Chapter 3: Single Variable Unconstrained Optimization optimization
In a sense nearly all economic problems are constrained because we are interested almost exclusively in non-negative numbers for variables like output and
Constrained and Unconstrained Optimization
Oct 10th 2017. C. Hurtado (UIUC - Economics). Numerical Methods. Page 2. On the Agenda. 1 Numerical Optimization. 2 Minimization of Scalar Function.
Mathematical Economics (ECON 471) Lecture 4 Unconstrained
That would be the Lagrangian Method. Consider now a constrained optimization problem with equality constraints. max x. F(x).
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Lecture 2: Optimization
There are two types of optimization problems depending on whether the set of feasible solutions is constrained or unconstrained. If the optimization problem is
Function of Several variables Quadratic forms and Optimization
Level curves in Economics: Example. A simple production function is the Cobb Unconstrained Optimization. Constrained Optimization: Equality constraints.
UNIT 9 CONSTRAINED OPTIMISATION WITH EQUALITY
In economic applications unconstrained optimisation is a relatively rare case. More often
Mathematical Economics Course Code: ECN 3083 (3 credit hours/5
The course covers unconstrained equality constrained and inequality constrained optimization problems including linear programming and a discussion of the Kuhn
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Constrained and Unconstrained Optimization Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign
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Mathematical Economics (ECON 471) Lecture 4 Unconstrained Constrained Optimization Teng Wah Leo 1 Unconstrained Optimization
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THEORY AND ECONOMIC EXAMPLES We begin with a constrained optimization problem of the type We then solve the unconstrained maximization problem
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This is not a very interesting case for economics which typically deals with problems where resources are constrained but represents a natural starting point
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economic decision analysis involves determining the action that best achieves a desired goal or objec- Constrained versus Unconstrained Optimization
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Unconstrained Optimization: Single Variable Basic Math for Economics – Refresher o Constrained Optimization on the other hand imposes limits
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Our agents may face constraints such as budget equations short-sale restric- tions or incentive-compatibility constraints There are also unconstrained
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Until now we have consider unconstrained problems Usually economic agents face natural constraints Consumer's problem: Suppose that a consumer has a
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Single Variable
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Eric Dunawayepdunaway@gmail.com1
Introduction
In both micro and macroeconomic contexts, optimization is a frequently relied upon tool. We use it to maximize profits, minimize costs, etc.We break optimization down into two types:
Unconstrained Optimization deals with situations where we seek to either maximize or minimize one or several variables without any restrictions on the values they can take. Constrained Optimization, on the other hand, imposes limits on the values that our variables can take. Things like budget constraints, production functions, etc.2Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
profit maximization function. Suppose a monopolist served a market that faced the inverse demand function of ൌʹͷͲെʹݍand a constant marginal cost of production of ܿWhat value of ݍ
What is the corresponding price and profit level?
We already know that the monopolist can set his
marginal revenue equal to his marginal costs and determine his optimal quantity that way.3Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
To properly set up an optimization problem, we need a few elements:First, we need to define the problem. This is a
Next, we need to list the choice variables; the ones that we are optimizing for. In this case, there is only one, ݍ. We list these variables under our optimization condition.4Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
Lastly, we list our objective function. This is the goal of the problem. In this case, the monopolist is maximizing his profits,Substituting,
of production is constant) to further expand on this, obtaining,5Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
Once here, we are ready to calculate our first-order (and second-order, if need be) conditions. Being able to set up a problem this way will keep you organized and reduce the likelihood of making mistakes going forward. As a tip, the first thing you should do when facing an optimization problem is to clearly define the problem. Make sure you consider all the variables of interest as well as properly defining the objective function.6Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
When we calculate first-order conditions, we take the derivative of the objective function for each of our choice variables, then set it equal to zero (since that is where a maximum or minimum occurs). Thus, we calculate the derivative of our profit function with respect to ݍ, since that is our only choice variable, then we set it equal to zero. marginal revenue equals marginal cost.7Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
From here, we simply solve this equation for ݍto find our equilibrium.Rearranging terms,
Lastly, dividing both sides by 4, we obtain our equilibrium quantity,If we wanted to, we could go back and calculate our equilibrium price and profit level from this information.
8Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
Recall that for this value of ݍto be a maximum (rather than a minimum), the second derivative must be negative.We can calculate the second derivative by just
differentiating this function one more time, Since this function is negative for every possible value of ݍ, we know that we have a maximum.9Eric Dunawayepdunaway@gmail.com
Unconstrained Optimization
Now we can calculate our equilibrium price and profit level.We find our equilibrium price by plugging the
equilibrium quantity back into the inverse demand function, Lastly, we obtain our profit level by plugging the equilibrium quantity back into the profit function,10Eric Dunawayepdunaway@gmail.com
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