Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
113
questions
2
votes
1answer
72 views
+50
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(...
0
votes
1answer
31 views
Typical Growth Model Social Planner Problem
Consider the following social planner's problem, hand-waving the usual assumptions on the preference, technology, endowment, and inelastic supply of labor:
$V(k_o^*)= max_{\{c_t,k_{t+1}\}_{t=0}^{\...
0
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0answers
22 views
optimization problem for two individuals
Two flat mates 1 and 2, rent a flat and play their own music on the only CD player owned by flat-owner. They both like their own music, but dislike the music played by the other. Given the timing ...
0
votes
1answer
39 views
Stokey, Lucas (1989) p 11 FOC
My Lagrangian is:
$L=\sum\limits_{t=0}^T \beta^tU(f(k_t)-k_{t+1})+\sum\limits_{t=0}^T\lambda_t(f(k_t)-k_{t+1}).$
My FOC for $[k_{t+1}]$ is:
$\beta^tU'(f(k_t)-k_{t_1}^*)(-1)-\lambda_t^*+\beta^{t+1}U'...
0
votes
0answers
26 views
Constrained Optimisation: Why is it that when I merge constraints, I get different results?
The problem that I am given is the following:
$ \max \ln c_0 + \beta \mathbb{E} [\ln c_1 ] \\
\text{ s.t. } c_0 + x_g q_g + x_b q_b = y_0\\
c_g = y_g + x_g\\
c_b = y_b + x_b $
Where $y_0$, $ y_b$ ...
0
votes
0answers
28 views
Finding the optimal consumption bundle given the strictly concave utility function $v(x,y) = U(x) +y$?
I am also finding it difficult to understand what are the fundamental differences between analysing optimal bundles between concave and convex functions ?
Does it also happen that the optimal bundle ...
0
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1answer
54 views
Maximization problem FOC and Euler equation
Can someone please help me with the Lagragian and the derivation of the following objective function ? Beneath I provide the objective function, the constraint and the Euler equation that results from ...
2
votes
1answer
24 views
Is anyone familiar with the following basic resource sharing model?
Here is a resource sharing model, I do not remember where I came across it, I am wondering if this is well known in econometrics.
Let $T > 0$ be the total quantity of resources. For example, ad ...
1
vote
1answer
64 views
Symmetric Cournot equilibrium: suffciency without second order conditon
Let $q_i \in Q = \mathbb R_+$ denote the quantity produced by firm $i \in \{1,2\}$. Further let $\pi_i(q_1,q_2) = (1-q_1-q_2)q_i$ denote the profits of $i$. A Nash equilibrium $(q_1^*,q_2^*) \in Q^2$ ...
0
votes
1answer
44 views
Pareto efficiency (optimality conditions) in simple New Keynesian model
I am looking for the pareto-optimal equilibrium for a central planner's problem in a simple New Keynesian model. The planner's problem is to choose $\{ C_{t}, H_{t}, Y_{t}, \pi_{t}, \{h_{t}(j)_{j=0}^{\...
0
votes
1answer
30 views
Neoclassical model with proportional taxes
In a certain economy, time is discrete with periods $t=0,1,2,...$. The economy is populated by many households and identical firms. The utility of a household is:
$\displaystyle\sum^{\infty}_{t=0}\...
0
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0answers
16 views
Monetary Policy under commitment. How to solve the optimization problem?
Under commitment the CB might follow this problem as Monetary Policy strategy:
$$
\min_{\pi_t,x_t}=E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( \pi_t^2 + \alpha x_t^2 )\right)
$$
$$
\text{s.t. }\...
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votes
1answer
61 views
How to find the optimal consumption basket? [closed]
A consumer has the following utility function and income.
𝑈(𝑥, 𝑦) =1/2 * ln 𝑥 + 1/2 * ln y
Price of 𝑥 = Price of 𝑦 = 100.
Income = 1000
Suppose that the consumer gets 2 redeemable coupons for ...
0
votes
1answer
46 views
Constrained Optimization using Lagrangian method
The stationary points that we derive by solving the first order conditions of the Lagrangian are those points global optimum points or local optimum points?
2
votes
1answer
36 views
Have I found the correct Emission Price
Let's say that there is a hotel owner $(H)$ and a woodworker $(W)$ working in close proximity to one another.
The woodworker produces $x$ units to sell at market at $p_{x}=6,5$. From the woodworking ...
0
votes
1answer
32 views
Why do we have to normalize the income of consumers when working with an Edgeworth Box in a simple trade model with Pareto optima?
I was studying microeconomics and I confess I am not the brightest person for maths and sorry if this is very dumb but I get that we CAN normalize the income and I get where it comes from and how it ...
1
vote
1answer
25 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
165 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
1answer
102 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
82 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
vote
1answer
89 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
vote
1answer
643 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 ...
-1
votes
1answer
127 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
143 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
304 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
1answer
117 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 ...
1
vote
1answer
76 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
votes
0answers
34 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
100 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
vote
0answers
101 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)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
1
vote
2answers
83 views
Economies of scale: when is it disadvantageous?
So, I watched a video on economies of scale. It makes sense to me but I'm wondering, is there a point where say doubling the production rate makes the product even more expensive? How can I figure out ...
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votes
1answer
326 views
Question about budget constraint and utility maximization [closed]
I have also following budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where y is income.
Assume that there are two stories. The agent can shop in both of them. The first store ...
2
votes
2answers
619 views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
2
votes
1answer
63 views
Linear programming, shadow price range
I'm not sure how to determine the range for which a shadow price is valid.
You might be able to skip straight to the question here.
I've been introduced to it using the following approach in 2D.
...
2
votes
2answers
2k views
Utility maximization question setting up.
Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$
Assume the agent's income is $y=5$. The price of one unit ...
1
vote
1answer
76 views
Constant MC - Monopolstic Competition
I would like to know if it is possible to have constant marginal costs (MC) in a business that is operating on a market, that is defined by monopolistic competition?
The company is a construction ...
2
votes
0answers
152 views
Solution to Dynamic Programming (Bellman Equation) Problem
Could someone please provide pointers on how to solve the below? If any theoretical approximations are possible, that would be very helpful. If numerical solutions are the right approach, could you ...
0
votes
1answer
57 views
How to prove that a point is a maximum point
I have the following function:
$$
\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
1answer
253 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
116 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
vote
1answer
243 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
votes
1answer
231 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 ...
1
vote
0answers
29 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
374 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
185 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
36 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
448 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....
3
votes
1answer
202 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
2answers
504 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
82 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 ...
