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
5
votes
3
answers
639
views
Constrained optimization problem
So for optimization problems we have only been given scenarios in which we can just solve by doing MRTS= -w/r and the quantity that wants to be produced is stated in the question.
However I was given ...
- 95
0
votes
0
answers
71
views
Comparing amount of investment under two situations
I am studying the following problem -
Investor has $1 and he has to split it between two assets. These are risk-free assets.
Under scenario 1 - These are normal times so he don't have to worry about ...
- 499
2
votes
1
answer
114
views
Find the Pareto Efficient set for 3 Leontiefs
I'm struggling with the following General Equilibrium exercise:
Find the Pareto Efficient set for this Pure Exchange Economy;
The consumers are $i = 1,2,3$ with these Leontief utilities:
$u_i(x_{1i},...
- 2,351
2
votes
0
answers
83
views
Nash Equilibrium as a Saddle Point
Consider the zero-sum game $A = \begin{bmatrix}5 & 3 \\ 4 & -3\end{bmatrix}$. The interpretation is that Row Player chooses a probability distribution over Top and Bottom $\vec x = \begin{...
- 455
2
votes
1
answer
58
views
Return to scale of a production function, $q = L^\lambda + K^\gamma$, is determining it possible in that general form?
Given the production function $q = L^\lambda + K^\gamma$, how do we determine the return to scale for different value of $\lambda$ and $\gamma$?
I know we have to determine the homogeneous degree of ...
- 41
1
vote
1
answer
37
views
Under what conditions would a quasilinear utility function in a function form exhibit diminishing marginal rate of substitution?
Let the utility function be: $U(x_1,x_2) = x_1 + x_2^\alpha$.
Diminishing MRS requires $\frac{dMRS}{dx_1} <0$, however, taking this derivative results in 0, as $MRS = \frac{1}{\alpha x_2^{\alpha -1}...
- 11
1
vote
2
answers
42
views
How to determine if a production function in a functional form has diminishing marginal rate of technical substitution?
For production function $q(L,K) = L^\lambda + K^\gamma$ The MRST is defined as
$\frac{\lambda L^{\lambda-1}}{\gamma K^{\gamma-1}}$. Is it correct and sufficient to say that in order for MRTS to be ...
- 41
0
votes
1
answer
45
views
How to solve for demand?
Hi I am fairly new to these kinds of optimization problems and I am not clear how from equation (1) the authors derive the demand function for a drug j.
I mean I think they used the Lagrangian and as ...
- 143
1
vote
1
answer
62
views
Cost minimisation for the production function $f(L,K) = L^\lambda + K^\gamma$
For the function: $f(L,K) = L^\lambda + K^\gamma $. The value for $\lambda$ and $\gamma$ is not given.
What type of production function is this (quasi-linear, CES)?
Is it true for there to be ...
- 41
0
votes
1
answer
25
views
Looking for tools to model an economy (like the US) and the effect of changes such as Fed Interest rate or inflation
I am leaning toward a python solution and can work in a Win or Linux environment. I have looked into the area a bit and looked at some of the modelling efforts of the Federal Reserve and the "...
- 1
2
votes
2
answers
59
views
Maximise Arbitrary Utility Subject to Budget Constraint
Suppose we have a general maximization problem of the form:
$$\max_{q_1,q_2} U(q_1,q_2) \text{ subject to } p_1 q_1 + p_2 q_2 = y$$
Suppose I allow $U$ to be concave, increasing and invertible. What ...
- 259
1
vote
0
answers
73
views
Convexity of indirect utility in probabilities
I am interested in the concavity in $p$ of the indirect utility function
$$V(p,W)=max_{x,y,z} pf_1(x,y)+(1-p)f_2(x,z)$$
under the constraint
$$x+py+(1-p)z=W$$
where $0<p<1$ and where $f_1,f_2$ ...
- 11
0
votes
0
answers
17
views
Breakeven analysis for computer upgrade decision making
I need to perform a break-even analysis of moving from one system design to another.
Definitions
$M_0, M_1$ = one-time initial manufacturing cost of currently deployed system design and new system ...
- 1
0
votes
1
answer
100
views
How to find marshallian demand for Leontief Utility with 3 goods. u(x 1 ,x 2 ,x 3 )=min{2x1 + x3, x2/2}
I have a utility function
$$u(x_1, x_2, x_3) = \min \{2x_1 + x_3, x_2/2\} $$
I would have assumed that the relationship established is $2x_1 + x_3 = x_2/2$ but my solution manual has it as $$x_1 + ...
