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
vote
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
vote
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
votes
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 ...
1
vote
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 ...
1
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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, ...
1
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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
votes
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
votes
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 ...
0
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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
vote
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 ...
1
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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)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
1
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2answers
61 views
Economies of scale: when is it disadvantageous?
So, I watched a video on economies of scale. It makes sense to me but I'm wondering, is there a point where say doubling the production rate makes the product even more expensive? How can I figure out ...
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1answer
216 views
Question about budget constraint and utility maximization [closed]
I have also following budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where y is income.
Assume that there are two stories. The agent can shop in both of them. The first store ...
2
votes
2answers
324 views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
2
votes
1answer
34 views
Linear programming, shadow price range
I'm not sure how to determine the range for which a shadow price is valid.
You might be able to skip straight to the question here.
I've been introduced to it using the following approach in 2D.
...
2
votes
2answers
964 views
Utility maximization question setting up.
Consider a consumer whose preferences can be represented by the following utility function: $$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 ...
1
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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 ...
0
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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 ...
0
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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
votes
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 ...
0
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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:...
0
votes
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 ...
0
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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}...
0
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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?
4
votes
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. ...
1
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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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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-...
1
vote
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
votes
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
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 $...
4
votes
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: ...
1
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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 ...
1
vote
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
votes
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
votes
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
votes
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 ...
1
vote
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 -...
