2.1 Introduction to multivariate calculus

2.1 Introduction to multivariate calculus

Economic theory consists of models designed to improve our understanding of economic phenomena. Many of these models have the following structure: each member of a set of economic agents optimizesgiven some constraints and, given the optimal actions, variables adjust to reach some sort of equilibrium.

Consider, for example, a model in which the agents are profit-maximizing firms. Suppose that there is a single input that costs w per unit, and that a firm transforms input into output using a (differentiable) production function  f  and sells the output for the price p. A firm's profit when it uses the amount x of the input is then

p f (x) − wx.

As you know, if the optimal amount of the input is positive then it satisfies the "first-order condition"

p f '(x) − w = 0.

Further, under some conditions on  f  this equation has a single solution and this solution maximizes the firm's profit. Suppose these conditions are satisfied. For any pair (w, p), denote the solution of the equation by z(w, p). Then the condition

p f '(z(w, p)) − w = 0 for all (w, p)

defines z(w, p) implicitly as a function of w and p.

What can we say about the function z? Is it increasing or decreasing in w and p? How does the firm's maximized profit change as w and p change?

If we knew the exact form of  f  we could answer these questions by solving for z(w, p) explicitly and using simple calculus. But in economic theory we don't generally want to assume that functions take specific forms. We want our theory to apply to a broad range of situations, and thus want to obtain results that do not depend on a specific functional form. We might assume that the function  f  has some "sensible" properties---for example, that it is increasing---but we would like to impose as few conditions as possible. In these circumstances, in order to answer the questions about the dependence of the firm's behavior on w and p we need to find the derivatives of the implicitly-defined function z. Before we do so, we need to study the chain rule and derivatives of functions defined implicitly, the next two topics. (If we are interested only in the rate of change of the firm's maximal profit with respect to w and p, not in the behavior of its optimal input choice, then the envelope theorem, studied in a later section, is useful.)

Having studied the behavior of a single firm we may wish to build a model of an economy containing many firms and consumers that determines the prices of goods. In a "competitive" model, for example, the prices are determined by the equality of demand and supply for each good---that is, by a system of equations. In many other models, an equilibrium is the solution of a system of equations. To study the properties of such an equilibrium, another mathematical technique is useful.

I illustrate this technique with an example from macroeconomic theory. A simple macroeconomic model consists of the four equations

Y	=	C + I + G
C	=	f (Y − T)
I	=	h(r)
r	=	m(M)

where Y is national income, C is consumption, I is investment, T is total taxes, G is government spending, r is the rate of interest, and M is the money supply. We assume that M, T, and G are "parameters" determined outside the system (by a government, perhaps) and that the equilibrium values of Y, C, I and r satisfy the four equations, given M, T, and G.

We would like to impose as few conditions as possible on the functions  f , h, and m. We certainly don't want to assume specific functional forms, and thus cannot solve for an equilibrium explicitly. In these circumstances, how can we study how the equilibrium is affected by changes in the parameters? We may use the tool of differentials, another topic in this section.

1.2 Matrices and solutions of systems of simultaneous equations

1.2 Matrices and solutions of systems of simultaneous equations

Matrices

I assume that you are familiar with vectors and matrices and know, in particular, how to multiply them together. (Do the first few exercises to check your knowledge.)

The determinant of the 2 × 2 matrix

a   b   c   d

is ad − bc. If ad − bc ≠ 0, the matrix is nonsingular; in this case its inverse is

1 ad − bc

d   −b   −c   a

.

(You can check that the product of the matrix and its inverse is the identity matrix.) We denote the inverse of the matrix A by A⁻¹.

The determinant of the 3 × 3 matrix

a   b   c   d   e    f    g   h   i

is

Δ = a(ei − h f ) − b(di − g f ) + c(dh − eg).

