Single Variable Calculus and ApplicationsSingle Variable Calculus A function is usefully thought of...

$: Single Variable Calculus and ApplicationsSingle Variable Calculus A function is usefully thought of as a \rule" which converts an input (denoted typically by x) into an output (denoted$
Single Variable Calculus and Applications

Dudley Cooke

Trinity College Dublin

Dudley Cooke (Trinity College Dublin) Single Variable Calculus and Applications 1 / 47

EC2040 Topic 1 - Single Variable Calculus and Applications

Reading

1 Chapters 7.1-7.3, 9.1-9.4 and 10 of CW

2 Chapters 6, 7, and 9 of PR

Plan

1 Review of single-variable calculus

2 Optimization

3 Concavity and Convexity

4 Applications


Single Variable Calculus

A function is usefully thought of as a “rule” which converts an input(denoted typically by x) into an output (denoted typically by y).

Notationally, we write y = f (x). In words, “y is a function of x .”

x is also called the independent variable or the exogenous variable.

y is also called the dependent variable or the endogenous variable.

In terms of economics, it is important to know what these functions‘look like’.


Economic Functions

Macro - National income accounting. The link between aggregateincome, consumption, investment, and government spending.

Y = C(

Y d)

+ I (r) + G

Micro - Utility functions. We think that individuals felicity depends onconsumption.

W = u (c)

Micro - Production functions. Individual firms output depends onlabor.

y = f (L)

Sometimes we leave the functions general and other times we assumea functional form. It all depends on how much structure we want toput on the underlying economics.


Slope of Linear Functions

The simplest function we consider is a linear function. A line is arelationship of the form y = ax + b.

Start at any point (x0, y0) on the line and move along the line so thatthe x-coordinate increases by one unit.

The corresponding change in the y -coordinate is called the slope ofthe line.

The slope tells us the rate of change: how much y changes when xchanges by a given amount.

The defining characteristic of a line is that this rate of change isconstant.


Slope of Linear Functions

Claim: the slope of y = ax + b is a.

Proof: pick x = x0; this implies that y0 = ax0 + b.

Increase x0 to x0 + ∆x . The corresponding new value of y isy1 = a(x0 + ∆x) + b.

The change in y is ∆y = y1 − y0 = a(x0 + ∆x) + b− (ax0 + b).

This simplifies to ∆y = a∆x .

Therefore,∆y

∆x= a

Rate of change (of a line) is constant.


Slope of a Non-Linear Function

Linear functions are useful because there are simple. However, notmany economic phenomena are linear. Consider the followingnon-linear function (a quadratic) y = x2.

If we start at x = 1 and increase x by 1, then y changes by 3 (i.e.,4− 1).

If we start at x = 2 however, then increasing x by 1 changes y by 5(i.e., 9− 4).

Thus the same change in x leads to different changes in y .

We cannot therefore define a global notion of the slope for non-linearfunctions.

However, it is possible to define a notion of the slope which is validwhen the change in x is “small.”


Tangent Line

Consider the curve y = x2. Also consider the line y = 4x − 4.

This line just touches the curve y = x2 at the point (x , y) = (2, 4).How? y = 22 and y = 8− 4.

Such a line is called a tangent line.

The tangent line has the property that it “looks the same as thefunction around the point that it just touches.”

The tangent line shows the rate of change at a point for smallchanges.

The slope of the tangent line is called the derivative at the point x0.


Diagram of the Tangent Line


The Tangent Line as a Limit of Secant Lines

An alternative way of thinking about the tangent line is that it is alimit of secant lines.

A secant line is a line joining any two points on the curve.

The tangent line is the line to which the secant lines get “closer” asthe points on the line get closer and closer.

This is shown in the diagram which follows for the curve y = x2 andthe point (x , y) = (2, 4).


Diagram with Secant Lines


Derivative

The derivative of a function f (x) at the point x0 is denoted as f ′(x0).

The total differential of f (x) at x0 is defined by the following.

dy = f ′(x0)dx

The total differential is a way of understanding the local rate ofchange of the function f (x) around the point x0.

That is, it is an algebraic way of denoting the slope of a function.


Economic Significance of the Derivative: Production Costs

Imagine that y represents the costs of production in (in $) and x thequantity produced by a firm.

The derivative at a given x0 tells us how costs change in response toa change in quantity, provided the change is small.

For example, if we know that the derivative at x = 2 is 4, this tells usthat if the quantity produced changes by a small amount dx , then theimpact on cost is given approximately by the total differentialdy = 4dx .

