Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
31 questions
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Marshallian Demand for Cobb-Douglas
When trying maximize the utility having a cobb-douglas utility function $u=x_1^ax_2^b$, with $a+b = 1$, I found the following formulas (Wikipedia: Marshallian Demand):
$x_1 = \frac{am}{p_1}\\
x_2 = \...
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Dynamic Optimization: What if the second order condition does not hold?
Consider the following dynamic optimization problem
\begin{align}
&\max_u \int^T_0{F(x,u)dt}\\
\text{s.t.}~& \dot{x} = f(x,u)
\end{align}
FOCs
The Hamiltonian is given by
\begin{align}
H(x,u,...
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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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Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They give the following reason as outlined in the problem below.
$Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$
&...
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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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Cost Minimization and Karush-Kuhn-Tucker
A firm produces an output $y$ using two inputs $x_1$ and $x_2$, where the production function is given by $y = \sqrt{x_1 x_2}$ for any $(x_1, x_2) \in \mathbb{R}^2_+$. Union agreements obligate the ...
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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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2
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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 $w_{t+1}$ ...
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Portfolio choice problem of a CARA investor with n risky assets
Ok, I am working on a problem that consists of the following:
I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
- 237
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Remainder term in Linear Approximations going to 0
A number of proofs in optimisation use the idea that the remainder term in either the differential or the Taylor Approximation go to zero. For example:
Some envelope theorem proofs:.
Necessity and ...
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4
votes
1
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How can I show convexity of this value function?
I have set up an optmization problem as follows:
$$V(A)=\max_{l, C} \quad u(C,l)$$
Where the only constraint is as follows:
$$C=f(l,A)$$
Here $u$ is the utility function which captures social welfare. ...
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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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Lagrange Multiplier Dual Meaning?
Is the Lagrange multiplier:
The marginal cost of the constraint?
The marginal benefit of relaxing the constraint?
Through duality, both interpretations imply the other?
If anyone were so kind, I ...
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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 ...
user17900
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1
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Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
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2
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1
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Solving utility maximization, and finding demand function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I want to maximize his utility function.
$$max: 2x_1x_2+x_1+2x_2. uc:p_1x_1+p_2x_2=y_A$$
Using Lagrange, I get
$$L(x_1,...
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0
answers
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Calculating the optimal portfolio for an investor with quadratic utility
The problem is from Asset Pricing and Portfolio Theory by Back and can be found here.
The relevant info from section 2.5 can be found here. Given that we have the Expected value and the variance of ...
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Missing Solutions in KKT Optimisation Problem
In the attached inequality, constrained, optimisation, problem. Looking at the specific case where $\lambda_1 = 0, \lambda_2 > 0$ that I am trying to solve, you can see that I have managed to find ...
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Mixed Partial Derivatives in Profit Function
$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$
Question 1:
Using the first order conditions, we get:
$x = \frac{pa}{w_x}$
$z = \frac{pb}{w_z}$
What do we call these Input demand functions as a ...
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Non-Negativity Constraints KKT
When we take our Lagrangian and we include non-negativity constraints. If a variable $x = 0$ do we take FOC first or set $x=0$ first?
E.g.
$Max \; L(x, y, λ) = f(x,y) - λ_1(g(x,y) - k) - λ_x(-x) - λ_y(...
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Solving first-order conditions to this social planner's problem
I am trying to derive the first-order condtions to this economic problem, where a unit mass of ex ante agents identical agents have preference given by
$$E_{0}\sum_{t=0}^{\infty} \beta^{t} \left\{ \...
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2
answers
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Missing Non-Negativity Constraint?
We have the constrained maximisation problem:
A perfectly competitive firm produces one output with two inputs, capital $(k)$ and labour $(l)$. The rental cost of capital is equal to $r >0$ and ...
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1
answer
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Find Pareto optimal allocations and the core for the following economies
Find Pareto optimal allocations and the core for the following economies.
There are two consumers and two goods. Utility functions are $u_1(x_1,y_1)= 10x_1-(y_1-2)^2$ and $u_2(x_2,y_2) = 10y_2 − (x_2 −...
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Calculating optimal level of negative externality
I am trying to solve the following question(s):
Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility ...
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1
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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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why is the MRS same for everyone?
If the consumers are optimizing and at interior solutions and facing the same prices, then the MRS=p1/p2 will be the same for everyone no matter the preferences and income. but why? I don't understand ...
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1
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How do I get to this demand function in the monocentric city model?
I need to get this resulting price and quantity (housing):
It's pretty clear that the denominator of the quantity function is just the price function.
From this utility function:
And this constraint:...
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2
answers
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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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1
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Quasiconvex Constraints in Maximisation
Why do we have to have Quasi-convex Constraints for constrained maximisation? I think i'm missing something pretty simple as this feels like a basic question:
My current Logic: If both the objective ...
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A basic cost-benefit analysis between two scenarios
I've got a basic question regarding cost-benefit analysis of two potential scenarios:
Scenario A
A firm has a monthly revenue of I dollars with no associated cost, such that its net profit P1 = I.
...
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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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