If Δ ≠ 0 the matrix is nonsingular; in this case its inverse is

1 Δ	D11   −D12   D13   −D21   D22   −D23   D31   −D32   D33

where D_ij is the determinant of the 2 × 2 matrix obtained by deleting the ith column and jth row of the original 3 × 3 matrix. That is, D₁₁ = ei − h f , D₁₂ = bi − ch, D₁₃ = b f  − ec, D₂₁ = di −  f g, D₂₂ = ai −cg, D₂₃ = a f  − dc, D₃₁ = dh − eg, D₃₂ = ah − bg, and D₃₃ = ae − db. (Again, you can check that the product of the matrix and its inverse is the identity matrix.)

The determinant of the matrix A is denoted |A|.

Let A be an n × m matrix (i.e. a matrix with n rows and m columns). The transpose A' of A is the m × n matrix in which, for i = 1, ..., n, the ith row of A' is the ith column of A. In particular, if x is a column vector (n × 1 matrix) then x' is a row vector (1 × n matrix).

Solutions of systems of simultaneous equations

Consider a system of two equations in two variables x and y:

ax + by	= u
cx + dy	= v.

Here are three ways to solve for x and y.

1. Isolate one of the variables in one of the equations and substitute the result into the other equation. For example, from the second equation we have

   y = (v − cx)/d.

   Substituting this expression for y into the first equation yields

   ax + b(v − cx)/d = u,

   which we can write as

   (a − bc/d)x + bv/d = u,

   so that

   x =	u − bv/d a − bc/d

   or

   x =

   ud − bv ad − bc

   .

   To find y we now use the fact that y = (v − cx)/d, to get

   y =

   va − cu ad − bc

   .

2. Use Cramer's rule (due to Gabriel Cramer, 1704-1752). Write the two equations in matrix form as

   a    b   c    d

   x   y

   =

   u   v

   (*)

   Cramer's rule says that the solutions are given by

   x =	u   b   v   d     ad − bc

   and

   y =

   a   u   c   v     ad − bc

   .

   Note that the denominator in these expressions, ad − bc, is the determinant of the matrix at the left of (*). The matrix in the numerator of each expression is obtained by replacing the column in the matrix on the left of (*) that corresponds to the variable for which we are solving with the column vector on the right of (*).Calculating the determinants in the numerators, we have
   

   x =

   ud − bv ad − bc

   and

   y =

   va − cu ad − bc

   .

3. Write the two equations in matrix form as

   a    b   c    d

   x   y

   =

   u   v

   (as when using Cramer's rule) and solve by inverting the matrix on the left hand side. The inverse of this matrix is

   1 ad − bc

   d    −b   −c    a

   so we have

   x   y

   =

   1 ad − bc

   d    −b    −c    a

   u   v

   so that

   x =

   ud − bv ad − bc

   and

   y =

   va − cu ad − bc

   .

Which of these three methods is best depends on several factors, including your ability to remember Cramer's rule and/or how to invert a matrix. If you have to solve for only one of the variables, Cramer's rule is particularly convenient (if you can remember it).

For a system of more than two variables, the same three methods are available. The first method, however, is pretty messy, and unless you are adept at inverting matrices, Cramer's rule is probably your best bet.

1.1 Logic

1.1 Logic

Basics

When making precise arguments, we often need to make conditional statements, like

if the price of output increases then a competitive firm increases its output

or

if the demand for a good is a decreasing function of the price of the good and the supply of the good is an increasing function of the price then an increase in supply at every price decreases the equilibrium price.

These statements are instances of the statement

if A then B,

where A and B stand for statements. We may alternatively write this general statement as

A implies B,

or, using a symbol, as

A ⇒ B.

Yet two more ways in which we may write the same statement are

A is a sufficient condition for B,

and

B is a necessary condition for A.

(Note that B comes first in the second of these two statements!!)