Economists have a special for the derivative of the cost function: it iscalled marginal cost.


Remarks

Since the rate of change along a curve is changing constantly, thederivative has to be computed separately at each possible value of x .

The derivative is thus a local phenomenon: it tells us somethingabout the rate of change in the neighborhood of a point, but it givesno information about the rate of change globally.

For example, the information that the derivative of y = x2 (i.e.,dy = 2xdx) at x = 2 is 4 tells us that the rate of change is 4 when xis “close” to 2.

It does not give any information about the rate of change at x = 10,and so on.


Derivative as a Function

Notice that we can also think of the derivative itself as a function ofx .

Given a function y = f (x), the derivative function simply associatesto every x the slope of the tangent line at x .

Typically, when we talk about the derivative, we mean the derivativeas a function. So when we want to talk about the value of thederivative at a point x , we shall mention it by saying “the derivativeat x is . . .”Notationally, given a function y = f (x), we shall denote thederivative function as,

f ′(x) ordy

dx


Rules for Computing Derivatives

There are a number of rules for computing derivatives of specificfunctions:

1 f (x) = a⇒ f ′(x) = 0. [constant]

2 f (x) = mx + b ⇒ f ′(x) = m. [linear function]

3 f (x) = xn ⇒ f ′(x) = nxn−1. [power function]

4 g(x) = cf (x)⇒ g ′(x) = cf ′(x).

5 h(x) = f (x) + g(x)⇒ h′(x) = f ′(x) + g ′(x).

6 h(x) = f (x)g(x)⇒ h′(x) = f ′(x)g(x) + f (x)g ′(x). [product rule]

7 h(x) = f (x)g (x) ⇒ h′(x) = f ′(x)g (x)−f (x)g ′(x)

[g (x)]2 , provided g(x) 6= 0.

[quotient rule]


A Simple Example

Consider the following: y = f (x) = αx−2. We can also write this asy = f (x) = α

x2 . In the first case we can use the power rule. In thesecond case, we use the quotient rule.

Case 1: y ′ = 0 · x−2 + (−2) x−2−1 · α = −2α/x3 < 0.

Case 2: y ′ = 0·x2−α(2x)(x2)2

= −2α/x3 < 0.

In either case we get the same result. As x rises, y lowers. We alsoknow, when x = 0, y = 0 and when x > 0, y > 0 and when x < 0,y < 0.

In terms of which rule you use, it is best to use the simplest one (case1 in this example).


Simple Economic Examples

Consider the production function, y = f (L). Assume the followingfunctional form: y = 12L2 − L3. By (3), (4) and (5),

dy

dL= 24L− 3L2

An alternative form is, y = 6L(3L2 + 4).

1 We can use (6) along with (1), (3), (4) and (5).

dy

dL= 6(3L2 + 4) + 6L(6L) = 54L2 + 24

2 Alternatively, note that y = 18L3 + 24L. Now apply rules (3), (4)and (5) to obtain the same result as above.

Do we expect output to go up or down as we use more labor?Probably up. This assumption rules out certain functional forms.That is, we think, dy

dL > 0.


Another Economic Example

Consider cost and revenue functions. We know the profits of a firmare given by revenue minus costs.

Π(y) = R (y)− c(y)

What would we expect the revenue and cost functions to look like?Probably the following:

R ′ (y) > 0 and c ′(y) > 0

As we produce more, we make more revenue. However, as weproduce more, costs rise.

Clearly, there is a trade-off. At some point marginal revenue (R ′(y))will equal marginal cost (c ′(y)). That is the profit maximizingcondition. We will revisit this idea repeatedly.


Chain Rule

Suppose now that y is a function of u and u itself is a function of x .Such a function is called a composite function.

As an example, consider y = (x2 + 1)2. Here, u = x2 + 1 andy = u2.

The chain rule is used to deal with such cases. Suppose we havey = f (u) where u = g(x), so that y = f (g(x)). The chain rule isgiven by,

dy

dx= f ′(u)g ′(x)

This says that the derivative of y with respect to x is equal to thederivative of y with respect to u times the derivative of y withrespect to x .


Proof of the Chain Rule

Let y = f (g(x)). Now let u = g(x) so that y = f (u). Using thetotal differential, we have

dy = f ′(u)du (1)

Since u = g(x), we can use the total differential here to get

du = g ′(x)dx (2)

Use (2) to substitute for du in (1):

dy = f ′(u)du = f ′(u)[g ′(x)dx

]= f ′(u)g ′(x)dx (3)

This implies that dy/dx = f ′(u)g ′(x).


