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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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22 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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27 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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1answer
107 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. $t_l$ ...
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1answer
53 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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12 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....
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37 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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1answer
36 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 ...
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1answer
101 views

Can be the duality theorem applied to not locally non-satiated utility functions?

I have the following not locally non-satiated utility function: $$U(x,y)=-(x-1)^2-(y-2)^2$$ where $U(x,y): \, \!R^n_+ \rightarrow \!R $ The 3D plot of this function is an infinite paraboloid; ...
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1answer
71 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 $(q_1^*,q_2^*) \in Q^2$ ...
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1answer
37 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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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 ...
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1answer
107 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
30 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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1answer
116 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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39 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
46 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 ...
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1answer
99 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(...
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1answer
41 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 ...
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2answers
482 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....
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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}^{\...
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2answers
2k views

Transformation Function

In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
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1answer
50 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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24 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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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$ ...
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39 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 ...
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1answer
93 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 ...
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1answer
48 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}^{\...
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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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1answer
89 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 ...
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1answer
48 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?
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1answer
37 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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2answers
931 views

Editing formula for finding Marshallian Demand with Cobb-Douglas utility function

Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$: $x_1 = \frac{am}{p_1}\\ x_2 = \frac{bm}{p_2}$ But what if the utility function looks like $u=...
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1answer
26 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 ...
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2answers
183 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 ...
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1answer
231 views

Optimization of Households' utility in “ Rule-of-Thumb Consumers and the Design of Interest Rate Rules ” (Gali et al., 2004)

I can't figure out how the calculation of first order conditions was carried out. I can't figure out where the stochastic discount factor came from.
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3answers
166 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 ...
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1answer
132 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,...
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1answer
95 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
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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1answer
138 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
341 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 ...
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1answer
251 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
87 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 ...
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2answers
92 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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0answers
39 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 ...
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2answers
144 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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1answer
364 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 ...
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0answers
113 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
739 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 ...
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1answer
74 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. ...