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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39 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 $...
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1answer
281 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 ...
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2answers
92 views

In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?

In lecture, my professor defined the following 2-period consumption model: $c_i = $ consumption in period $i$. $y =$ endowed income in period 1. $r = $ interest rate in perfect credit markets. $h = $ ...
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1answer
490 views

Is this Cost function concave or convex?

Given the following cost function, where t is the quantity of some product. $$C(t) = 1/3t^3 - 7t^2 +11t + 50$$ here is a graph between $t= 0$ and $t = 25$ We are asked if this function is convex or ...
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1k 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 ...
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1answer
513 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 ...
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1answer
29 views

When the global optimal is outside of the constraint set, what will be the demand?

$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex. $a,b\in\mathbb R^n$ are two points. Our budget set is the segment $[a,b]$. Given: $$x^*=\arg\max_{...
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1answer
2k views

Concave utility functions corner solution explanation

I seem to not be getting this. Could someone explain me the mathematical way to show a concave utility function [like (ax^2+by^2)] subject to a budget constraint has a corner solution. I get the ...
2
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1answer
208 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. ...
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2answers
1k views

Intermediate macroeconomics: optimal bundle for quasilinear utility?

How would I go about solving this question: Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. ...
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1answer
56 views

Can the weierstrass and the Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?

Question: An agent who consumes three commodities has a utility function given by: $u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$ Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’...
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1answer
50 views

Have I found the correct Emission Price

Let's say that there is a hotel owner $(H)$ and a woodworker $(W)$ working in close proximity to one another. The woodworker produces $x$ units to sell at market at $p_{x}=6,5$. From the woodworking ...
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2answers
54 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,...
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2answers
974 views

Editing formula for finding Marshallian Demand with Cobb-Douglas utility function

Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$: $x_1 = \frac{am}{p_1}\\ x_2 = \frac{bm}{p_2}$ But what if the utility function looks like $u=...
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2answers
75 views

Complementary slackness conditions (Kuhn-Tucker)

Consider the problem of maximising a smooth function subject to the inequality constraint that $g(x) \leq b$. The complementary slackness condition says that $$ \lambda[g(x) - b] = 0$$ It is often ...
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42 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 ...
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408 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 ...
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64 views

Does this conditional increase in income affect a budget line in the same way as an unconditional increase in income would?

It's been awhile since I've taken introductory microeconomics. I remember increases in income move budget line outward. What if the increases have some condition? The problem: Jill has $I$ to ...
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1answer
251 views

Constraints on a symmetric pareto allocation under uncertainty

I've been trying to figure out how the author came up with the constraints for this liquidity model in a textbook I'm reading. details: http://imgur.com/TpVjg4w $U = \pi_1u(C_1) + \pi_2u(C_2)$ where ...
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248 views

Optimizing Cobb-Douglas like functions

Economics isn't my home field, but I'm looking for references for a paper I'm working on and I'm hoping one of you can help. Are there many good references for optimizing a Cobb-Douglas like utility ...
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1k views

First and Second Order Lagrangian-Multiplier Conditions for Optimization

The Statement of the Problem: Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier ...
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263 views

Calculating the optimal portfolio for an investor with quadratic utility

The problem is from Asset Pricing and Portfolio Theory by Back and can be found here. The relevant info from section 2.5 can be found here. Given that we have the Expected value and the variance of ...
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2answers
186 views

Dynamic programming, optimal consumption-savings (finite horizon) problem

Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$, the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ ...
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1answer
158 views

Derivative of CARA utility

Can someone help explain the passage here? I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much ...
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1answer
38 views

Can I Upload my Preprint on Arxiv Before Submitting it to JPE

I wrote a paper relating the optimal deterrence strategy for crime to concepts on statistical physics, and I am considering submitting the paper to the Journal of Political Economy. I'm wondering if ...
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1answer
100 views

Utility Function Implies Consumption of Not All Goods

Suppose we have a utility function with three inputs, $j, k,$ and $s$ described by $$u(j,k,s) = A\ln(k^\alpha + \beta j^\alpha) + B\ln(s).$$ The price of $j, k,s$ are $p_j, p_k, p_s$, respectively, ...
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1answer
192 views

