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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Policy function iteration method in continuous time (with shocks)

Is there any reference available on the algorithm of policy function iteration method in continuous time, when we have uncertainty in the model? Currently, my conclusion is that the combination of ...
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utility maximization problem

Consider a simple two-period model in which consumer utility is a function of consumption over two periods.At this time, the utility function of the consumer is assumed as follows. u(x1,x2) = x1 times ...
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Concave utility functions solution example

In the following post an example is given of the corner solution for a concave utility function. I tried solving it but got stuck. I have no idea how these types of problems are solved so if you could ...
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How to define the market and the clearance conditions of a general sectoral computable equilibrium model?

I am trying to implement a general multiproduct market (partial/sectoral) computable equilibrium model, where "general" refers to the fact that the relation of complementarity/...
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System of first order partial differential equation

I have following function - $$ \max_{x, y} ~ u(x, y)^{3}x + (1-u(x, y))^{3}y$$ FOC: $$u_{x}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +u(x,y)^{3} = 0$$... (1) $$u_{y}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +(1-u(...
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3 votes
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Nash Equilibrium with Constraints on Decision Variables

I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this: $$\max_{x_1} f(x_1, x_2)$$ $$\max_{x_2} g(x_1, x_2)$$ subject to $$x_1 + ...
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Market price of interest rate risk under the CIR model

My goal is to find the market price of risk associated with the interest rate under the CIR model whose stochastic differential equation under the physical measure is given: \begin{eqnarray}\label{...
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Unified growth theory optimization problem

I'm trying to understand the standard unified growth theory model as summarized on page 60 here: https://www.econstor.eu/bitstream/10419/80210/1/481894578.pdf;Unified The basic household optimization ...
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What assumptions can be made to ensure convexity in this optimization problem?

This question is a continuation of the question I asked at: How can I show convexity of this value function? Where I came to the conclusion that more assumptions are required to show that the ...
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4 votes
1 answer
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How can I show convexity of this value function?

I have set up an optmization problem as follows: $$V(A)=\max_{l, C} \quad u(C,l)$$ Where the only constraint is as follows: $$C=f(l,A)$$ Here $u$ is the utility function which captures social welfare. ...
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How can I formulate the following optimization problem?

I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I ...
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Distinguishing Between Different Terms in Economics

I have no background and Economics and am trying to teach myself about some basic things in Economics. For example, I am trying to understand the following terms: Nash Equilibrium Optimal Strategy ...
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1 answer
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Differentiability of value of convex optimization problem

Setup: Consider the problem $$ V(y) \quad = \quad \min_{x \in \mathbb R^N} f(x) \quad \text{s.t.} \quad g(x+y) \leq 0 $$ where $f$ and $g$ are convex functions and $y \in \mathbb R^N$ is a parameter ...
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Solving this budget constraint tangent to indifference curve without a graphical approach

Question: Let $V(x, y) = (1-\overline{p}) U(x) + \overline{p} U(y) - \overline{p} U(F)$ where $U$ is a strictly concave function ($U'>0$ and $U''<0$) with $U(0)=0$ and $0<\overline{p}<1$ ...
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Approximating Optimal Choices and Income Inequality

It seems that at least the basic microeconomic theory assumes we are optimizing over quantities that can take continuous magnitudes, but in practice one can often only purchase goods in discrete ...
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Conditions to use the Lagrangian method

I have seen that the prices and $\text{MU}_{i}$ are assumed to be positive (or, the preferences monotonic). This is always mentioned when a utility maximization problem is being solved with the ...
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Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?

This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts. To preface suppose we have the ...
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Two-Stage Utility Maximization Problem

Actually I don't know how to solve such utility maximization problem, only know using FOC and budget constraint to solve for demand. I will appreciate it if someone tell me the procedure facing such ...
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Calculating cost-minimizing inputs with 3 inputs in production function [closed]

How can I determine the cost-minimizing input bundle with a standard Cobb-Douglas production function with three inputs. despite its simple process, the algebra becomes very hard as you go through the ...
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2 votes
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How to derive utility function from indirect utility and Marshallian demand?

c is composite good with normalised price, q is good with price p. y is income. I have this indirect utility function: $$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)}$$ And ...
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Optimization problem - wage determination

Could someone please show how the author derives the the first order condition (14) of this optimization problem using the expressions shown here ? For context this is taken from a labour market model ...
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Merging constraints in an optimization formulation

I am having trouble merging two constraints into one in an optimization formulation. If $x\geq y\geq 0$, and $0\leq z \leq 1$, how can we add the following (merge) the following statements in the ...
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1 answer
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How do I get to this demand function in the monocentric city model?

I need to get this resulting price and quantity (housing): It's pretty clear that the denominator of the quantity function is just the price function. From this utility function: And this constraint:...
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7 votes
5 answers
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Real Life Examples of Optimization in Economics

I am trying to find some real life ("non trivial") examples of optimization related to economics. So far, most of the examples that I come across are from introductory economics textbooks ...
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Visualizing the expenditure minimization problem

I can easily visualize the utility maximization problem ie. $$v(\mathbf{p},m^{*})= \max_{\mathbf{x}} \ u(\mathbf{x}) \ \ s.t \ \ \mathbf{px}\leq m$$ Since it is pretty easy to graph the indifference ...
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Deriving factor allocation of production function

I am trying to solve an allocation problem for a nested CES production function with three factors. The production function we posit is: $$ F(K, \mathbf L, \mathbf C) = [\alpha K^\rho + \sum_{i\not\in ...
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3 votes
1 answer
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Proof of a comparative statics result in a maximization problem

I am thinking about the following question. Let $f(x,\theta)$ be a strictly concave real-valued function in $x$ and $c(x)$ be a strictly convex real-valued function in $x$, both $x$ and $\theta$ ...
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Necessary conditions in overlapping generations model (OLG)

The consumer at each period maximizes \begin{equation} \displaystyle\sum_{i=1}^{I}\beta^{i-1}U(c^i_{t+i},l^i_{t+i}) , t=0,1,2,3,... \end{equation} subject to \begin{equation} (1+\eta_t)c^{i}_t+...
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2 votes
1 answer
121 views

How do I find the socially optimum and equilibrium value?

