Single Variable Optimization - Study Questions | ECON 302, Study notes of Microeconomics

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LECTURE 2

SINGLE VARIABLE OPTIMIZATION

Q

UESTIONS

/I^ SSUES TO

A

DDRESSED

1. How did calculus made its way into Economics?2. Why is the optimization hypothesis widely used?3. How should one view optimization-based models?4. Why is single variable optimization not enough?

H

OW

D

ID

C

ALCULUS

M

AKE IT

I^ NTO

E

CONOMICS

The Assumption of Optimal Behavior: Economic agents are seeking to achieve something that is “best” or “optimal”from their point of view.

Examples: – Managers maximize profits or utility.– Consumers maximize utility.– Governments maximize output or chances of re-election.

Do these concerns invalidate the optimization hypothesis? –^

If the economic agents’ deviations from optimality are not systematic, theoptimal solution may describe the average behavior.

-^

If economic players arrive at a near optimum behavior via heuristics,optimization-based economics would still have descriptive power.

-^

Optimization based economics can be used in a normative manner, even itsometimes does not describe actual behavior well.

WHY IS THE

O

PTIMIZATION

H

YPOTHESIS SO

WIDELY

U

SED

a.^

This assumption, though not literally true, is a good approximation for“average” behavior in many cases.

b.^

The concept is precise.

c.^

There are a lot of mathematical techniques that are of widespread usethat can be readily used to explore problems that are written asmaximization problems.

What are some reasonable properties of this profit function?

R^

At zero output the firm makes zero profit.

R^

As output increases, its profits rise.

R^

At large levels of output the firm can only sell it all the output by droppingits price.In turn, this results in lower profits.

THEREFORE:

There must a particular level of output which results inmaximum profits.

Not yet. What is needed is a behavioral assumption.Assumption: The manager wants to maximize the firm’s profits.Other assumptions consistent with an optimization framework are possible.

I^ MPLEMENTATION

:^ F

INDING THE

O

PTIMAL

O

UTPUT

• Trial and Error.– Direct Optimization.

Observation

The profit maximizing level of output corresponds to the level of output wherethe slope of the profit function is zero.

Idea

If we could find a function for the slope, then we could solve the equation

slope (q) = 0

and find the optimum output.This function for the slope exists and is called the

derivative

Key Result

Slope = 0 is a necessary condition for a maximum [if we are at a maximum,then the slope must be equal to zero], but not a sufficient condition [sometimesthe slope will be equal to zero, but we will not be at a maximum]We can gain some insight by looking at the slope of the slope.This is known as the

second derivative

of a function.

A.

For a maximum, say maximizing profits, we needi.

First Order Condition: ii.^

Second Order Condition: