Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
141
questions
0
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1answer
68 views
How to find the Utility Possibility Frontier when there are Perfect Substitutes?
I am trying to derive the Utility Possibility Frontier (UPF) when both utility functions display perfect substitutes (in an Edgeworth economy with to consumers and two goods).
The specific problem:
$...
1
vote
1answer
33 views
Expectational stability: adaptive learning of RE equilibria in dynamic systems
There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand.
Step 1.
What does this formula below mean, ...
-1
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2answers
45 views
Economics of Justifying N95 masks and Mass COVID testing [closed]
The US has shutdown a significant fraction of its economy because of COVID-19. Eventually we will all migrate in a pre-COVID direction. Obviously, too fast would be a medical disaster, too slow ...
0
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1answer
29 views
Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line?
This a person's budget line with various points, and their consumption, C*, and their endowment e, which is worth $5000 (unimportant). Also shows is their initial indifference curve. The difference ...
3
votes
1answer
40 views
Concavity of Cobb-Douglass Utility Function on Non-Open set
My textbook argues that the Cobb-Douglass utility function $u=(x1)^a(x2)^b$ with $a,b>0$ and $a+b<1$ is concave on $R2+$ by computing the Hessian and showing it to be negative semidefinite for ...
0
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0answers
17 views
Taking the partial derivative of the demand function
Define the demand function which maximizes x -> U(x) as:
$\sum_{i=1}^n$$p_i$$\zeta_i$(p, I) = I
According to my textbook if I differentiate this with respect to $p_j$ I will obtain,
$\zeta_j$(p, ...
0
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0answers
7 views
'Constrained optimisation' for mutually exclusive goods?
Taking the standard approach to constrained optimisation, where we maximise utility subject to a budget constraint with some allocation on the consumption of two goods, does it make apply the same ...
1
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0answers
30 views
Optimization problem of a Cobb-Douglas function with 3 inputs
A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function:
$$
Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3}
$$
...
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0answers
27 views
Is optimizing revenue and expense objectives simultaneously better than optimizing profit as composite objective?
In the profit maximization problem, I am curious if co-optimizing revenue and expense objectives simultaneously are better than optimizing profit (revenue - expense) as a single composite objective? I ...
2
votes
2answers
63 views
Complementary slackness conditions (Kuhn-Tucker)
Consider the problem of maximising a smooth function subject to the inequality constraint that $g(x) \leq b$. The complementary slackness condition says that
$$ \lambda[g(x) - b] = 0$$
It is often ...
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0answers
29 views
On Demand Functions and Engel Curves
A consumer has utility function $U(x,y)=(xā2)y$, where $xā„2$ and $yā„0$. The price of $x$ is $P_x$, the price of $y$ is $P_y$ and the consumer's income is $I>2P_x$. ($x$ and $y$ do not have to be ...
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0answers
28 views
Kuhn-Tucker conditions in linear cost minimization
Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by
$$
f(x,y) = ax + by
$$
and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} ...
0
votes
1answer
53 views
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
25 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
41 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 ...
0
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0answers
28 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
vote
1answer
68 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
2answers
170 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
votes
2answers
206 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
97 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.
...
1
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0answers
30 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
17 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
119 views
Central bank loss function (I did a solution, but it doesnāt 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 $$i=r+\pi^e$$...
1
vote
1answer
149 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
vote
0answers
48 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 ...
0
votes
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
vote
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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votes
2answers
193 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 ...
1
vote
2answers
113 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
votes
1answer
104 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
votes
1answer
40 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
37 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
57 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
68 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
votes
1answer
318 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
vote
1answer
43 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
85 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
74 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
53 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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votes
1answer
169 views
How to find the optimal consumption basket? [closed]
A consumer has the following utility function and income.
š(š„, š¦) =1/2 * ln š„ + 1/2 * ln y
Price of š„ = Price of š¦ = 100.
Income = 1000
Suppose that the consumer gets 2 redeemable coupons for ...
0
votes
1answer
51 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
50 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
43 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
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
vote
2answers
241 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 ...
3
votes
1answer
151 views
Can $u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ be solved by KuhnāTucker 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
177 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 Aā product B, are there ...
1
vote
1answer
99 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
2k 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 ...
