Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
255
questions
0
votes
0
answers
27
views
Max and Min with $\leq$ and $=$ constraints. General questions
I wrote this question on Maths.stackexchange but perhaps this community suits better (?)
I need to ask you for this question, which is a rather general one, in order to understand how to behave when ...
- 101
3
votes
3
answers
109
views
The formula for expansion path
Is there a way how to precisely compute the expansion path?
I know a consumer's utility function $U(\boldsymbol{x})$, I know the budget constraint $\sum P_i x_i \leq M$, I am able to compute the ...
- 680
1
vote
1
answer
32
views
Use of zero profit condition in determining unique solution
Assume that we have a profit function of the form:
$$
\pi=pf(Q)-wl-rk
$$
where $Q$ represents total volume of output, $w$ represents wages
and $r$ represents rental rate on capital $k$. One way to ...
- 167
3
votes
1
answer
86
views
Kuhn-Tucker(KT) conditons EMP
How should I formally solve the expenditure min.problem (EMP) by using KT conditions?
Since I should follow the notation of the Mas-Colell, I should write:
$\min~$ $p \cdot x$ , s.t. $u(x) \ge u$
...
- 71
4
votes
1
answer
102
views
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 ...
- 343
1
vote
1
answer
57
views
Notation for ∆y used in Necessary Second Order Condition Proof
Edit: I have updated the link so that it works!
I was watching a lecture for a proof that if $x^*$ is a local maximiser of $f$ then necessarily the hessian is negative semi definite.
However i've got ...
- 343
0
votes
1
answer
99
views
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 ...
- 343
2
votes
1
answer
72
views
Quasi-convex constraints using monotonic functions
I believe i have a major misunderstanding surrounding quasi-convex constraints in maximisation, when using monotone functions. Can you help me spot my errors please?
The definition of a quasi-convex ...
- 343
2
votes
0
answers
32
views
Is the Jelly Shape Quasi-concave?
Studying concavity for our constrained optimisation problems made me think of the jelly shape...I believe this shape (see Jelly image) isn't quasi-concave and hence isn't concave. Because it would be ...
- 343
1
vote
1
answer
36
views
Are other 'variables' in demand functions always fixed?
My question is whether our demand functions e.g. Hicksian (compensated) demand, are ever functions of 3 or more variables, or if the other price variables and utility are always fixed, and hence just ...
- 343
0
votes
1
answer
25
views
Compensating Variation - Interpreting the formulae
Assume $U(x,y) = x^{1/2}y^{1/2}$ s.t. $P_xx + P_yy = m$
And a price increase from $P_x$ to $P'_x$:
$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$
Compensation variation formulae is: $\frac{M + ∆M}{2(P_x'P_y)^{1/2}...
- 343
0
votes
0
answers
42
views
Compensating Variation for $U = (xy)^{1/2}$ and $U = xy$
I want to check my calculations for these compensating variations regarding an increase in $P_x$ to $P'_x$.
Below I have used $-∆M$, this is how my course first laid it out but I appreciate that it's ...
- 343
2
votes
2
answers
63
views
What is the Lagrangian a function of?
I understand the role of Lagrangian in constrained optimisation, and that we could conceptualise it as for example, a penalty function.
What I don’t understand is the notation, and perhaps any deeper ...
- 343
0
votes
2
answers
63
views
Decoding Endogenous vs Exogenous - Parameter vs Decision Variable - and Independent vs Dependent
this is a topic that i feel is very implicit in a lot of economics, but is some times brushed over in interest of getting strait to the model or the maths. But often i realise i don't actually know ...
- 343
0
votes
1
answer
74
views
Ruling out corner solution in portfolio maximization problem
I am new to the econometric world.
I have a portfolio maximization problem
$$
\max \sum_{i}^ n a_{i} x_{i} \quad \text{s.t.} \quad \sum_{i}^n a_{i}=1, a_{i} \geq 0.
$$
I solved the problem but I had a ...
4
votes
2
answers
169
views
Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Complementary slackness condition (CSC) state that
$\lambda_j[g_j(x) − c_j] = 0 \hspace{5pt} \text{for} \hspace{5pt} j = 1, ..., m.$ Therefore, every constraint either needs to be an equality ...
- 173
2
votes
1
answer
53
views
Budget-feasible set in a portfolio choice problem
I am going through Duffie's Dynamic Asset Pricing book, and already ran into something that confused me on the third page. First, some definitions.
