Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
130
questions
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1answer
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Budget Constraint in Utility Maximisation Problem with Lagrange Multipliers
Lets say we have a utility function $U: \mathbb{R}^{2} \to \mathbb{R}$ given by $U(x,y)$ and a binding budget constraint $p_{x} x + p_{y} y = m$, where $p_{x}, p_{y}$ are prices of goods $x,y$ and $m$ ...
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0answers
20 views
What does the elasticity say about the fraction of total cost used on input 1?
A firm have the following production function
$$
y=x_{1}^{\alpha} x_{2}^{1-\alpha}, \quad 0< \alpha < 1
$$
$w_1>0$ is the cost of input 1 and $w_2 > 0$ is the cost of input 2.
(1.1) ...
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0answers
39 views
What is the economic interpretation of the solution of this optimal control problem?
I have the following optimal control problem
$$\max_{c_t} \int^{\infty}_0 e^{-p_it}\ln(c_t(i))dt$$
subject to $$\dot{w_t}(i)=rw_t(i) -n_ic_t(i)$$
$$w_0(i)=w_0>0$$
I have some wealthy and ...
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0answers
27 views
Second-Order Conditions under Kuhn-Tucker Formulation
How should I address second-order conditions if I use the Kuhn-Tucker formulation of constrained optimization as opposed to the usual one?
For instance, suppose an agent wishes to maximize $f(x_1, ...
1
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1answer
66 views
Kuhn Tucker Maximization
I have to maximize following expected utility function using Kuhn tucker conditions -
Since expected utility function are increasing $C_{1,t}$ and $C_{2,t}$ so constraints (i) and (ii) will hold with ...
2
votes
1answer
97 views
Dynamic programming, optimal consumption-savings (finite horizon) problem
Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$,
the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ ...
0
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2answers
175 views
Corner solution of the maximization problem
Answer
Hello, I upload the actual question with my 8-pages answer. Please can you check it. Is there a corner dissolution for $c=\gamma$. Please share your ideas. Thanks.
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0answers
85 views
Revenue maximization problem
There are $N>0$ Households in an economy.
The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones.
...
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0answers
29 views
The centralized shift from barter to currency economy
Suppose some ancient king of small bronze age city-state wants to introduce universal currency instead of barter that is currently in overwhelming practice in his kingdom. In order to smooth the shift,...
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0answers
15 views
Numerical Solution Using Excel about optimal consumption of households
I'm not sure how to solve this problem. I'm given the discount factor, interest rate, probability of high income shock, and various income shock sizes that I need to use to compute optimal consumption....
0
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1answer
80 views
Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)
I have question on central bank loss function.
We know that the central bank loss function is
$$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$
And we know that fisher equation is $$i=r+\pi^e$$...
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1answer
145 views
A profit maximization problem (whole problem has been solved, I just have question about interpretation)
I would like to discuss with you about the following production function.
$$y=f(t_m, t_l)=\rho t_m^m(n+t_l)$$
where $0<m<1 $ and $n>0$ are fixed parameters.
$t_m$ is manager time.
$t_l$ ...
1
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0answers
42 views
derive value function from utility function
We have the utility function.
$$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$
And I am trying to find the value function.
$U$ is utility function. $c_t$ is consumption at ...
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0answers
11 views
Designing the payment function in Mechanism Design problems
Suppose we have a network in which agents request access to its resources. Thus we have a resource allocation problem.
Ideally, we want to incentivize agents to send social-welfare supporting ...
1
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1answer
37 views
Can I Upload my Preprint on Arxiv Before Submitting it to JPE
I wrote a paper relating the optimal deterrence strategy for crime to concepts on statistical physics, and I am considering submitting the paper to the Journal of Political Economy. I'm wondering if ...
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2answers
93 views
Cobb Douglas, Budget Line, Demand function question
use the general form of the Cobb Douglas utility function $U(x,y)= (x^a)(y^b)$ and the budget constraint in the form $B=p_{x}X + p_{y}Y$ to find the demand functions for good x and good y.
Is this ...
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2answers
55 views
Optimal point and MRS
I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that ...
3
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1answer
100 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(...
0
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1answer
33 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}^{\...
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0answers
26 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 ...
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1answer
53 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'...
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0answers
27 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
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0answers
44 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
190 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 ...
1
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1answer
37 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
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1answer
75 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
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1answer
55 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
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1answer
41 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}\...
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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
116 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
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1answer
50 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
42 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
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1answer
38 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 ...
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1answer
29 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
206 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
129 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
132 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
97 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, ...
2
votes
1answer
1k 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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votes
1answer
133 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
198 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
375 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
412 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
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1answer
106 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
40 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
177 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
128 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)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
1
vote
2answers
96 views
Economies of scale: when is it disadvantageous?
So, I watched a video on economies of scale. It makes sense to me but I'm wondering, is there a point where say doubling the production rate makes the product even more expensive? How can I figure out ...
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votes
1answer
402 views
Question about budget constraint and utility maximization [closed]
I have also following budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where y is income.
Assume that there are two stories. The agent can shop in both of them. The first store ...
2
votes
2answers
896 views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$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 ...
