Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
0
votes
1answer
31 views
General Equilibrium with Linear Production
I don't think I understand how optimization problems with a linear function work as of now. If you have a production economy with two agents, two goods and Cobb-Douglas utility representation, and you ...
2
votes
0answers
14 views
Finding savings in an Overlapping Generations model
I have not seen this question asked anywhere, so I'm posing it here in case anybody else (hopefully) can help me get to the answer. In a nutshell, my question is: how do we arrive at the saving ...
1
vote
2answers
37 views
Difficulty in an economics' optimization problem using Kuhn-Tucker conditions (interpretation difficulty)
I am having troubles in solving correctly the following problem:
A company wants to minimize its total costs, on the condition that the income obtained from the sale of the quantities $x_1, x_2$ of ...
1
vote
0answers
27 views
Natural borrowing/debt limit and other borrowing constraints
When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
1
vote
2answers
49 views
Economies of scale: when is it disadvantageous?
So, I watched a video on economies of scale. It makes sense to me but I'm wondering, is there a point where say doubling the production rate makes the product even more expensive? How can I figure out ...
-4
votes
1answer
130 views
Question about budget constraint and utility maximization [closed]
I have also following budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where y is income.
Assume that there are two stories. The agent can shop in both of them. The first store ...
2
votes
2answers
118 views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
2
votes
1answer
22 views
Linear programming, shadow price range
I'm not sure how to determine the range for which a shadow price is valid.
You might be able to skip straight to the question here.
I've been introduced to it using the following approach in 2D.
...
2
votes
2answers
446 views
Utility maximization question setting up.
Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$
Assume the agent's income is $y=5$. The price of one unit ...
1
vote
1answer
37 views
Constant MC - Monopolstic Competition
I would like to know if it is possible to have constant marginal costs (MC) in a business that is operating on a market, that is defined by monopolistic competition?
The company is a construction ...
0
votes
0answers
17 views
Broyden-Fletcher-Goldfarb-Shanno, L-BFGS-B, PORT
I was looking for advantages and disadvantages of the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS), the L-BFGS-B and PORT algorithm in optimization. Which one promises the best results and why?
0
votes
0answers
27 views
Optimization of supply between multiple distinct demand 'markets'
I am attempting to solve the following optimization problem, and am curious if it is doable without an iterative process.
Consider the following:
1) Two distinct markets (Market X and Market Y) with ...
0
votes
0answers
21 views
Stackelberg/seqential entry in Hotelling and numerical analysis
I am trying to solve a Stackelberg type of game with three firms in the framework of the Hotelling line. The game is as follows for n=3 firms:
sequential entry in the market
compete in prices in the ...
2
votes
0answers
56 views
Solution to Dynamic Programming (Bellman Equation) Problem
Could someone please provide pointers on how to solve the below? If any theoretical approximations are possible, that would be very helpful. If numerical solutions are the right approach, could you ...
0
votes
0answers
19 views
PORT Routines Advantages/Disadvantages
I was using the PORT optimization routines as proposed by Gay (1990)“Usage Summary for Selected Optimization Routines”. I searched online and in the paper, but I couldn't find any particular listed ...
0
votes
1answer
55 views
How to prove that a point is a maximum point
I have the following function:
$$
\Pi =\int_{0}^{z}[x_{1} + \alpha y + \alpha \frac{N-2}{2}y - \beta(z) - \gamma ( \beta(z) - \beta(y))](N-1)y^{N-2} dy $$
The first derivative with respect to $z$ is:...
0
votes
0answers
28 views
Suppose Jon experiences anticipatory utility and is dealing in the current period (t=0) when to consume a good, what is the optimal time
Suppose Jon experiences anticipatory utility and is dealing in the current period (t=0) when to consume a good, what is the optimal time to consume as a function of $\delta$ and $a$? Recall the ...
0
votes
0answers
46 views
Dornbusch-Fischer-Samuelson model optimization
I have a basic utility function in Dornbusch-Fischer-Samuelson model to be optimized: $u=\int_{0}^{1}b(z)\ln c(z)dz$ with budget constraints as follows $\int_{0}^{1}b(z)c(z)dz\leq Y$ and $\int_{0}^{1}...