Let $\{1, \cdots, S\}$ be a finite set of states, $D$...
4
votes
1
answer
58
views
Calculating the Compensating Variation with $M^2$
We can calculate the compensating variation (CV), which (to my understanding) is the amount of money we would need to give back to a consumer to keep them at the same level of Utility after a price ...
- 343
1
vote
0
answers
30
views
Consumption smoothing via Euler equation
A representative cnsumer maximizes their lifetime utility function:
$$
U=\sum_{t=0}^{\infty}\beta^{t}\text{ln}\left(C_{t}\right)
$$
defined over consumption. They supply one unit of labour ...
- 167
1
vote
0
answers
30
views
How is the constrained maximization problem in the Ramsey model solved?
Assuming that utility is forward discounted $ \rho $, lifetime utility = $ \int_{0}^{\infty} e^{-\rho t}u(c) \,dt $.
And in continuous time, we also know that $ \dot{k} = y - (\delta + n)k - c$.
...
- 69
2
votes
2
answers
45
views
Why does $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$?
I just started learning economics and the textbook says $\frac{MU_X}{P_X}=\frac{MU_Y}{P_Y}$ for a buyer with a fixed budget to spend on two goods, $X$ and $Y$.
Let's say goods $X$ and $Y$ both cost $\\...
- 121
0
votes
0
answers
19
views
Does decreasing a player's external regret always (monotonically?) decrease that player's cost function?
In a game in which, at each time step, a player declares a mixed strategy, then an adversary assigns a cost to each of that player's pure strategies, and then the mixed strategy is applied and that ...
- 389
1
vote
0
answers
27
views
Optimization Model for Market Clearing using Uniform Pricing
Dear all, can someone please share a simple example for market clearing via Uniform Pricing using an Optimization Model?
I am trying to simulate a market using bid values with quantity and price, from ...
0
votes
1
answer
58
views
Looking for a term I'm pretty sure exists
Let me describe the situation:
Company is selling a product; they buy it at x, sell it at some % over for profit. Taken on a monthly scale, you can see the profit of that particular object by ...
1
vote
1
answer
41
views
How to find the e(p,u) of u(x) = x1 + x2 + x3
If I do the LaGrangian for the Expenditure minimization problem, it comes as p1 = p2 = p3, how do I substitute it back in the constraint and find the Hicksian demand to find e(p,u)?
- 63
1
vote
1
answer
96
views
how to derive marshallian demand functions from leontief preferences?
For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a ...
- 121
0
votes
0
answers
43
views
Solving Spatial General Equilibrium Models - Redding and Ross-Hansberg
I am reading the article "Quantitative Spatial Economics" by Redding and Rossi-Hansberg 2017. The article is very interesting and builds out a rather complex mathematical model for ...
- 385
2
votes
0
answers
29
views
2 dimensional optimization using fminsearch
I have a complicated capital - debt capital structure optimization problem but to start off simple I just added an extra parameter to the stochastic neoclassical growth model to see how the ...
- 31
0
votes
0
answers
52
views
Finding cost function
The production function is as follows
$$f(z)=(z_1+z_2)(z_3+z_4)$$
Find the cost function?
What I did is as follows. But I am not sure about my solution. How do you solve it?
*duplicated question
- 96
0
votes
1
answer
24
views
How is production managed with respect to the long run vs the short run?
Assuming perfect competition, I think that firms are price takers in the labor/capital markets as well (in the short and long run), correct?
And I know that the Long-run total cost curve is derived by ...
- 51
1
vote
1
answer
30
views
Utility maximization across yield curves?
I'm attempting to solve a utility maximization problem for return-on-investment (ROI) across two different products, where each product experiences a different linear ROI curve.
For product one, the ...
- 11
1
vote
1
answer
75
views
Can following these marginal conditions have the net utility function converge to a maximum? [Edited]
I have the following net utility function which is made up of one positive utility (with a bliss point) and two negative utility (i.e., disutility) functions;
$$Y(a_1,a_2)=y(a_1)+v(a_1,a_2)+w(a_2).$$
...
- 419
1
vote
0
answers
96
views
Find cost function for given production function
I have the following production function
$$f(x_1,x_2,x_3,x_4)=max\{\min\{x_1, x_2), x_3+2x_4\}\}\ge q$$
And I want to find the cost function.
What I think
(1) $P_1+P_2 <P_3$ and $P_3/P_4<1/2$
...