- 1
3
votes
1
answer
99
views
A question on the optimisation problem and FWL theorem
Let's say we have the following model:
$$(\beta^{\star},f^{\star}) := \arg\min_{\beta,f \in \mathcal{F}} \mathbb{E}[\left(Z_i - f(X_i, E_i) - \beta^\top \boldsymbol{\tau}_{i,E_i}\right)^2|S_i^{tr} = ...
0
votes
1
answer
60
views
Why is incidence not included in social welfare maximization?
I am very confused on why incidence is not included in social welfare maximization of one good. Typically, I see the optimization over price done something like this:
$C$ ~ production cost function
$...
1
vote
0
answers
52
views
Help with Deriving Hicksian Demand in the Monocentric City Model?
I have a fairly standard Alonso-Muth-Mills model, but struggling to derive the Hicksian demand.
Starting with the basic utility function:
And this Budget Constraint:
Housing Floor-space is ...
- 11
2
votes
2
answers
120
views
Do standard consumer theory axioms rule out corner solutions?
By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.
If these axioms are not sufficient ...
2
votes
1
answer
61
views
A question about Lagrangian, KKT theorem, consumer's problem
Suppose we want to maximise a expected utility function: $$E_1(u(C_1,C_2,C_3))
$$ subject to following constraints. There are two possible situations each with probability $\frac{1}{2}$.
$$C_1 + S_1 = ...
- 223
0
votes
0
answers
33
views
Derivation of Euler Equation in presence of the Dixit-Stiglitz Aggregator
I reading the working paper of Sebastian Banz (2012). I have an issue with the derivation of the Euler equation.
The author models the demand side of the economy as follows
The representative consumer ...
- 430
9
votes
3
answers
1k
views
Cost Minimization and Karush-Kuhn-Tucker
A firm produces an output $y$ using two inputs $x_1$ and $x_2$, where the production function is given by $y = \sqrt{x_1 x_2}$ for any $(x_1, x_2) \in \mathbb{R}^2_+$. Union agreements obligate the ...
- 371
2
votes
2
answers
155
views
Proving quasi-concavity for a utility function
I have a utility function, and I want to prove that it is a quasi-concave function:
$$ u(x_1,x_2)= 2x_1x_2+x_1+2x_2 $$
I do this by showing that the set of points where the utility is larger than or ...
- 65
2
votes
1
answer
99
views
Using lagrange on a quasi-concave utility function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I have maximized his utility function, and found its demand functions, for $x_1$ and $x_2$, using Lagrange. However, is it ...
- 65
2
votes
1
answer
352
views
Solving utility maximization, and finding demand function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I want to maximize his utility function.
$$max: 2x_1x_2+x_1+2x_2. uc:p_1x_1+p_2x_2=y_A$$
Using Lagrange, I get
$$L(x_1,...
- 65
3
votes
1
answer
241
views
When is argmax increasing in some multiplier of the objective function?
Let $f : R_+ \to [0,1]$ be continuously differentiable and strictly increasing with $f(0)=0, \lim_{x\to\infty}f(x)=1$ and let $c : R_+ \to R_+$ be continuously differentiable, strictly increasing, ...
2
votes
0
answers
54
views
Irrelevance of Heterogeneous Agent Modelling
This is a question from a previous year PhD entrance exam. I have outlined how I have tried to tackle the problem as well:
N.B. 1 This exam is of 100 points and this particular problem is of 25 points....
- 51
4
votes
2
answers
184
views
Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization
It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign.
If the above is ...
- 53
1
vote
0
answers
33
views
question about production optimization
the question is,
if Q = AK^a(HL)^b
and the parameters are:
(A =100) (K = 10000$) (H = 1) (L = 100 person) (a = 0.5) (b = 0.5)
P = 5 per unit, R = interest rate of 3 percent per year , W = 3 per ...
- 11
1
vote
2
answers
74
views
Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)
Could you please help me evaluate this logic?
I've been told that "if preferences are strongly monotonic, duality holds."
In the case of utility u(x,y) = x^2 + y^2, we will get a corner ...
- 53
0
votes
0
answers
37
views
Modeling Approach to Adjust linear Elasticity Effect in Pricing Optimization
I am working on a pricing optimization model for a product where the price depends on the competition as well as our costs. The current formulation of the model is:
...
- 11
1
vote
0
answers
39
views
Missing Solutions in KKT Optimisation Problem
In the attached inequality, constrained, optimisation, problem. Looking at the specific case where $\lambda_1 = 0, \lambda_2 > 0$ that I am trying to solve, you can see that I have managed to find ...
- 1,011
1
vote
0
answers
27
views
Linear Dependence of Binding Constraint Qualifications in Karush-Kuhn-Tucker
When checking whether the CQ are satisfied in KKT, i.e. checking for Linear Independence amongst all combinations of the constraints.
