Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
333 questions
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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, \...
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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.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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-...
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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 ...
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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:
$$
\max_{...
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Where can I find a calculator for constrained optimization of general form of algebraic equation?
I'm working with a fairly complex equation and I need to carry out constrained optimization of the same. The first order differential equations are very messy to solve by hand and hence I thought to ...
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GE with an intermediate good
intro
I'm looking at a simple model with 1 consumer, 2 goods and 2 firms.
I'm trying to get a price vector [p0, p1] that makes it work.
By makes it work, I mean, ...
user34331
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Quantity restriction in model with fixed factor of production
I'm trying to see the effect of a restriction on production in a model where one factor of production is perfectly elastic and the other is fixed.
Specifically, suppose the production function is Cobb-...
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Solve the Ben-Porath Model (Optimal Control Problem)
Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
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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 ...
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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 ...
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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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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}...
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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 ...
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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 $i \in [n]$. ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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0
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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 ...
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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 ...
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What does binding mean?
I am curious how to solve the utility maximization problem if the representative agent has borrowing constraint.
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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 ( \...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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_{...
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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 ...
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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 ...
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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?
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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-...
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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$...
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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 ...
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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:
...
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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 ...
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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)
$\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
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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}...
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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 = $ ...
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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
$c_1 = w +(1-\nu)\theta_1 h_1^a$ ...
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3
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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’...
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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 ...
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0
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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 ...
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