Find Indifference curve/s and Marginal Rate/s of Substitution given only one point

Arka likes fries. She wants to consume as much as possible. She consumes either regular (1 oz) or large sizes (5 oz). Draw her indifference curve through $(x_R, x_L) = (10,0)$ and her indifference ...
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1answer
1k views

Duality of cost minimization and profit maximization

The firm tries to maximize profits $\Pi$ \begin{align} \max_{K,L}\{\Pi(K,L) = F(K,L) - RK - wL\} \end{align} where $F$ is the linear homogeneous production function, $R$ the rental rate of capital $K$...
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1answer
85 views

Solving Constrained Optimization Problem with Two-Period Model of Human Capital

I'm trying to solve a constrained optimization problem in human capital model. The objective function is $\max_{c_1,c_2,\nu} U = u(c_1) + \beta u(c_2)$, subjected to $c_1 = w +(1-\nu)\theta_1 h_1^a$ ...
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1answer
70 views

Kuhn Tucker Maximization

I have to maximize following expected utility function using Kuhn tucker conditions - Since expected utility function are increasing $C_{1,t}$ and $C_{2,t}$ so constraints (i) and (ii) will hold with ...
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1answer
29 views

What is the “bequest condition” in a finite-horizon discrete optimization problem?

For a finite-horizon discrete time optimization problem, my textbook provides a condition called the "bequest condition", which I'm not familiar with. Specifically, where the state at time $t$ is ...
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2answers
249 views

Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$?

Will all the solutions be in the corner or will the cusp in the middle give us any interior solution? This is by the intersection of the budget line. I am getting this type of a shape: But I am not ...
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1answer
83 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 ...
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1answer
92 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 ...
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2answers
157 views

Profit Maximisation if MC is still falling after intersecting MR

really basic one here, I am just in the process of re-covering old ground. I understand that for any profit maximising firm FOC: MC=MR SOC: MR'< MC' But suppose MC has a minimum beyond the ...
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1answer
35 views

Expectational stability: adaptive learning of RE equilibria in dynamic systems

There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand. Step 1. What does this formula below mean, ...
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1answer
151 views

A profit maximization problem (whole problem has been solved, I just have question about interpretation)

I would like to discuss with you about the following production function. $$y=f(t_m, t_l)=\rho t_m^m(n+t_l)$$ where $0<m<1 $ and $n>0$ are fixed parameters. $t_m$ is manager time. $t_l$ ...
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1answer
127 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 ...
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2answers
241 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 ...
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2answers
107 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 ...
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1answer
378 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 ...
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1answer
42 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 ...
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1answer
174 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 - $\beta$...
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Simplex Lp interpretation of dual problem´s solution

I am wondering whether my interpretation of my simplex dual problem result is correct. The primal problem is: ...
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0answers
17 views

Solving a HJB with additional constraints on control and state variables

I am trying to solve a Hamiltonian-Jacobi-Bellman equation with additional constraints on the state and control variables, but I am a bit confused on how to do that. In Intrilligator 2002, it is ...
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1answer
44 views

Generalizing demand for perfect substitutes utility function

I have the utility function: $U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero) $\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
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31 views

Analytical approach to estimate equilibrium price for Real Estate Property

I am looking to calculate the equilibrium price, i.e an optimal price that I can set without affecting demand and maximize revenue. I've gathered historical data: occupancy rates, asking rents for ...
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37 views

Optimization problem of a Cobb-Douglas function with 3 inputs

A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function: $$ Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3} $$ ...
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41 views

Kuhn-Tucker conditions in linear cost minimization

Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by $$ f(x,y) = ax + by $$ and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} ...
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25 views

What does the elasticity say about the fraction of total cost used on input 1?

A firm have the following production function $$ y=x_{1}^{\alpha} x_{2}^{1-\alpha}, \quad 0< \alpha < 1 $$ $w_1>0$ is the cost of input 1 and $w_2 > 0$ is the cost of input 2. (1.1) ...