Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
193
questions
1
vote
1answer
24 views
Deriving factor allocation of production function
I am trying to solve an allocation problem for a nested CES production function with three factors.
The production function we posit is:
$$
F(K, \mathbf L, \mathbf C) = [\alpha K^\rho + \sum_{i\not\in ...
3
votes
1answer
61 views
Proof of a comparative statics result in a maximization problem
I am thinking about the following question.
Let $f(x,\theta)$ be a strictly concave real-valued function in $x$ and $c(x)$ be a strictly convex real-valued function in $x$, both $x$ and $\theta$ ...
2
votes
0answers
32 views
Necessary conditions in overlapping generations model (OLG)
The consumer at each period maximizes
\begin{equation}
\displaystyle\sum_{i=1}^{I}\beta^{i-1}U(c^i_{t+i},l^i_{t+i}) , t=0,1,2,3,...
\end{equation}
subject to
\begin{equation}
(1+\eta_t)c^{i}_t+...
2
votes
1answer
101 views
How do I find the socially optimum and equilibrium value?
I am struggling with the following question. Could someone please explain how to do it?
People are trying to go to the city centre. A bus takes 1 hour and will always take 1 hour,
unaffected by ...
2
votes
1answer
37 views
Arguments of the Marshallian demand system of a Cobb-Douglas utility function
For a utility function of the form $U(x_1,x_2) = x_1^\alpha x_2^\beta$ and the standard budget constraint, the utility maximisation problem gives us a demand system characterised by:
$x_1(\alpha, \...
2
votes
1answer
85 views
Comparative Statics: Income Effect
Much of this is setting up the problem. So if you're familiar it's likely best to start from the very bottom and work up if needed. The question is asking about the income and substitution effect.
...
0
votes
1answer
74 views
Excel Solver, how to solve optimization problem?
I have to perform an optimization tast by using excel solver. The case study has two parts and I already managed to complete the first one.
However, I couldn't solve the second one so far.
Both parts ...
0
votes
1answer
58 views
Maximization with a binding boundary constraint
I have following profit function -
$$ max_{x} ~ mx^{2a} - rx $$
$\text{Subject to,}$
$$ p \geq mx^{2a} - rx \geq q $$
Where, $ m>r, p>0, q>0$ and $a< \frac{1}{2}$
Since firm always want to ...
2
votes
0answers
41 views
Constrained optimization problem written as minimax Lagrange
I am reading a note on moral hazard and it shows that a problem like $\max _{s(x)} \mathbb{E}[V(x-s(x))]$ s.t. $\mathbb{E}[U(s(x))] \geq \bar{H}+c\left(a^{*}\right)$ can be solved by setting up a ...
1
vote
3answers
181 views
Graphing indifference curves to visualize solutions?
I am having trouble with being able to graph indifference curves. This is a particularly important skill to have especially when trying to visualize corner solutions, and when the Lagrangian method ...
4
votes
1answer
65 views
Solving the following cost minimization problem using Kuhn-Tucker conditions
I am currently getting my Masters in Economics. I did not get any exposure to optimization with inequality constraints in my undergrad. I would like to ensure that I am doing this problem correctly. ...
2
votes
1answer
119 views
Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?
When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition).
As per the calculus wiki:
Link, when the Hessian is negative semi-...
1
vote
1answer
51 views
Optimal consumption for infinite number of periods and exogenous income
I have the following optimization problem:
$\max_{\{c_t, s_{t+1}\}} \Pi_{t=0}^\infty c_t^{\beta^t}$
$\text{subject to } \space c_t + s_{t+1} = y_t + (1 + r) s_t \text{ and } s_0 = 0$
How do I find ...
3
votes
1answer
109 views
Does global maximum of CRS Cobb-Douglas profit exist
In most macroeconomic papers it is taken as given that the aggregate prodution function is $Y=AK^{\alpha}L^{1-\alpha}$, and that the optimality conditions for inputs determine input demands:
$$
\max_{...
0
votes
0answers
58 views
Stochastic Optimization
I am working on a Data Panel project that is about macro volatility, taking into consideration some quality indicators. According to my model, indicators definitely determine macro variables. Data are ...