- 96
2
votes
1
answer
71
views
Simplfying Euler equation under expectations
I am working through a textbook that derives the following first order
(Euler) equation between periods $t$ and $t+1$. We end up with:
$$
-C_{j,t}^{-\sigma}+\beta E_{t}\left\{ \left(\frac{C_{j,t+1}^{-\...
- 167
1
vote
0
answers
32
views
Find the utility of each agent whenever the social welfare is maximized
Question:
Suppose that the utility possibilities curve of 2 people economy is given by the equation $u_1^2 + Au_2^2=20$ where $A\in R_+$ and the social welfare function of the economy is $W(u_1,u_2)=...
- 96
2
votes
0
answers
21
views
Relation between KKT necessary conditions
I am trying to understand the relationships in the KKT theorem between being a maximizer, satisfying the first order conditions (FOCs) and complementary slackness (CSC), and the linearly independent ...
- 198
5
votes
1
answer
219
views
What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?
I am new to Macroeconomics, but I understand the basics of Recursive Macroeconomic models--following the Ljungqvist and Sargent book. So I get the basic recursive problem to find a vector of ...
- 385
5
votes
1
answer
113
views
When does it make sense to use variational methods, versus dynamic programming, versus nonlinear control methods so solve DSGE models
I come from a statistics and applied math background, but have been looking at some problems related to macroeconomic DSGE models lately. So I am still trying to understand the ideas and economics ...
- 385
0
votes
0
answers
24
views
Policy function iteration method in continuous time (with shocks)
Is there any reference available on the algorithm of policy function iteration method in continuous time, when we have uncertainty in the model?
Currently, my conclusion is that the combination of ...
- 1
4
votes
1
answer
256
views
Concave utility functions solution example
In the following post an example is given of the corner solution for a concave utility function. I tried solving it but got stuck. I have no idea how these types of problems are solved so if you could ...
- 43
1
vote
0
answers
28
views
How to define the market and the clearance conditions of a general sectoral computable equilibrium model?
I am trying to implement a general multiproduct market (partial/sectoral) computable equilibrium model, where "general" refers to the fact that the relation of complementarity/...
- 277
1
vote
1
answer
64
views
System of first order partial differential equation
I have following function -
$$ \max_{x, y} ~ u(x, y)^{3}x + (1-u(x, y))^{3}y$$
FOC:
$$u_{x}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +u(x,y)^{3} = 0$$... (1)
$$u_{y}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +(1-u(...
- 475
3
votes
1
answer
78
views
Nash Equilibrium with Constraints on Decision Variables
I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this:
$$\max_{x_1} f(x_1, x_2)$$
$$\max_{x_2} g(x_1, x_2)$$
subject to
$$x_1 + ...
1
vote
0
answers
63
views
Market price of interest rate risk under the CIR model
My goal is to find the market price of risk associated with the interest rate under the CIR model whose stochastic differential equation under the physical measure is given:
\begin{eqnarray}\label{...
- 11
0
votes
1
answer
66
views
Unified growth theory optimization problem
I'm trying to understand the standard unified growth theory model as summarized on page 60 here: https://www.econstor.eu/bitstream/10419/80210/1/481894578.pdf;Unified
The basic household optimization ...
- 41
4
votes
0
answers
41
views
What assumptions can be made to ensure convexity in this optimization problem?
This question is a continuation of the question I asked at:
How can I show convexity of this value function?
Where I came to the conclusion that more assumptions are required to show that the ...
- 183
4
votes
1
answer
243
views
How can I show convexity of this value function?
I have set up an optmization problem as follows:
$$V(A)=\max_{l, C} \quad u(C,l)$$
Where the only constraint is as follows:
$$C=f(l,A)$$
Here $u$ is the utility function which captures social welfare. ...
- 183
3
votes
2
answers
188
views
How can I formulate the following optimization problem?
I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I ...
user40867
0
votes
1
answer
47
views
Distinguishing Between Different Terms in Economics
I have no background and Economics and am trying to teach myself about some basic things in Economics. For example, I am trying to understand the following terms:
Nash Equilibrium
Optimal Strategy
...
- 81
2
votes
1
answer
43
views
Differentiability of value of convex optimization problem
Setup:
Consider the problem
$$
V(y) \quad = \quad \min_{x \in \mathbb R^N} f(x) \quad \text{s.t.} \quad g(x+y) \leq 0
$$
where $f$ and $g$ are convex functions and $y \in \mathbb R^N$ is a parameter ...
- 121