Important note: The statement A ⇒ B does not make any claim about whether B is true if A is NOT true! It says only that if A is true, then B is true. While this point may seem obvious, it is sometimes a source of error, partly because we do not always adhere to the rules of logic in everyday communication. For example, when we say "if it's fine tomorrow then let's play tennis" we probably mean both "if it's fine tomorrow then let's play tennis" and "if it's not fine tomorrow then let's not play tennis" (and maybe also "if it's not clear whether the weather is good enough to play tennis tomorrow then I'll call you"). When we say "if you listen to the radio at 8 o'clock then you'll know the weather forecast", on the other hand, we do not mean also "if you don't listen to the radio at 8 o'clock then you won't know the weather forecast", because you might listen to the radio at 9 o'clock or check on the web, for example. The point is that the rules we use to attach meaning to statements in everyday language are subtle, while the rules we use in logical arguments are absolutely clear: when we make the logical statement "if A then B", that's exactly what we mean---no more, no less.

We may also use the symbol "⇐" to mean "only if" or "is implied by". Thus

B ⇐ A

is equivalent to

A ⇒ B.

Finally, the symbol "⇔" means "implies and is implied by", or "if and only if". Thus

A ⇔ B

is equivalent to

A ⇒ B and A ⇐ B.

If A is a statement, we write the claim that A is not true as

not(A).

Note that

not(not(A)) ⇔ A.

If A and B are statements, and both are true, we write

A and B,

and if at least one of them is true we write

A or B.

Note, in particular, that writing "A or B" includes the possibility that both statements are true.

Two rules

Rule 1

If the statement

A ⇒ B

is true, then so too is the statement

(not B) ⇒ (not A).

The first statement says that whenever A is true, B is true. Thus if B is false, A must be false---hence the second statement.

Rule 2

The statement

not(A and B)

is equivalent to the statement

(not A) or (not B).

Note the "or" in the second statement! If it is not the case that both A is true and B is true (the first statement), then either A is not true or B is not true.

Quantifiers

We sometimes wish to make a statement that is true for all values of a variable. For example, denoting the total demand for tomatoes at the price p by D(p), it might be true that

D(p) > 100 for every price p in the set S.

In this statement, "for every price" is a quantifier.

Important note: We may use any symbol for the price in this statement: "p" is a dummy variable. After having defined D(p) to be the total demand for tomatoes at the price p, for example, we could write

D(z) > 100 for every price z in the set S.

Given that we just used the notation p for a price, switching to z in this statement is a little odd, BUT there is absolutely nothing wrong with doing so! In this simple example, there is no reason to switch notation, but in some more complicated cases a switch is unavoidable (because of a clash with other notation) or convenient. The point is that every statement of the form

A(x) for every x in the set Y,

where x is any symbol, has exactly the same content.

Another type of statement we sometimes need to make is

A(x) for some x in the set Y,

or, equivalently,

there exists x in the set Y such that A(x).

"For some x" (alternatively "there exists x") is another quantifier, like "for every x"; my comments about notation apply to it.

Intervals and functions

1.3 Intervals and functions

Intervals

An interval is a set of (real) numbers between, and possibly including, two numbers. The interval from a to b is denoted as follows:

• [a, b] if a and b are included (i.e. [a, b] = {x: a ≤ x ≤ b})
• (a, b) if neither a nor b is included (i.e. (a, b) = {x: a < x < b})
• [a, b) if a is included but b is not
• (a, b] if b is included but a is not.

We use the special symbol "∞" ("infinity") in the notation for intervals that extend indefinitely in one or both directions, as illustrated in the following examples.

• (a, ∞) is the interval {x: a < x}.
• (−∞, a] is the interval {x: x ≤ a}.
• (−∞, ∞) is the set of all numbers.

Note that ∞ is not a number, but simply a symbol we use in the notation for intervals that have at most one endpoint. We alternatively denote the set (−∞, ∞) of all (real) numbers by the symbol ℝ.

The notation (a, b) is used also for an ordered pair of numbers. The fact that the notation has two meanings is unfortunate, but the intended meaning is usually clear from the context. (If it is not, complain to the author.)

The interior of an interval is the set of all numbers in the interval except the endpoints. Thus the interior of [a, b] is (a, b); the interval (a, b) is the interior also of the intervals (a, b], [a, b), and (a, b).

We say that (a, b), which does not contain its endpoints, is an open interval, whereas [a, b], which does contain its endpoints, is a closed interval. The intervals (a, b] and [a, b) are neither open nor closed.