0
votes
2answers
258 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
138 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, ...
4
votes
1answer
40 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
66 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: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
0
votes
0answers
13 views
Pricing optimisation when elasticities are positives? [duplicate]
I’m performing some exercises in order to get the optimal price of some product such a potato chips, biscuits, drinks, etc. (I’m taking price per unit and sales in units) But I’ve found that some of ...
2
votes
1answer
73 views
Regression Optimization problem under constraints
To estimate a simple linear regression:
$$ y = \beta_0 + \beta_1 x + \epsilon $$
I have the assumptions that a researcher $A$ can only sample individuals with a value $y < y^A$. Similarly, a ...
1
vote
1answer
56 views
Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function
I am trying to solve the household's optimization problem in the New Keynesian model framework, where utility is given by
$$
E_0\sum_{t=0}^\infty \beta^t \mathcal{U}(C_t,L_t,N_t;Z_t)
$$
and period ...
1
vote
1answer
53 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
51 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
81 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
251 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
votes
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
79 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
82 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
110 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
119 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
81 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
36 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)
$$
I need to take the total derivative of this, in my ...
4
votes
1answer
28 views
Illustrating Karesh Kuhn Tucker with two non-nonnegativy constraints binding
I'm teaching Karesh-Kuhn-Tucker, and looking for papers, ideally in the fields of development, agricultural or environmental economics, and ideally in good journals, that I can use to illustrate the ...
1
vote
1answer
156 views
What does binding mean?
I am curious how to solve the utility maximization problem if the representative agent has borrowing constraint.
3
votes
2answers
122 views
setting of Lagrangian function
Consider a simple consumer's problem:
Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda ( \...
1
vote
0answers
88 views
Conic optimization in economics
Are there any mainstream economic models that rely on conic optimization to solve for decision variables? Conic optimization is a type of convex optimization problem, different from linear and ...
2
votes
1answer
45 views
Is a binding ZLB a binding constraint?
Usually, in an optimisation problem, a binding constraint is one at which the optimal solution holds at the constraint with equality, i.e. it's a boundary solution.
However, in many articles, for ...
4
votes
1answer
451 views
Perfect substitutes and Lagrange
How does one solve utility maximization of perfect substitutes using Lagrangian function?
Consider the problem
$$\max_{x,y} ax +by $$
subject to the constraint that
$$px + qy \leq I$$
where $a,b,p,q,...
1
vote
1answer
20 views
The optimal price for a demand curve with a steep slope
Given the demand function,
$$D(p)=A-ap$$
I've found the optimal price,
$$p=\frac{A+ac}{2a}$$
Where $c$ is cost and $A,a >0$.
My question is how is the optimal price is dependent of $a$ (1) - what ...
0
votes
0answers
53 views
Cost-optimal p2p-trade in a community of households
I’m trying to solve the following problem and I’ve been working on it for a long time already:
I want to optimize electricity-costs in a smart grid. There’s producer and consumer households in the ...
7
votes
2answers
147 views
Applications of Optimal Transport in Economics
The 1975 Nobel Prize winner in Economics was Kantorovich who reformulated the optimal transportation theory of Monge and applied it to optimal resource allocation. The Wasserstein distance is central ...
3
votes
0answers
131 views
What are the boundary value conditions for generic HJBs in economics?
Consider a routine continuous time optimization problem:
$
V(t,a_{t}) :=
\max \int_{\tau=t}^{\tau = T}
e^{-\rho (\tau -t)} u(c_{\tau})d\tau
$ $\text{ s.t. }$
$\dot{a}_{t} = y + ra_{t} - c_{t}$,
$a_{...
2
votes
1answer
53 views
Practice question on Correspondences and maximization
We're learning about Theory of the Maximum. I tend to struggle with correspondences in this context, so I'm trying to work through some practice questions. I will start with some general notation of a ...
3
votes
0answers
75 views
How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?
Statement: Let the dynamics of wealth of the agent satisfy
$$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
65 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
41 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
63 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:
...