0
votes
1answer
80 views
Determining the elasticity of Hicksian Demands
If we have Hicksian (compensated) demand functions, how can we determine the income elasticity and own price elasticity?
Is the procedure the same as for Marshallian (uncompensated) demands?
1
vote
0answers
43 views
Karush-Kuhn-Tucker in infinite dimension
Does the Karush-Kuhn-Tucker theorem on sufficient conditions for optimality of a convex program apply in countable dimension?
For precisions, see Definition 4.1.1 and Theorem 4.1.4 of this course. ...
1
vote
1answer
97 views
Reservation utility
I am self-studying contract theory using Bolton and Dewatripont (2005). It is meant for grad students, which might be why I am having a difficult time understanding basic terminology. Here is the ...
0
votes
1answer
65 views
Social planner's first order condition (current-value Lagrangian)
Here is (one of the ways to state) social planner's problem:
Eric Sims' notes then immediately gives the solution:
I am trying to connect these two lines. This is what I get after taking a ...
1
vote
0answers
28 views
Optimal population allocation layout over the Earth
Does exist some model of optimal people occupancy over whole our planet?
Something accounting for climate, resources (with and without existing settlements) availability, progress and grows ...
2
votes
0answers
286 views
Second order condition for symmetric game
Denote by $i \in \{1, \ldots, n\}$ an economic agent.
Let $\mathbf x \in \mathbb R^n$ denote a vector of actions and $x_i \in \mathbf x$ a typical element.
Let further $f_i : \mathbb R^n \to \mathbb ...
7
votes
1answer
132 views
Optimization: Dynamic Programming vs Kuhn-Tucker
Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to ...
0
votes
1answer
34 views
Optimal taxing in case of negative externalities
Suppose an individual $i$ has the utility function
$U= f(x(i)) - k$(sum of all $x$ with index not equal to $i$)
Where $x(i)$ denotes the miles driven by $i$, and $k$ is a positive constant.
The ...
0
votes
1answer
197 views
Price optimization with demand forecast
I have one year sales data of a retail company and lets say I am forecasting the next month sales for the product. I have got the sales using time series in R. Now I want to forecast the price as well....
4
votes
1answer
130 views
Topkis' Theorem
Suppose my optimization problem is stated as follows
$\max\limits_x f(x,t)$
$s.t.$ $g(x,t) \leq 0$
I am interested in finding the direction $x^*$ changes with the parameter $t$. Can someone ...
4
votes
2answers
335 views
Solving a maximization problem by substitution when the constraint is in implicit form
I am trying to understand how the first order conditions for an interior solution of a maximization problem were derived using the substitution method.
The problem is:
$$\max\limits_{x\ge0,y\ge0}P(a-...
1
vote
1answer
78 views
Can 10% of the population provide super-abundance for the entire world?
In a discussion at the Watson Institute for International and Public Affairs on Nov 9, Mark Blyth, a political scientist, said that
We live in a world where literally 10% of the population could ...
3
votes
1answer
218 views
Adding a non-binding constraint to the objective function
I am dealing with a constrained optimization problem found in Tirole's Theory of corporate finance. My question is not related to the details of this model, but just to provide some context, we are ...
3
votes
1answer
38 views
Computing optimum efforts
Consider the following cost function:
$$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$
The value function is:
$$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$
How do I compute the optimum efforts $e_1$ and $...
4
votes
1answer
209 views
Externalities - First order conditions
I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell on my own. I'm reading the externalities chapter, and i found an interesting example:
There are just two firms: ...
1
vote
1answer
39 views
What does this condition for a profit of a firm exist mean?
What is the intuition behind this?
Let $Y=zF(K,N)$ be a production function.
For a profit to exist Indana conditions must hold and :
$$\frac{∂^2zF(K,N^d)}{\partial K \partial N} > 0$$
I ...
1
vote
0answers
17 views
Perfect complement outputs with each output being composed of substitutable inputs
How does one solve the following maximization problem?