Examples using the Chain Rule

In the above case, we have u = g(x) = x2 + 1 and f (u) = u2.

Then, f ′(u) = 2u and g ′(x) = 2x , and by the chain rule, we have,

dy

dx= 2u × 2x = 4x(x2 + 1)

Another example: suppose y = (x1/2 + 3x2)5. Then (show foryourself),

dy

dx= 5(x1/2 + 3x2)4 ×

[(1/2)x−1/2 + 6x

]Notice the difference. In the first case, we can multiply outy = (x2 + 1)2 = x4 + 2x2 + 1 and use the usual rules. In the secondcase, we can do that, but it is not a good idea as it becomes verymessy.


Economic Example

Again, consider the total revenue of a firm, R. It also depends on thetechnology (i.e., the shape of the production function) and laborinput. That is,

R = k (y) ; y = f (L)⇒ R = k (f (L))

The derivative of this function is:

dR

dl=

dR

dy

df

dL= k ′ (y) f ′ (L)

Economic meaning: dRdy is marginal revenue (as above). df

dL is themarginal product of labor.

We conclude that the MRPL is equal to the MR times the MPL.


Derivative of the Inverse Function

Consider the functions y = f (x) = x2 and x = g(y) =√

y . We canthink of these two functions as inverses.

If we take x as the input, apply f to it and then pass this outputthrough the function g , we get back x . The derivatives of inversefunctions are related to each other.

If y = f (x) and x = g(y) are inverse functions, we have,

g ′(y) =1dydx

=1

f ′(x)

In the above case, f ′(x) = 2x . Therefore,

g ′(y) =1

f ′(x)=

1

2x=

1

2√

y


Exponential and Logarithmic Functions I

Consider the following function:

y = bx

y is the dependent variable, x is independent, and b is the (fixed)base of the exponent.

We can change the base. A common base is e = 2.71828.... This iscalled the natural base.

Why do we care about this? Because ex is it’s own derivative it iseasy to work with.


Exponential and Logarithmic Functions II

Now consider the relationship between 8 and 64. Clearly, 82 = 64.

The exponent 2 is the logarithm of 64 to the base 8. That is,log8 (64) = 2.

Likewise, if we have y = bx then x = logb (y).

We also note that if y = ex then x = loge (y) ≡ ln (y).

Again, there is a useful result, as d ln (x) /dx = 1/x .


Rules for Exponential and Logarithmic Functions

Logarithmic and exponential functions are very common in economics.

The basic rules for differentiating exponential and logarithmicfunctions are the following.

1 f (x) = ln x ⇒ f ′(x) = 1/x

2 f (x) = ln u(x)⇒ f ′(x) = u′(x)/u(x)3 f (x) = ex ⇒ f ′(x) = ex

4 f (x) = eu(x) ⇒ f ′(x) = eu(x)u′(x)


Examples of Functions

Some straightforward examples of exponential and logarithmicfunctions are:

f (x) = ln (xα)⇒ f ′(x) = αx (1/xα) = α/x

f (x) = ln(x2 + 3x + 1)⇒ f ′(x) = (2x + 3)/(x2 + 3x + 1)f (x) = e5x ⇒ f ′(x) = 5e5x

f (x) = 5ex2 ⇒ f ′(x) = 10xex2

You should be comfortable with all of these.


Logarithmic Differentiation

Suppose we want to differentiate y =(x2)x2

. We proceed as follows:

1 Take logs on both sides:

ln y = x2 ln x2 = 2x2 ln x

2 Differentiate both sides with respect to x . On the left-hand side, bythe chain rule, we get (1/y)(dy/dx). On the right-hand side, bythe product rule, we have 4x ln x + 2x . After re-arranging, we obtain

dy

dx= y(4x ln x + 2x) = (x2)x2

[4x ln x + 2x ]


Economic Example: Elasticity of Demand

Suppose we have a demand function q = D(p). We feel happyassuming D ′(p) < 0; that is, the demand curve slopes downwards.However, how does quantity change as price changes along the curve?

The elasticity of demand is defined as,

εpq ≡dqq

dpp

=p

q

dq

dp

It is interesting to note that we can represent the elasticity of demandin a more concise manner as

εpq =d(ln[q])d(ln[p])

These expressions are equivalent.


Economic Example: Elasticity of Demand

To see this, note that by using the chain rule, we have

d ln[q]d ln[p]

=d ln[q]

q× dq

d ln[p]=

d ln[q]dq

× dq

dp× dp

d ln[p]

Since d ln[q]/dq = 1/q and dp/d ln[p] = p (by the inverse rule), theresult follows.