Is it true to say we only need to check combinations that could be ...
- 1,011
5
votes
1
answer
110
views
Optimisation problem and KKT conditions (unsatisfied?)
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a ...
- 153
1
vote
2
answers
186
views
Why this optimisation problem cannot be solved with "usual" KT conditions?
I have this optimisation problem:
$$f(x, y, z) = 2xy + yz \qquad \text{subject to} \qquad \begin{cases} x+y+2z \leq 1 \\ x \geq 0, y \geq 0, z\geq 0 \end{cases}$$
I solved it with "a certain ...
- 111
1
vote
1
answer
64
views
Why do multiple investment funds exist?
Say that you are a head of an investment fund. Your goal is to maximise the return on money entrusted to you by investing in various enterprises. You look to your left, and see another investment fund....
- 39
3
votes
0
answers
57
views
When is a value function twice differentiable?
Consider the optimization problem
$$
V(\gamma) \ \equiv \ \max_x \ f(x,\gamma) \quad \text{s.t.} \quad g(x,\gamma) \leq 0
$$
Roughly, the Benveniste-Scheinkman theorem implies that if $f$ and $g$ are ...
- 191
0
votes
1
answer
70
views
How should I add my period by period constraint in lagrangian?
This question just suddenly comes to my mind, and I'm not sure what i'm thinking is correct:
suppose I want to maximize my cumulative expected utility $E\left[ \sum_{t=0}^{T}\beta^tU(c_t) \right]$ by ...
- 11
2
votes
1
answer
93
views
Finding the optimal trading strategy
Suppose there is one asset that you can buy and sell, and that you know what was the selling price and buying price at all times between times t0 and t1, both in the past. If you would have had an ...
- 121
2
votes
1
answer
77
views
Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?
Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable.
...
- 23
1
vote
1
answer
111
views
Cost function from a weighted CES production function
I want to find the cost function given the CES production function:
$$
Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho}
$$
with $0<\rho<1$.
So far I have set up the ...
- 11
0
votes
1
answer
188
views
Assumption of interior solution in the Lagrangian method
Why do we need to assume an interior solution before using Lagrangian method for utility maximization problems?
1
vote
0
answers
18
views
Normalization for model comparisons
I have a time series applying the Markov Switching model, which is estimated in about 15 different versions. One or two of the time series had to be normalized in order to converge. That is 1-2 out of ...
- 21
0
votes
1
answer
105
views
Profit maximization; how to derive
Can someone expalin me the math process step by step behind this? I haven't been able to figure it out and I cannot find anything in google
- 1
1
vote
1
answer
249
views
Conditions for an interior solution to the UMP
I was wondering under what set of conditions one is allowed to assume an interior solution to the Utility Maximisation Problem. In most of my classes and lecture notes, interior solutions are assumed ...
- 13
1
vote
0
answers
61
views
Durable goods in a (two sector) necolassical growth model
i want to add a firm to a neoclassical growth model that produces a durable good which it rents out in each period to the consumers.
Right now i'm using the following approach:
The firm maximizes: $\...
- 11
1
vote
1
answer
103
views
FOCs for profit maximization using a transformation function
I'm (still) reading the microeconomics textbook of Mas-Colell et al. On p. 135, the profit maximization problem (PMP) for producers is introduced; characterizing the technology as $Y = \{ y \in \...
- 335
1
vote
1
answer
185
views
How do I show that the minimization problem has a solution?
Consider an inner product space $X$ with the induced metric $d$ (induced by the inner product). Suppose that the induced metric space $(X,d)$ is complete. Moreover, for all $x,y,z\in X$,
$$[d(x,y)]^2+[...
user45416
0
votes
0
answers
189
views
why is the MRS same for everyone?
If the consumers are optimizing and at interior solutions and facing the same prices, then the MRS=p1/p2 will be the same for everyone no matter the preferences and income. but why? I don't understand ...
- 1
2
votes
0
answers
155
views
help with nested integrals in common agency public goods paper
I'm trying to derive the example function used in a paper I am reading and am stuck. Please help. Below is the equation. For now, all I am requesting is someone to help me solve for p. More details ...
- 21
2
votes
1
answer
100
views
Help with a proof for an quite intuitive Utility optimization problem
Assume $U(x,y,a,c )= - c x + B(x,y,a)$, with $\frac{\partial B(x,y,a)}{\partial c }=0$, and with $a$ and $c\geq 0$ being parameters, and with $x$ and $y$ being variables. Further, $B(x,y,a)$ is ...
- 105