Questions tagged [optimization]
Mathematical techniques for the selection of a best element (with respect to some criteria) from the set of available alternatives.
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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 \...
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Differentiation of a composite utility function
Taking the utility function $U: ln(c_1)+\beta ln(c_2)$ with $c_t=\sqrt{c_{Nt}c_{Tt}}$, t indicating a time period 1,2. Ignoring any b.c.'s.
I am having issues interpreting the marginal utility wrt $c_{...
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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+[...
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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 ...
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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 ...
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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 ...
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Is there a name for this generalized assignment problem variant with an additional indicator constraint
I have an optimization problem that's closely related to the generalized assignment problem with an additional constraint. Each knapsack $i$ has capacity $b^i$. There is a number of tasks $j$, each ...
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Effect of a change in prices when the consumer starts from a bundle of goods rather than wealth
I'm struggling to make progress with the following problem:
Assume that instead of income, the agent starts with an endowment of goods c (not necessarily her optimal bundle) & can buy and sell ...
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Inequality and effect on pricing
Suppose there’s an island that for whatever reason has a finite amount of US Dollars; 1000 residents live on the island and use this fixed amount of currency to exchange goods and services.
Suddenly, ...
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Problem with Optimizing Profit in Log-Linear Demand Model
I've trained a linear model to predict log(volume) to capture the elasticity of demand with respect to price difference between my product and a competitor's:
$$log(volume)= constant+elasticityCoef×(...
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Habit formation ala Constantinides (1990)
Consider the following problem, from Constantinides (1990).
\begin{align}
V(W_0, x_0) \equiv \max_{c, \alpha} \mathrm{E}_0 \int_0^\infty e^{-\rho s}\gamma^{-1}[c(s) - x(s)]^\gamma \mathrm{d}s,
\end{...
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Question About Non-Degenerated Constraint Qualification (NDCQ)
I am studying constrained optimization using Mathematics for Economists by Simon and Blume, and I have some difficulties understanding the Non-Degenerated Constraint Qualification (NDCQ). I would like ...
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Leontief function nested in a cobb-douglas function for a computable general equilibrium
I am currently trying to build a CGE model, and I'm stuck with the specification of the agriculture sector. I'm trying to understand how to do nested production functions and also how to solve them. I ...
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Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32
I am studying Implicit Function Theorem and its application on comparative statics using Mathematics for Economists by Simon and Blume. Here is the question:
Consider a pure exchange economy with two ...
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Intertemporal Utility Optimization For Multiple Goods
I'm building an economic simulation game and I'm trying to solve for the values that a person will spend on each good and the amount they will save in the current period, taking into account all ...
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Is there a labor vs leisure model with work experience?
I find the labor-leisure model with utility functions interesting, but I find it lacks the factor of work experience, which is very important in the real life labor market.
This is a reason people why ...
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Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7
I am working on Mathematics for Economists by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that ...
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Signalling a relation
I have a very basic, trivial question to ask. (I apologise if it is not a worthy problem)
I have been trying to formulate an international Economics model where a larger country is trying to signal a ...
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Can a business's search for profits be considered as movement in state space?
When businesses make decisions to increase profits, they have to adjust several "parameters" of the business such as quantity to output, pricing, choosing a production mix, brand positioning,...
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Expected value in budget constraint
I have been reading this paper by Yang Liu, Lukas Schmid and Amir Yaron, which contains a very elegant mechanism that generates an endogenous liquidity premium for US government debt. However, I got ...
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Why do we omit the integral when deriving the f.o.c.’s in long-run growth models such as Romer (1990)?
For example when solving Romer’s model (1990) in continuous time, for the firm producing final goods, its production function is:
$ Y(t) = \int_{0}^{M(t)} (A(t) L_Y)^{1-\alpha} {x(i,t)}^{\alpha} di$,
...
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Technology Parameter In Converted Minimisation Problem
Question:
I want to understand what's going on with respect to the technology parameter $A$ when i convert this minimisation problem into a maximisation problem. The issue is only revealed when i use ...
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Profit Functions are Homogeneous of Degree 1 in all prices
I'm struggling to understand the intuition behind why the profit function is homogenous of degree one jointly in all prices (i.e. input prices and output prices). the Intuition feels like it should be ...
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Hessian Matrix Test - When does it fail?
When does the hessian matrix test fail. I understand we are testing the definiteness of the Matrix, and i also understand that because it's a symmetric $n•n$ matrix, we have a principal minor ...
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Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They given the following reason as outlined in the problem below.
$Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$
&...
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Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
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Perfect Complement Utility Function Maximisation
When we have the function $U(x_1,x_2) = min\{x_1,3x_2\}$
S.t. $p_1x_1 + p_2x_2 = m$
What's the economic, and mathematical intuition for assuming this constraint is binding, i.e. not having to ...
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Stochastic control of jumps of random size
Consider the problem of maximizing expected lifetime utility
$$
V(a_t) \equiv \max_c\mathrm{E}_t \int_t^\infty e^{\rho (s - t)}u(c_t)\mathrm{d}t
$$
subject to a state process $\mathrm{d}a_t$ which is ...
