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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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 ...
Henry's user avatar
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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 ...
Heidegger's user avatar
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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....
Iron filings's user avatar
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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 ...
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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 ...
Eileen's user avatar
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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 ...
user171780's user avatar
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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. ...
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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 ...
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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?
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Relation between second-order derivatives and corner solutions

Imagine I have the simple maximisation programme: $$\max_{x,y}U(x,y)$$ $$\text{subject to: } B=x+y$$ $U(\cdot)$ satisfies the usual properties: it is increasing and concave in both arguments. This ...
ju_pi_car's user avatar
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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 ...
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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
matias's user avatar
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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 ...
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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: $\...
mfba's user avatar
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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 \...
chsk's user avatar
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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 ...
KatLeigh11's user avatar
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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 ...
retVI23's user avatar
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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 ...
TomUcl2003's user avatar
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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×(...
MarcM's user avatar
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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{...
Wittgenstein's Poker's user avatar
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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 ...
Beerus's user avatar
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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 ...
Meg's user avatar
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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 ...
Beerus's user avatar
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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 ...
Aidan Loten's user avatar
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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 ...
Nicolas Torres's user avatar
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2 answers
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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 ...
Beerus's user avatar
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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 ...
Elina Gilbert's user avatar
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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,...
Joebevo's user avatar
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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 ...
Wittgenstein's Poker's user avatar
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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$, ...
Nicolas Torres's user avatar
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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 ...
CormJack's user avatar
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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 ...
CormJack's user avatar
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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 ...
CormJack's user avatar
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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. $Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$ &...
CormJack's user avatar
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2 votes
1 answer
227 views

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 ...
CormJack's user avatar
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1 vote
2 answers
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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 ...
CormJack's user avatar
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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 ...
Wittgenstein's Poker's user avatar
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1 answer
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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 ...
CormJack's user avatar
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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}...
CormJack's user avatar
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1 vote
2 answers
186 views

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{...
Debbie's user avatar
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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 ...
CormJack's user avatar
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2 votes
1 answer
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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 ...
CormJack's user avatar
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1 vote
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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 ...
Nicolas Torres's user avatar
2 votes
1 answer
60 views

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 ...
CormJack's user avatar
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1 vote
1 answer
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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 ...
CormJack's user avatar
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3 votes
4 answers
345 views

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 ...
CormJack's user avatar
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