Sequences and Series. Limits. Differential Calculus. Partial derivatives. Lagrangian

multiplier.

September 2011

Sequencies and Series

Differential Calculus

Exercises

Practice with Exercise 1 – Questions 1 to 3

Exercises

Practice with Exercise 1 – Questions 4 and 5

Exercises

Practice with Exercises 1 - 2 - 3

Let's imagine the case where the formularepresents the distance, y, traveled by a moving object at a time, x.

Exercises

Practice with Exercise 4

Convex function can also be called a concave UP function!!

Concave function can also be called a concave DOWN function!!

A recession means that the output of an economy is declining. The rate of decline is the slope of the downward curve at some moment in time. That slope is mathematically the first derivative of the line at that time point.

The second derivative is the change in the slope. When an economy peaks out, at first the slope is flat as output changes from growing to shrinking. Then the rate of decline gets bigger as the economy falls faster. Last fall, the economy was plunging like a boulder thrown off a cliff.

But now there are signs that the rate of decline is going to get less steep.

Economic meaning...

Partial differentiation

Exercises

Practice with Exercise 5

On the Cobb-Douglas production function...

In a more general form the Cobb-Douglas production function is:

where:Y = total production (the monetary value of all goods produced in a year);L = labor input the total number of persons-hours worked per year);K = capital input the monetary worth of all machinery, equipment, and buildings);A = total factor productivity.α and β are the output elasticities of labor and capital, respectively.

These values are constants determined by available technology.Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example if α = 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output.

The CRS property occurs in Cobb-Douglas function because the sum of the exponents on the L and K input variables sum to one. In more general forms of this production function, the fractional exponents on the input variables could sum to less than one (decreasing returns to scale) or sum to greater than one (increasing returns to scale or economies of scale).

Exercises

Practice with Exercise 6

Isoquants

Marginal rate of technical substitution (or just TRS)

Profit maximization and production sets

Exercises

Practice with Exercises 7 – 8 – 9

Comparing unconstrained and constrained profit maximization: A firm’s production function is given by Q=2L1/2+3K1/2, wage = £2, interest rate = 1 and price = £8. Find the maximum profit and the values of K and L that maximize it (a) without constraint, (b) if there is a cost constraint of £99.

Example

A consumer can be characterized by many factors and aspects such as sex, age, lifestyle, wealth, parentage, ability, intelligence, etc.

In modern consumer theory it is assumed that the key characteristic of

a consumer consists of three essential components:

the consumption set,

initial endowments,

and the preference relation.

Consumer's characteristic together with the behavior assumption are

building blocks in any model of consumer theory.

Consumer theory

The consumption set represents the set of all individually feasible alternatives or consumption plans and sometimes also called the choice set.

An initial endowment represents the amount of various goods the

consumer initially has and can consume or trade with other individuals.

The preference relation specifes the consumer's tastes or satisfactions

for the different objects of choice.

The behavior assumption expresses the guiding principle the consumer

uses to make final choices and identifies the ultimate objects in choice. It

is generally assumed that the consumer seeks to identify and select

an available alternative that is most preferred in the light of his/her

personal tasts/interests.

In the basic problem of consumer's choice, not all consumptions bundles are affordable in

a limited resource economy, and a consumer is constrained by his/her wealth.

Budget constraint

Preferences and utility

Example

Prices and elasticities

Exercises

Practice with Exercise 11