range for which a shadow price is valid. You might be able to skip straight to the question here. I've been introduced to it using the following approach in 2D. ...
2
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2answers
964 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 ...
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1answer
41 views

Constant MC - Monopolstic Competition

I would like to know if it is possible to have constant marginal costs (MC) in a business that is operating on a market, that is defined by monopolistic competition? The company is a construction ...
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0answers
23 views

Broyden-Fletcher-Goldfarb-Shanno, L-BFGS-B, PORT

I was looking for advantages and disadvantages of the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS), the L-BFGS-B and PORT algorithm in optimization. Which one promises the best results and why?
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0answers
28 views

Optimization of supply between multiple distinct demand 'markets'

I am attempting to solve the following optimization problem, and am curious if it is doable without an iterative process. Consider the following: 1) Two distinct markets (Market X and Market Y) with ...
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0answers
30 views

Stackelberg/seqential entry in Hotelling and numerical analysis

I am trying to solve a Stackelberg type of game with three firms in the framework of the Hotelling line. The game is as follows for n=3 firms: sequential entry in the market compete in prices in the ...
2
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0answers
109 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 ...
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0answers
39 views

PORT Routines Advantages/Disadvantages

I was using the PORT optimization routines as proposed by Gay (1990)“Usage Summary for Selected Optimization Routines”. I searched online and in the paper, but I couldn't find any particular listed ...
0
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1answer
56 views

How to prove that a point is a maximum point

I have the following function: $$ \Pi =\int_{0}^{z}[x_{1} + \alpha y + \alpha \frac{N-2}{2}y - \beta(z) - \gamma ( \beta(z) - \beta(y))](N-1)y^{N-2} dy $$ The first derivative with respect to $z$ is:...
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0answers
32 views

Suppose Jon experiences anticipatory utility and is dealing in the current period (t=0) when to consume a good, what is the optimal time

Suppose Jon experiences anticipatory utility and is dealing in the current period (t=0) when to consume a good, what is the optimal time to consume as a function of $\delta$ and $a$? Recall the ...
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0answers
62 views

Dornbusch-Fischer-Samuelson model optimization

I have a basic utility function in Dornbusch-Fischer-Samuelson model to be optimized: $u=\int_{0}^{1}b(z)\ln c(z)dz$ with budget constraints as follows $\int_{0}^{1}b(z)c(z)dz\leq Y$ and $\int_{0}^{1}...
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1answer
187 views

Determining the elasticity of Hicksian Demands

If we have Hicksian (compensated) demand functions, how can we determine the income elasticity and own price elasticity? Is the procedure the same as for Marshallian (uncompensated) demands?
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1answer
80 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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1answer
166 views

Reservation utility

I am self-studying contract theory using Bolton and Dewatripont (2005). It is meant for grad students, which might be why I am having a difficult time understanding basic terminology. Here is the ...
0
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1answer
117 views

Social planner's first order condition (current-value Lagrangian)

Here is (one of the ways to state) social planner's problem: Eric Sims' notes then immediately gives the solution: I am trying to connect these two lines. This is what I get after taking a ...
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0answers
28 views

Optimal population allocation layout over the Earth

Does exist some model of optimal people occupancy over whole our planet? Something accounting for climate, resources (with and without existing settlements) availability, progress and grows ...
2
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0answers
324 views

Second order condition for symmetric game

Denote by $i \in \{1, \ldots, n\}$ an economic agent. Let $\mathbf x \in \mathbb R^n$ denote a vector of actions and $x_i \in \mathbf x$ a typical element. Let further $f_i : \mathbb R^n \to \mathbb ...
7
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1answer
157 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 ...
0
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1answer
35 views

Optimal taxing in case of negative externalities

Suppose an individual $i$ has the utility function $U= f(x(i)) - k$(sum of all $x$ with index not equal to $i$) Where $x(i)$ denotes the miles driven by $i$, and $k$ is a positive constant. The ...
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2answers
350 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....
4
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1answer
173 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 ...
4
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2answers
415 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-...
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1answer
79 views

Can 10% of the population provide super-abundance for the entire world?

In a discussion at the Watson Institute for International and Public Affairs on Nov 9, Mark Blyth, a political scientist, said that We live in a world where literally 10% of the population could ...
3
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1answer
337 views

Adding a non-binding constraint to the objective function

I am dealing with a constrained optimization problem found in Tirole's Theory of corporate finance. My question is not related to the details of this model, but just to provide some context, we are ...
3
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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 $...
4
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1answer
254 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: ...
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1answer
39 views

What does this condition for a profit of a firm exist mean?

What is the intuition behind this? Let $Y=zF(K,N)$ be a production function. For a profit to exist Indana conditions must hold and : $$\frac{∂^2zF(K,N^d)}{\partial K \partial N} > 0$$ I ...
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0answers
18 views

Perfect complement outputs with each output being composed of substitutable inputs

How does one solve the following maximization problem? $\underset{K_1, K_2, L_1, L_2}{\text{maximize }} min\{K_1 + L_1,K_2 + L_2\}$ subject to $c(K_1 + \mu K_2) + \beta c(L_1 + \mu L_2)$ where $c(...
3
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2answers
143 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 $x$ ...
2
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1answer
232 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 ...
3
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3answers
649 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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1answer
127 views

Optimal Pricing with Advertising

Below are three different demand curves (i) - (iii), which depend on advertising (A). (i) Q(P,A) = A $\times$ ($\alpha$ - $\beta$P), where $\alpha$, $\beta$ > 0 (ii) Q(P, A) = $\alpha$ + A -...