I am struggling with the following question. Could someone please explain how to do it? People are trying to go to the city centre. A bus takes 1 hour and will always take 1 hour, unaffected by ...
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2 votes
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Arguments of the Marshallian demand system of a Cobb-Douglas utility function

For a utility function of the form $U(x_1,x_2) = x_1^\alpha x_2^\beta$ and the standard budget constraint, the utility maximisation problem gives us a demand system characterised by: $x_1(\alpha, \...
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Comparative Statics: Income Effect

Much of this is setting up the problem. So if you're familiar it's likely best to start from the very bottom and work up if needed. The question is asking about the income and substitution effect. ...
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Excel Solver, how to solve optimization problem?

I have to perform an optimization tast by using excel solver. The case study has two parts and I already managed to complete the first one. However, I couldn't solve the second one so far. Both parts ...
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Maximization with a binding boundary constraint

I have following profit function - $$ max_{x} ~ mx^{2a} - rx $$ $\text{Subject to,}$ $$ p \geq mx^{2a} - rx \geq q $$ Where, $ m>r, p>0, q>0$ and $a< \frac{1}{2}$ Since firm always want to ...
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Constrained optimization problem written as minimax Lagrange

I am reading a note on moral hazard and it shows that a problem like $\max _{s(x)} \mathbb{E}[V(x-s(x))]$ s.t. $\mathbb{E}[U(s(x))] \geq \bar{H}+c\left(a^{*}\right)$ can be solved by setting up a ...
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1 vote
3 answers
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Graphing indifference curves to visualize solutions?

I am having trouble with being able to graph indifference curves. This is a particularly important skill to have especially when trying to visualize corner solutions, and when the Lagrangian method ...
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4 votes
1 answer
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Solving the following cost minimization problem using Kuhn-Tucker conditions

I am currently getting my Masters in Economics. I did not get any exposure to optimization with inequality constraints in my undergrad. I would like to ensure that I am doing this problem correctly. ...
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2 votes
1 answer
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Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?

When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition). As per the calculus wiki: Link, when the Hessian is negative semi-...
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Optimal consumption for infinite number of periods and exogenous income

I have the following optimization problem: $\max_{\{c_t, s_{t+1}\}} \Pi_{t=0}^\infty c_t^{\beta^t}$ $\text{subject to } \space c_t + s_{t+1} = y_t + (1 + r) s_t \text{ and } s_0 = 0$ How do I find ...
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3 votes
1 answer
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Does global maximum of CRS Cobb-Douglas profit exist

In most macroeconomic papers it is taken as given that the aggregate prodution function is $Y=AK^{\alpha}L^{1-\alpha}$, and that the optimality conditions for inputs determine input demands: $$ \max_{...
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Stochastic Optimization

I am working on a Data Panel project that is about macro volatility, taking into consideration some quality indicators. According to my model, indicators definitely determine macro variables. Data are ...
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Where can I find a calculator for constrained optimization of general form of algebraic equation?

I'm working with a fairly complex equation and I need to carry out constrained optimization of the same. The first order differential equations are very messy to solve by hand and hence I thought to ...
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6 votes
1 answer
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GE with an intermediate good

intro I'm looking at a simple model with 1 consumer, 2 goods and 2 firms. I'm trying to get a price vector [p0, p1] that makes it work. By makes it work, I mean, ...
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4 votes
1 answer
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Quantity restriction in model with fixed factor of production

I'm trying to see the effect of a restriction on production in a model where one factor of production is perfectly elastic and the other is fixed. Specifically, suppose the production function is Cobb-...
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Solve the Ben-Porath Model (Optimal Control Problem)

Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
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Pricing optimisation when elasticities are positives? [duplicate]

I’m performing some exercises in order to get the optimal price of some product such a potato chips, biscuits, drinks, etc. (I’m taking price per unit and sales in units) But I’ve found that some of ...
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2 votes
1 answer
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Regression Optimization problem under constraints

To estimate a simple linear regression: $$ y = \beta_0 + \beta_1 x + \epsilon $$ I have the assumptions that a researcher $A$ can only sample individuals with a value $y < y^A$. Similarly, a ...
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1 vote
1 answer
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Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function

I am trying to solve the household's optimization problem in the New Keynesian model framework, where utility is given by $$ E_0\sum_{t=0}^\infty \beta^t \mathcal{U}(C_t,L_t,N_t;Z_t) $$ and period ...
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1 vote
1 answer
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Calculating optimal level of negative externality

I am trying to solve the following question(s): Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility ...
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Optimization problem of the firm

I have been reading an Economics working paper and trying to derive the first-order conditions of a seemingly complicated optimization problem. The optimization problem with choice variables $P_{t}^{R}...
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5 votes
1 answer
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When can one drop time subscripts? Example from Angrist and Kugler (2003)

Not the first time I am asking myself, but in this paper they actually start with a time dependent maximisation problem and then drop all time subscripts. Background: They have profit maximisation ...
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