The alternative formula for the elasticity can be useful in somecircumstances. Suppose we have the demand function q = pa wherea < 0 is a constant. (Why must we have a < 0?)

We can compute the elasticity very easily as follows. Since q = pa,we have ln q = a ln p. Thus d ln[q]/d ln[p] = a showing that theelasticity of demand is a constant everywhere.


Higher Order Derivatives

We call f ′(x) the first derivative of the function f .

Since the derivative itself is a function, we can take it’s derivative –this is called the second derivative and denoted d2y

dx2 or f ′′(x).

Formally,

d2y

dx2=

d( dydx )

dx

Suppose f (x) = x5.

Then, f ′(x) = 5x4 and f ′′(x) = 20x3.


Higher Order Derivatives

Since the second derivative is also a function, we can also take itsderivative.

This is called the third derivative and denoted f ′′′(x) to indicate thatthis function is found by three successive operations of differentiation,starting with the function f .

One can continue this process but we will typically not go beyond thesecond derivative.

Example continued: Suppose f (x) = x5.

Then, f ′(x) = 5x4, f ′′(x) = 20x3, f ′′′(x) = 60x2.


Concave and Convex Functions

A function f (x) is called concave if f ′′(x) ≤ 0 at all points of itsdomain.

An economic example of a concave function is the productionfunction (usually).

A function f (x) is called convex if f ′′(x) ≥ 0 at all points of itsdomain.

If the inequalities are strict, then the function is called strictlyconcave or strictly convex.

Note: Concavity and convexity do not say anything about the firstderivative (increasing or decreasing functions).


Examples of Concavity and Convexity

The function f (x) = x2 is convex on the domain x ≥ 0 becausef ′′(x) = 2 > 0 for all x ≥ 0.

The function g(x) = ln x is concave on the domain x > 0 becauseg ′′(x) = −1/x2 < 0 for all x > 0.

A function may be either concave nor convex on a given domain.

For example, consider f (x) = −(2/3)x3 + 10x2 + 5x on the domainx ≥ 0.

In this case, f ′′(x) = −4x + 20; so,

1 f ′′(x) ≥ 0 if x ≤ 5 (convex)2 f ′′(x) < 0 if x > 5 (concave)


Economic Examples of Concavity and Convexity

Consider the following production function: y = lα. We know,y ′ = αlα−1 > 0. But y ′′ = (α− 1) αlα−2. This is only positive whenα < 1. That is consistent with diminishing marginal returns to labor.

Consider the revenue and costs functions from before. Now we mightsuppose the following.

R ′ (y) > 0; R ′′ (y) < 0; c ′(y) > 0; c(y)′′ > 0

As we produce more, we make more revenue, but as production risesever higher, the additional unit generate less revenue than the last.

However, as we produce more, costs rise, and as we continue toproduce, each extra unit is more costly that the last.


Introduction to Unconstrained Optimization

We can also relate second-order conditions to necessary and sufficientconditions for maximization problems.

That is, to find the maximum, it is not enough to “set the firstderivative to zero”. We need to check second order conditionsexplicitly.

Two familiar examples:

1 Suppose that a monopolist faces a demand curve p = 100− q andhas production costs given by c(q) = q2. If the monopolist isinterested in maximizing profits, how much should he produce?

2 Suppose that a firm has fixed cost of 100 and variable costs given byv(q) = q2. If the firm wants to minimize average costs of production,how much should it produce?


Necessary Conditions for Optimization

If a differentiable function f (x) reaches its maximum or minimum atx∗ then f ′(x∗) = 0. Reason: Consider the total differential

dy = f ′(x)dx

If the function reaches a maximum or minimum at x∗ then it must beimpossible to increase or decrease the value of the function by smallchanges in x . However, if f ′(x∗) 6= 0, then it is always possible tomake dy larger or smaller by making (small) appropriate changes in x .Therefore, we must have f ′(x∗) = 0 at a maximum or minimum.

This is called a necessary condition because it cannot guarantee thatx is indeed a maximum or minimum. It is entirely possible thatf ′(x∗) = 0 but x∗ is neither a maximum nor a minimum.


Global and Local Maxima

A point x∗ is called a global maximum of the function f (x) iff (x∗) ≥ f (x) for all x in the domain of f .

A point x∗ is called a local maximum of the function f (x) if there is a“small interval” centered at x∗ such that f (x∗) ≥ f (x) for all x inthe small interval.

Note the necessary condition only identifies a local maximum orminimum, but not a global maximum or minimum.