Functions

A function is a rule that associates with every member of some set, a single member of another set. The first set is called the domain of the function. A function with domain A is said to be defined on A.

To specify a function we need to specify the domain and the rule. Here are some examples.

• Domain: [−2, 1]. Rule:  f (x) = x².
• Domain: (−∞, ∞). Rule:  f (x) = x².
• Domain: (−1, 1). Rule:  f (x) = x if x ≥ 0;  f (x) = 1/x if x < 0.
• Domain: union of (0, 1) and (4, 6). Rule:  f (x) = √x.
• Domain: set of all pairs of numbers. Rule:  f (x, y) = xy.
• Domain: set of n-vectors (x₁, ..., x_n) for which 0 ≤ x_i ≤ 1 for i = 1, ..., n. Rule:  f (x₁, ..., x_n) = ∑_i=1ⁿa_ix_i, where a₁, ..., a_n are nonnegative constants.

(Note that the symbols x and y are arbitrary. We could, for example, define the first function equally well by  f (z) = z² or  f (g) = g². We generally use x and y for variables, but we may use any other symbols.)

These examples have two features in common:

• the domain is a subset of the set of n-vectors of numbers, for some positive integer n (where n may of course be 1, as in the first four examples)
• the rule associates a real number with each point in the domain.

A function is not restricted to have these features. For example, the domain of a function may be a set of complex numbers and the function may associate a set with each member of its domain. All the functions in this tutorial, however, have the two features, and the word "function" throughout means a rule that associates a number with every point in some subset of the set of n-vectors of numbers (where n may be 1). I refer to a function whose domain is a set of 1-vectors (i.e. numbers) as a function of a single variable, and one whose domain is a set of n-vectors for n ≥ 1 as a function of many variables. (Note that a function of a single variable is a special case of a function of many variables.)

The number that a function associates with a given member x of its domain is called the value of the function at x. As x varies over all points in the domain of a function, the value of the function may (and generally does) vary. The set of all such values of the function is called the range of the function. Here are the ranges of the examples given above.

• Domain: [−2, 1]. Rule:  f (x) = x². Range: [0, 4].
• Domain: (−∞, ∞). Rule:  f (x) = x². Range: [0, ∞).
• Domain: (−1, 1). Rule:  f (x) = x if x ≥ 0;  f (x) = 1/x if x < 0. Range: union of (−∞, −1) and [0, 1).
• Domain: union of (0, 1) and (4, 6). Rule:  f (x) = √x. Range: union of (0, 1) and (2, √6).
• Domain: set of all pairs of numbers. Rule:  f (x, y) = xy. Range: (−∞, ∞).
• Domain: set of n-vectors (x₁, ..., x_n) for which 0 ≤ x_i ≤ 1 for i = 1, ..., n. Rule:  f (x₁, ..., x_n) = ∑_i=1ⁿa_ix_i, where a₁, ..., a_n are nonnegative constants. Range: (0, ∑_i=1ⁿa_i).

In formal presentations of mathematical material, the notation  f : A → B is used for a function given by the rule  f  and the domain A whose range is a subset of B. We might say, for example, "consider the function  f : [0, 1] → ℝ defined by  f (x) = √x", or "for every function  f : ℝ → ℝ." (Remember that the symbol ℝ denotes the set of all (real) numbers.) The set B in this notation is called the co-domain ortarget of the function. Note that this set is not part of the definition of the function, and may be larger than the range of the function. When we say, for example, "for every function  f : ℝ → ℝ", we mean every function whose domain is ℝ and whose range is a subset of ℝ.

Graphical illustrations aid the understanding of many functions. A function of a single variable, for example, may be represented on x−y coordinates by plotting, for each value of x, the value of  f (x) on the y-axis. An example is shown in the following figure. In this diagram, the small circle indicates a point excluded from the graph: the value of the function at x₀ is y₀, whereas the value of the function at points slightly greater than x₀ is y₁.