$\underset{K_1, K_2, L_1, L_2}{\text{maximize }} min\{K_1 + L_1,K_2 + L_2\}$
subject to $c(K_1 + \mu K_2) + \beta c(L_1 + \mu L_2)$
where $c(...
3
votes
2answers
89 views
Indirect changes in Marshallian Demand
Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$.
It can be shown that the Marshallian demand for $x$ ...
2
votes
1answer
193 views
CV, EV for additive utility; confirm or deny
I'm currently a TA for a class and recently graded a midterm. I gave the answer key back to the teacher, after going over part of the exam in a study hall. I was going to go over the rest of it ...
3
votes
3answers
561 views
Method of Lagrange multipliers with random variables
I'll illustrate the issue I'm having with a simple problem.
Let $c_1, c_2 \in \mathbb{R}$, and $Z$ a real-valued random variable. Let $u:\mathbb{R} \rightarrow \mathbb{R} $ be a differentiable ...
1
vote
1answer
83 views
Optimal Pricing with Advertising
Below are three different demand curves (i) - (iii), which depend on advertising (A).
(i) Q(P,A) = A $\times$ ($\alpha$ - $\beta$P), where $\alpha$, $\beta$ > 0
(ii) Q(P, A) = $\alpha$ + A -...
4
votes
0answers
117 views
Dynamic demand model in many good competitive markets and price optimization
This is a question about demand models, price optimization, dynamic pricing, big data, online learning, so I will cross-post in other communities.
$\mathbf{Background}$
I am interested in dynamic ...
1
vote
0answers
34 views
Local maximum when Hessian is negative semi-definite?
If it possible to have a local maximum when the Hessian is only negative semi-definite (i.e., there is one zero eigenvalue and all other eigenvalues are negative).
If not, what it the ultimate ...
2
votes
2answers
36 views
Finding the minimum # of items to be sold to meet a goal, based on likelihood
Let's say that I have a project and I need to earn some target to be successful, let's say $50,000
The way I earn is by one time donations at various levels. 1, 5, 7.5 (some discounted item), 10, 15,...
5
votes
3answers
359 views
Kuhn Tucker Conditions with fewer non-negativity constraints than number of variables
I have a following type of problem:
$Maximize\,\, F(s,x,y,z)$
$s,x,y,z$
s.t. (i) $g(x,y,z) \le I$
(ii) $x \ge 0$
(iii) $y \ge 0$
(iv) $s > 0$
That is there is no non negativity constraint on ...
3
votes
2answers
210 views
Interpretation of lagrange multiplier
A student wishes to minimize the time required to gain a given expected average
grade, 𝑚, in her end-of-semester examinations. Let $\displaystyle {t}_{i}$ be the time spent studying
subject i ∈ {1,2}....
8
votes
1answer
80 views
Mythbusters - Determine optimal boarding strategy based on time and satisfaction score
Most airlines board passengers starting from the back of the plane and then working their way towards the front (after boarding priority classes and passengers).
In an episode of Mythbusters, Adam ...
3
votes
1answer
91 views
Monetary policy optimization
I was wondering if anyone could give me some advice / lectures / introduction to stochastic optimization that could be applied to monetary policy. I have heard of the Dynamic stochastic general ...
3
votes
1answer
101 views
Constrainted optimization: merge two constraints into one
Consider the following problem
\begin{align}
&\max_u F(x,u)\\
\text{s.t. }& u \in [0,\bar u].
\end{align}
Any idea how to merge the two constraints $u \geq 0$ and $\bar u - u \geq 0$ into one ...
7
votes
1answer
58 views
Overlaping jurisdictions Model: Proof of Lemma 1; The Size of Nations
I've been reading the book 'The Size of Nations' by Alberto Alesina and Enrico Spolaore (can be found on the net if you know where to look) and I'm having trouble following their "proof" of the first ...
5
votes
1answer
953 views
Transformation Function
In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
-1
votes
1answer
155 views
Price Optimization from Data
How can I find the optimal price that maximizes profits, given past sales data? I thought I could do this, but I've been running into problems.
Data:
...
