Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
180
questions
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2answers
38 views
Where can I find a calculator for constrained optimization of general form of algebraic equation?
I'm working with a fairly complex equation and I need to carry out constrained optimization of the same. The first order differential equations are very messy to solve by hand and hence I thought to ...
6
votes
1answer
135 views
GE with an intermediate good
intro
I'm looking at a simple model with 1 consumer, 2 goods and 2 firms.
I'm trying to get a price vector [p0, p1] that makes it work.
By makes it work, I mean, ...
0
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0answers
39 views
Optimization with tax and deduction constraint
I want to find optimal demands for a utility function in the form:
$$U_{1}(c_{1},c_{2})=c_{1}^{1-a}\cdot c_{2}^{a}$$
It is related to an altruistic behaviour, where $c_1$ is the consumption of the ...
4
votes
1answer
37 views
Quantity restriction in model with fixed factor of production
I'm trying to see the effect of a restriction on production in a model where one factor of production is perfectly elastic and the other is fixed.
Specifically, suppose the production function is Cobb-...
4
votes
0answers
48 views
Solve the Ben-Porath Model (Optimal Control Problem)
Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
0
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0answers
13 views
Pricing optimisation when elasticities are positives? [duplicate]
I’m performing some exercises in order to get the optimal price of some product such a potato chips, biscuits, drinks, etc. (I’m taking price per unit and sales in units) But I’ve found that some of ...
2
votes
1answer
69 views
Regression Optimization problem under constraints
To estimate a simple linear regression:
$$ y = \beta_0 + \beta_1 x + \epsilon $$
I have the assumptions that a researcher $A$ can only sample individuals with a value $y < y^A$. Similarly, a ...
1
vote
1answer
50 views
Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function
I am trying to solve the household's optimization problem in the New Keynesian model framework, where utility is given by
$$
E_0\sum_{t=0}^\infty \beta^t \mathcal{U}(C_t,L_t,N_t;Z_t)
$$
and period ...
1
vote
1answer
48 views
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 ...
3
votes
0answers
49 views
Optimization problem of the firm
I have been reading an Economics working paper and trying to derive the first-order conditions of a seemingly complicated optimization problem.
The optimization problem with choice variables $P_{t}^{R}...
5
votes
1answer
76 views
When can one drop time subscripts? Example from Angrist and Kugler (2003)
Not the first time I am asking myself, but in this paper they actually start with a time dependent maximisation problem and then drop all time subscripts.
Background: They have profit maximisation ...
5
votes
2answers
237 views
Arrow-Debreu Theorem of Existence: Non satiation
Let $n$ be the number of consumers and $m$ be the number of commodities.
The Arrow-Debreu theorem requires closed and convex consumption sets $X_i \subset \mathbb{R}^m$ for all buyers $i \in [n]$. ...
0
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0answers
27 views
Solving for parameter value
I have the following maximization function -
$\max_{x \in (0,1)} (((p_1e_1x^2)^{r} + (p_2e_2(1-x)^2)^{r})/2)^{1/r}$
where, $p_1$ and $p_2$ are drawn from uniform distribution [0,1] and are considered ...
1
vote
1answer
47 views
Lagrangian multiplier and optimal bundle
I would like to know where I am wrong (if I am) and why I am wrong here please:
If a consumer has an income of 600 euros to spend for good x (Px = 10 euros) and good y (Py = 5 euros).
What is the ...
1
vote
1answer
68 views
Contradictory FOC and maximizing solution
I have to maximize the following function -
$\max_{x \in (0,1)} (((p_1x)^{2r} + (p_2(1-x))^{2r})/2)^{1/r}$
where, $p_1$ and $p_2$ are drawn from uniform distribution [0,1] and are considered to be ...
3
votes
0answers
108 views
Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?
I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving in Matlab and Python and could not find any) so that I cannot get an explicit expression ...
1
vote
1answer
95 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
56 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
vote
0answers
26 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)
$$
I need to take the total derivative of this, in my ...
4
votes
1answer
25 views
Illustrating Karesh Kuhn Tucker with two non-nonnegativy constraints binding
I'm teaching Karesh-Kuhn-Tucker, and looking for papers, ideally in the fields of development, agricultural or environmental economics, and ideally in good journals, that I can use to illustrate the ...
1
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1answer
150 views
What does binding mean?
I am curious how to solve the utility maximization problem if the representative agent has borrowing constraint.
2
votes
2answers
80 views
setting of Lagrangian function
Consider a simple consumer's problem:
Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda ( \...
1
vote
0answers
83 views
Conic optimization in economics
Are there any mainstream economic models that rely on conic optimization to solve for decision variables? Conic optimization is a type of convex optimization problem, different from linear and ...
2
votes
1answer
41 views
Is a binding ZLB a binding constraint?
Usually, in an optimisation problem, a binding constraint is one at which the optimal solution holds at the constraint with equality, i.e. it's a boundary solution.
However, in many articles, for ...
4
votes
1answer
144 views
Perfect substitutes and Lagrange
How does one solve utility maximization of perfect substitutes using Lagrangian function?
Consider the problem
$$\max_{x,y} ax +by $$
subject to the constraint that
$$px + qy \leq I$$
where $a,b,p,q,...
0
votes
0answers
9 views
Algorithms/Models to solve minimal Matchings for consumer producer household pairs
I’m working on the following problem:
Minimising the electricity price for household trading pairs. There’s producer and consumer households. Trades are just possible between producers and consumers.
...
1
vote
1answer
19 views
The optimal price for a demand curve with a steep slope
Given the demand function,
$$D(p)=A-ap$$
I've found the optimal price,
$$p=\frac{A+ac}{2a}$$
Where $c$ is cost and $A,a >0$.
My question is how is the optimal price is dependent of $a$ (1) - what ...
0
votes
0answers
53 views
Cost-optimal p2p-trade in a community of households
I’m trying to solve the following problem and I’ve been working on it for a long time already:
I want to optimize electricity-costs in a smart grid. There’s producer and consumer households in the ...
6
votes
1answer
109 views
Applications of Optimal Transport in Economics
The 1975 Nobel Prize winner in Economics was Kantorovich who reformulated the optimal transportation theory of Monge and applied it to optimal resource allocation. The Wasserstein distance is central ...
3
votes
0answers
119 views
What are the boundary value conditions for generic HJBs in economics?
Consider a routine continuous time optimization problem:
$
V(t,a_{t}) :=
\max \int_{\tau=t}^{\tau = T}
e^{-\rho (\tau -t)} u(c_{\tau})d\tau
$ $\text{ s.t. }$
$\dot{a}_{t} = y + ra_{t} - c_{t}$,
$a_{...
2
votes
1answer
44 views
Practice question on Correspondences and maximization
We're learning about Theory of the Maximum. I tend to struggle with correspondences in this context, so I'm trying to work through some practice questions. I will start with some general notation of a ...
3
votes
0answers
71 views
How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?
Statement: Let the dynamics of wealth of the agent satisfy
$$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
54 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
votes
1answer
40 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
59 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
25 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
26 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
172 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
183 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
125 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
139 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
71 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
58 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
44 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
1answer
216 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
40 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
71 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
98 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 ...
