Derivatives and Differentiation Chapter 6

6.1 Introduction Slopes of the function graphs are important in economics Total cost curves: how fast do costs increase with production? Demand curves: how sensitive are consumers to changes in price? PPC curves: how much bread must be given up in order to produce more milk? Definition. The technique of measuring the slope of a function (in case it exists) is called differentiation. It is also known as differential calculus, or just calculus. Note: differentiation is about more than just measuring slopes, but at this stage the ‘slope’ definition will just suffice. Objective: learn how to calculate slopes.

6.2 Difference Quotient We use Greek letter delta to denote changes in variables. Definition. Consider two variable values and . The difference is called an increment of variable y.

General Setting Since equilibrium values of the endogenous variables can be represented as functions of the exogenous variables, it would be handy to give an answer to the following general question: given a change in exogenous variable x, what would be the resulting change in the endogenous variable y? Current value of the exogenous variable x Corresponding value of the endogenous variable y A disturbance to the exogenous variable The resulting change in the endogenous variable New value of the endogenous variable

Difference Quotient A change in the initial value of by results in a corresponding change in the level of y by Definition. Consider a function y=f(x). A change in y per unit of a change in x is called difference quotient, and is computed as follows: Note 1. Difference quotient is also known as Newton quotient. Note 2. Difference quotient represents the amount of change in y per unit of change in x, so that it can be interpreted as the rate of change of y with respect to changes in x. Note 3. In case the increase in x results in a decrease in y the difference quotient is negative.

6.3 Calculating the Difference Quotient Consider the following quadratic function: Initial value of x: Initial value of y: The new value of y: The difference quotient is:

6.4 The Slope of a Curved Line It is often important to know exactly how steep a curve is. For the linear function, we know that its slope is equal to a single parameter p if the function is The precise value of the slope of any function is given by what we will call a derivative function: = the slope of the tangent to the curve f(x)at point (a,f(a)). is the slope of the tangent line to the graph of function f(x) at point

Difference Quotient and the Slope Definition. Consider any two points P and Q on a curve. A line segment connecting points P and Q is called a chord. The difference quotient is measuring the slope of a chord connecting P and Q. Fixing point P, there exist infinitely many points Q, and hence infinitely many chords and the associated slopes. Which one is the slope of the curve at point P?

Tangents and Slopes Consider the following cost function: C A straight line passing through A and B B is crossing the horizontal axis at angle whose tangent is equal to A As is getting smaller, this straight line will be approaching the tangent line to C(Q) at Q

Tangent Line Definition. A straight line that has only one single point in common with a curve is called a tangent to the curve at that point. Straight line AB is tangent to the curve at point .

6.5, 6.6 The Derivative Definition. Consider a function at point . Consider an increment of the function’s argument taken at point . The limit of a difference quotient computed for function at point for is called a derivative of function f at point . Notation: Note 1. The derivative taken at point is a function of point alone, while the difference quotient is a function of both and . Note 2. Derivatives may not be computed for certain functions. For example, functions with kinks do not have derivatives everywhere in their domain. Note 3. When computing a derivative, remember it only makes sense to do so with a particular point in the function’s domain in mind—”loose” derivatives are nonsensical.

A Straightforward Recipe for Computing the Derivative Choose point a in the function’s domain and increase it by a (small) Compute the corresponding change in the function’s value: Compute the difference quotient Try to simplify the Newton quotient as much as possible Infer the value of the derivative as the value of the Newton Quotient for very small

Straightforward Recipe: an Example Denote . A) B) C-D) As h grows smaller, the difference quotient above “loses” and so that its value becomes equal to While the straightforward recipe works all right on simple functions, it becomes very difficult to apply it on more complicated functions like

6.7 Constant-Function Rule Consider ! There is an important difference between and means that the derivative of function f is equal to zero in the domain of f—in that case function f is a constant. means that function f has a zero derivative at a particular point --in that case the tangent to function f’s graph is parallel to the horizontal axis at that point.

Power Rule In case a is an arbitrary constant, For example, consider Clearly, differentiating a polynomial of degree n the number of times equal to n+1 will result in a function that is identically equal to 0.

Additive and Multiplicative Constants Additive constant rule: Remember adding a constant means shifting the graph vertically up or down, which doesn’t change the slope. Multiplicative constant rule: This rule follows from the constant function rule and the definition of a derivative.

Sums, Products, and Quotients Suppose we know how to differentiate each one of the two functions and . How do we differentiate, say, Sum-difference rule: Note that from now on we will not indicate at a specific point at which a derivative is taken, unless we really need to. That is, we are going to write assuming Product rule: Quotient rule:

Sum-Difference Rule This rule can be extended to any number of summand functions: For example, The constant in this equation drops out when we differentiate the function.

Product Rule By definition of the derivative, Add and subtract in the numerator: Then collect the terms:

Marginal and Average Revenues Suppose we know that a firm’s average revenue is given by some function What can we say about the difference between average and marginal revenues? We can apply the product rule of differentiation to answer that question. Marginal revenue is by definition a derivative of the total revenue function which in turn is equal to the product of average revenue and quantity. In case of perfect competition we know that average revenue is the same (since the competitive price is constant), so and the marginal revenue is equal to the average revenue. On the other hand, when the average revenue (i.e. the demand curve) is downward sloping, reflecting the fact that under imperfect competition, the marginal revenue curve always lies below the demand curve.

Primitive and Marginal Functions Consider a function . Definition. A derivative function is called a marginal function of function f(x). Definition. Function is called a primitive function with respect to the marginal function . Example. Consider a demand function where P is the price, and Q is quantity. Revenue is defined as the product . The revenue function R(Q) is a primitive function with respect to the marginal revenue function defined as .

Quotient Rule By definition: Reducing to a common denominator: Adding and subtracting Rearranging:

An Example Consider a derivative of the following rational function: Identify functions and : Applying the quotient formula: Note that at x=0 neither the primitive nor the marginal function are defined!

Chain Rule We need chain rule when we want to differentiate a function of one variable that is in turn a function of another variable such as Function z is called a composite function since in fact it is a combination of two other functions. The chain rule: Examples: Suppose a firm’s total revenue function, TR(Q)=f(Q), while the output Q itself is a function of labor: Q=g(L). Additional revenue due to one more worker is equal to the additional output produced by the worker times the money at which additional output is sold.

Application 1: a Simple Market Model Equilibrium Q and P Demand; a>0, b>0 Supply; c>0, d>0 Q S a What happens to the equilibrium quantity when parameter a increases? ? a D The equilibrium price level increases as well since P

Application 1: a Simple Market Model Q Consider an increase in the demand parameter b: S D D P