Second Order or Sufficient Conditions for Optimization

We also have the following conditions:

1 If f ′(x∗) = 0 and f ′′(x∗) < 0, then x∗ is a local maximumof f (x).

2 If f ′(x∗) = 0 and f ′(x∗) > 0, then x∗ is a local minimum of f (x).

Note that it is possible that a point may be a maximum or minimumand yet not satisfy the sufficient condition for maximization orminimization.


Economic Example

We now return to the economic examples above.

Suppose the monopolist’s profit function is given by,

Π(y) = py − c(y) = (100− y)y − y2

The necessary condition implies that Π′(y) = 100− 2y − 2y = 0 ory = 25.

The firm’s average cost is given by AC (y) = 100/y + y . Thenecessary condition AC ′(y) = 0 gives −100/y2 + 1 = 0 ory2 − 100 = 0. Therefore, y = 10 or y = −10. Since q = −10 is notpossible, we must have y = 10.

To check that these are indeed the maximum and minimum, we caneither plot the two functions, or check the second order conditions foroptimization.


Economic Example Continued ...

In the monopolist’s problem, note that the second derivativeΠ′′(y) = −4 for all y , (in particular, for y = 25) and hence y = 25 isa local maximum. In the firm’s problem, AC ′′(y) = 200/q3;substituting y = 10 gives AC ′′(10) = 200/1000 = 1/5 > 0. Hence,y = 10 is a local minimum.

We do not know whether these are local or global minima. Theimportance of concave and convex functions stem from the fact thatthe local maxima of concave functions are also global maxima; thelocal minima of convex functions are also global minima.

In the monopolist’s problem, the profit function is concave; hencey = 25 is a global maximum. In the firm’s problem, the AC functionis convex on the domain qy > 0 and hence it is a global minimum.


Another Example: Supply, Demand, and Taxes

Suppose we have a market with demand function qd = a0 − a1p andsupply function qs = b0 + b1p.

The government wants to impose an excise tax t on this market.What rate should the government choose if it wants to maximize taxrevenue?

Without taxation, market equilibrium is determined by the conditionqd = qs . Thus, a0 − a1p = b0 + b1p, which gives,

p∗ =a0 − b0

a1 + b1and q∗ =

a0b1 + a1b0

a1 + b1

When a tax t is imposed, the equilibrium conditions require that (i)qd = qs and (ii) pb = ps + t.


Supply and Demand Continued ...

Equilibrium now requires a0 − a1pb = b0 + b1(pb − t).

This gives

p∗b =a0 − b0 + b1t

a1 + b1, p∗s =

a0 − b0 − a1t

a1 + b1, q∗t =

a0b1 + a1b0 − a1b1t

a1 + b1

The tax revenue as a function of the tax rate t is:

T (t) = tq∗t = t

[a0b1 + a1b0 − a1b1t

a1 + b1

]We want to find t to maximize T (t). First, we identify t such thatdT /dt = 0.


Supply and Demand Continued ...

Differentiating T (t) and equating to zero gives

dT

dt=[

a0b1 + a1b0 − 2a1b1t

a1 + b1

]= 0

This gives us a unique value of t:

t∗ =a0b1 + a1b0

2a1b1

We now have to check whether t∗ corresponds to a maximum or not.We have

T ′′(t) = − 2a1b1

a1 + b1

Thus, the second derivative is a constant, and since a1 > 0, b1 > 0 itis always negative. This shows that the tax revenue function T (t) isconcave, and hence t∗ is not only a local but also a global maximum.


Remark on the Necessary Condition for Optimization

The condition f ′(x) = 0 at a maximum or minimum is valid only if xis in the “interior” of the domain of the function. Basically, if we lookat the argument given for showing that if x∗ is a maximum orminimum, then f ′(x) = 0, it should be noticed that the argumentdepended on the ability to make small changes in x .

However, at a “boundary point” we cannot make certain changes. Forinstance, if the function is defined for all x in the interval [a, b], thenat a, we can only increase x ; similarly at b, we can only decrease x .Thus, the argument that f ′(x) = 0 is not true if the maximum occursat a or b.

We can modify the necessary condition f ′(x) = 0 to account forthese “boundary points” but we will deal with such situations whenwe discuss constrained optimization.


Roundup

You should now be able to do the following:

1 Differentiate functions (inc. log and exp.).

2 Check for concavity/convexity.

3 Apply these ideas to microeconomic problems.

4 Understand the importance of necessary versus sufficient conditions.


Single Variable Calculus and ApplicationsSingle Variable Calculus A function is usefully thought of...

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