Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
166
questions
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0answers
23 views
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 −...
1
vote
1answer
48 views
Essential goods: How does one restrict the utility function?
I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure ...
1
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0answers
25 views
Total Derivative of a Max Function: Maximizing Social Welfare Function
I'm studying public economics but my question here is purely mathematical in nature. I have a function:
$$
V(1-\tau, R) = \max_zu((1-\tau)z+R,z)
$$
I need to take the total derivative of this, in my ...
4
votes
1answer
22 views
Illustrating Karesh Kuhn Tucker with two non-nonnegativy constraints binding
I'm teaching Karesh-Kuhn-Tucker, and looking for papers, ideally in the fields of development, agricultural or environmental economics, and ideally in good journals, that I can use to illustrate the ...
2
votes
1answer
132 views
What does binding mean?
I am curious how to solve the utility maximization problem if the representative agent has borrowing constraint.
3
votes
2answers
51 views
setting of Lagrangian function
Consider a simple consumer's problem:
Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda ( \...
2
votes
0answers
81 views
Conic optimization in economics
Are there any mainstream economic models that rely on conic optimization to solve for decision variables? Conic optimization is a type of convex optimization problem, different from linear and ...
2
votes
1answer
36 views
Is a binding ZLB a binding constraint?
Usually, in an optimisation problem, a binding constraint is one at which the optimal solution holds at the constraint with equality, i.e. it's a boundary solution.
However, in many articles, for ...
4
votes
1answer
56 views
Perfect substitutes and Lagrange
How does one solve utility maximization of perfect substitutes using Lagrangian function?
Consider the problem
$$\max_{x,y} ax +by $$
subject to the constraint that
$$px + qy \leq I$$
where $a,b,p,q,...
0
votes
0answers
8 views
Algorithms/Models to solve minimal Matchings for consumer producer household pairs
I’m working on the following problem:
Minimising the electricity price for household trading pairs. There’s producer and consumer households. Trades are just possible between producers and consumers.
...
1
vote
1answer
18 views
The optimal price for a demand curve with a steep slope
Given the demand function,
$$D(p)=A-ap$$
I've found the optimal price,
$$p=\frac{A+ac}{2a}$$
Where $c$ is cost and $A,a >0$.
My question is how is the optimal price is dependent of $a$ (1) - what ...
0
votes
0answers
51 views
Cost-optimal p2p-trade in a community of households
I’m trying to solve the following problem and I’ve been working on it for a long time already:
I want to optimize electricity-costs in a smart grid. There’s producer and consumer households in the ...
6
votes
1answer
87 views
Applications of Optimal Transport in Economics
The 1975 Nobel Prize winner in Economics was Kantorovich who reformulated the optimal transportation theory of Monge and applied it to optimal resource allocation. The Wasserstein distance is central ...
3
votes
0answers
103 views
What are the boundary value conditions for generic HJBs in economics?
Consider a routine continuous time optimization problem:
$
V(t,a_{t}) :=
\max \int_{\tau=t}^{\tau = T}
e^{-\rho (\tau -t)} u(c_{\tau})d\tau
$ $\text{ s.t. }$
$\dot{a}_{t} = y + ra_{t} - c_{t}$,
$a_{...
2
votes
1answer
38 views
Practice question on Correspondences and maximization
We're learning about Theory of the Maximum. I tend to struggle with correspondences in this context, so I'm trying to work through some practice questions. I will start with some general notation of a ...
4
votes
0answers
56 views
How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?
Statement: Let the dynamics of wealth of the agent satisfy
$$dX_{t} =
\pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$
where $(\pi,c)$ is an ...
1
vote
1answer
53 views
How do you formulate a distance constraint and a budget constraint?
Everybody knows about budget constraints and how they are represented:
but what if I want to represent a distance constrain from the shop you buy the goods? How can I build that?
0
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1answer
38 views
Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]
A consumer with a utility function given by $u(x_1, x_2) = \sqrt{x_1} + x_1x_2$ has an income of $100$. The unit prices of $x_1$ and $x_2$ are $4$ and $5$, respectively.
(a) Compute the utility-...
2
votes
1answer
71 views
How can this be proved? (Convex optimization)
Consider the following maximization problems:
$\max_{x} x -\gamma p(x)$
subject to $x \in \Omega_1$
$\max_{x} x-\gamma (p(x) + q(x) )+K$
subject to $x \in \Omega_2$
where $\Omega_1 $ and $ \Omega_2$...
2
votes
1answer
58 views
When the global optimal is outside of the constraint set, what will be the demand?
$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex.
$a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that ...
1
vote
0answers
24 views
Simplex Lp interpretation of dual problem´s solution
I am wondering whether my interpretation of my simplex dual problem result is correct.
The primal problem is:
...
1
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0answers
25 views
Solving a HJB with additional constraints on control and state variables
I am trying to solve a Hamiltonian-Jacobi-Bellman equation with additional constraints on the state and control variables, but I am a bit confused on how to do that.
In Intrilligator 2002, it is ...
1
vote
1answer
86 views
Generalizing demand for perfect substitutes utility function
I have the utility function:
$U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero)
$\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
4
votes
2answers
171 views
Solving a HJB with a probability to transit to a new state
I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time.
The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}...
2
votes
2answers
101 views
In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?
In lecture, my professor defined the following 2-period consumption model:
$c_i = $ consumption in period $i$.
$y =$ endowed income in period 1.
$r = $ interest rate in perfect credit markets.
$h = $ ...
1
vote
1answer
101 views
Solving Constrained Optimization Problem with Two-Period Model of Human Capital
I'm trying to solve a constrained optimization problem in human capital model. The objective function is
$\max_{c_1,c_2,\nu} U = u(c_1) + \beta u(c_2)$,
subjected to
$c_1 = w +(1-\nu)\theta_1 h_1^a$ ...
2
votes
1answer
70 views
Can the weierstrass and the Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?
Question: An agent who consumes three commodities has a utility function given by:
$u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$
Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’...
1
vote
1answer
45 views
A maximization problem with multiple goods and integrated markets
Update:
I will try to clarify the question: Let us say that the total harvest of the fish population at time t is $H_t$. Every harvest produce three types of fish: salmon ($f_1$), which is valuable ...
1
vote
0answers
40 views
Analytical approach to estimate equilibrium price for Real Estate Property
I am looking to calculate the equilibrium price, i.e an optimal price that I can set without affecting demand and maximize revenue.
I've gathered historical data: occupancy rates, asking rents for ...
0
votes
0answers
32 views
Using ML to estimate demand function
Say, I am looking to estimate the demand curve for rental of a real estate property. The demand varies depending on time of the year, location, economic and demographic variables.
I'd like to ...
0
votes
1answer
143 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
39 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
votes
2answers
58 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
votes
1answer
41 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
68 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
votes
0answers
37 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
votes
0answers
10 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
vote
0answers
73 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:
$$
Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3}
$$
...
0
votes
0answers
30 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 ...
3
votes
3answers
189 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 ...
0
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0answers
39 views
On Demand Functions and Engel Curves
A consumer has utility function $U(x,y)=(x−2)y$, where $x≥2$ and $y≥0$. 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 ...
1
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0answers
86 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
102 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 $m$ ...
1
vote
0answers
26 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) ...
1
vote
1answer
77 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
214 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
279 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.
1
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0answers
104 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,...
1
vote
0answers
20 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....