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Logarithmic Utility function Algebra
Question:
I'm told the following (by an exam mark scheme):
Using $a + b =1$
$a[ln(\frac{am}{p_1})] + b[ln(\frac{bm}{p_2})] = ln(m) - aln(p_1) - bln(p_2)$
I can't get this to hold without the ...
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Quadratic Form Single Summation notation
I'm trying to translate what feels like a double summation notation , compacted into a single summation. It's the summation definition of quadratic forms.
$Q(x_1, x_2.....x_n)=\sum_{i \le j}a_{ij}...
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How to derive the short run cost function
Given the production function $f(K, L)=\min\{3K,2L\}$, the procedure to find the long-run cost function would be to use the condition: $3K=2L=Y$ where $K=\frac{\overline{Y}}{3}$ and $L=\frac{\overline{...
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Envelope Theorem and Factor Mix Intuition
Brief Summary of question:
I'm frustrated with the intuition of the envelope theorem that when input costs change, our envelope theorem tells us we do not need to re-optimise demand levels.
Context:
I ...
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CRS, Homothetic Functions, and constant MRTS
Questions
When our Isoquant map exhibits constant MRTS along a ray from the origin making. Why do we make specific reference to.
Constant returns to scale
Homothetic Functions
I'm asking because it ...
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Is it possible to get back the consumer’s utility function from their demand functions?
I am curious about if it’s possible to reverse the utility maximization process, i.e. given the consumer’s Marshallian demand functions, find their utility function.
I was thinking of trying to find ...
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Inappropriate use of Calculus in estimating ΔCost?
I have the following model, and i solve for my optimised conditional factor demands, and minimised cost functions $C$. (Note: I have turned a minimisation problem into a maximisation problem). Let's ...
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Mixed Partial Derivatives in Profit Function
$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$
Question 1:
Using the first order conditions, we get:
$x = \frac{pa}{w_x}$
$z = \frac{pb}{w_z}$
What do we call these Input demand functions as a ...
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Lagrange Multiplier Dual Meaning?
Is the Lagrange multiplier:
The marginal cost of the constraint?
The marginal benefit of relaxing the constraint?
Through duality, both interpretations imply the other?
If anyone were so kind, I ...
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Non-Negativity Constraints KKT
When we take our Lagrangian and we include non-negativity constraints. If a variable $x = 0$ do we take FOC first or set $x=0$ first?
E.g.
$Max \; L(x, y, λ) = f(x,y) - λ_1(g(x,y) - k) - λ_x(-x) - λ_y(...
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Utility maximization for a household consisting of a woman and a man, with gender discrimination
Consider a household consisting of a woman and a man, with preferences over leisure and consumption given by:
$U(\overrightarrow{c},\overrightarrow{l}) = \ln{c} + \ln{l^F} + \ln{l^M}$
where $\...
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Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
Encountered the following problem during self study:
My take on the problem is that if we can show that the equation of the income expansion path is $x_1=x_2$ for all such $U(x_1,x_2)$ then we have ...
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Missing Non-Negativity Constraint?
We have the constrained maximisation problem:
A perfectly competitive firm produces one output with two inputs, capital $(k)$ and labour $(l)$. The rental cost of capital is equal to $r >0$ and ...
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Semi Definite Test of a Matrix:
I have a problem. My linear algebra course pack says that there is no simple test to check whether a matrix is positive or negative semi-definite. But my mathematical economics course-pack says.
A is ...
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Convex Combination of pairs of points
Is it appropriate/meaningful to write vector/points $(a,b) \le (c,d)$, where i would mean component wise each component is $\le$
Specifically is my example below with reference to concavity ...
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Habit forming Model & State Variables
Taking the habit formation model of consumption, as a standard dynamic programming problem.
Bellman Value Function for Habit Model
Max$\sum_{t=1}^Tβ^tu(c_t - γc_{t-1})$ $\qquad \qquad (1)$
s.t.
$w_{t+...
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Bellman Equation & Envelope Theorem
I'm unsure where the envelope theorem comes into play when i differentiate the Bellman Equation with respect to $k_t$.
To me it looks like the regular chain rule and in fact the exact opposite of the ...
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Solving first-order conditions to this social planner's problem
I am trying to derive the first-order condtions to this economic problem, where a unit mass of ex ante agents identical agents have preference given by
$$E_{0}\sum_{t=0}^{\infty} \beta^{t} \left\{ \...
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Testing for Concavity - Local Maximum & Global Maximum
My question is under which contexts Negative Definitness (ND) vs Negative Semi-Definitness (ND) is required for classifying a global maximiser. And also Global vs Local.
I also want to understand what ...
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Second Order Condition - Always means second derivative?
In optimisation, does First Order Condition (FOC) always mean a condition for a max/min related to the first derivative.
Similarly, is Second Order Condition (SOC), called second order because it ...
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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 ...
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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 ...
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