The red line in this figure is called the graph of the function. Techniques for drawing graphs are discussed in a later section.

Logarithms and exponentials

You need to be comfortable working with the logarithm function and with functions of the form x^y (where y is known as an exponent). In particular, you should know the following rules.

• x^yx^z = x^y+z
• (x^y)^z = x^yz (so that in particular (x^y)^1/y = x)
• ln e^x = x and e^ln x = x
• a ln x = ln x^a (so that e^a ln x = x^a)

Continuous functions

A function of a single variable is continuous if its graph has no "jumps", so that it can be drawn without lifting pen from paper. In more precise terms, a function  f  is continuous at the point a if we can ensure that the value  f (x) of the function is as close as we wish to  f (a) by choosing x close enough to a. Here is a completely precise definition for a function of many variables. (The distance between two points (x₁, ..., x_n) and (y₁, ..., y_n) is the Euclidean distance √[∑_i=1ⁿ(x_i − y_i)²].)

Definition Let  f  be a function of many variables and let a be a point in its domain. Then  f  is continuous at a if, for any number ε > 0, there is a number δ > 0 such that for any value of x in the domain of  f  for which the distance between x and a is less than δ, the difference between  f (x) and  f (a) is less than ε. The function  f  is continuous if it is continuous at every point in its domain.

The function whose graph is shown in the previous figure is not continuous at x₀. The value of the function at x₀ is y₀, but the values at points slightly larger than x₀ is much larger than y₀: no matter how small we choose δ, some points x within the distance δ of x₀ yield values of the function far from y₀ (=  f (x₀)).

The following result is useful in determining whether a function is continuous.

Proposition If the functions  f  and g of many variables are continuous at x0 then the function h defined by h(x) =  f (x) + g(x) for all x is continuous at x0. If the functions  f  and g of many variables are continuous at x0 then the function h defined by h(x) =  f (x)g(x) for all x is continuous at x0. If the functions  f  and g of many variables are continuous at x0 and g(x0) ≠ 0 then the function h defined by h(x) =  f (x)/g(x) for all x with g(x) ≠ 0 is continuous at x0. If the function  f  of many variables is continuous at x0 and the function g of a single variable is continuous at  f (x0), then the function h defined by h(x) = g( f (x)) for all x is continuous at x0.

An implication of the second part of this result is that if  f  is continuous at x₀ then the function h defined by h(x) = ( f (x))^k, where k is a positive integer, is continuous at x₀.

A polynomial is a function of a single variable x of the form a₀ + a₁x + a₂x² + ... + a_kx^k, where k is a nonnegative integer and a₀, ..., a_k are any numbers. Because the function  f  defined by  f (x) = x for all xis continuous, all polynomials are continuous.

The next result gives an important property of continuous functions. It says that if the function f of a single variable on the domain [a, b] is continuous, then f(x) takes on every value from f(a) to f(b).

Proposition (Intermediate value theorem) If  f  is a continuous function of a single variable with domain [a,b] and  f (a) ≠  f (b), then for any number y between  f (a) and  f (b), or equal to  f (a) or  f (b), there is a value of x (in [a,b]) such that  f (x) = y.

This result is illustrated in the following figure.

In this figure, the set of values from  f (a) to  f (b) is shown in red; for every value y in this set, there is a value of x such that  f (x) = y. For example,  f (x₁) = y₁. Two points to note:

• For some values y between  f (a) and  f (b) there may be more than one value of x such that y =  f (x). For example, in the figure  f (x₂) =  f (x₃) =  f (x₄) = y₂.
• The result does not say that for values y less than  f (a) or greater than  f (b) there is no x such that  f (x) = y. Indeed, in the figure we have  f (x₅) = y₃.

An important implication of the result is that if  f (a) is positive and  f (b) is negative, then  f (x) = 0 for some x.

Example Consider the function  f  defined by  f (x) = x4 − 4x2 + 2. Does the equation  f (x) = 0 have a solution between 0 and 1? The function is a polynomial, and thus is continuous. We have  f (0) = 2 and  f (1) = −1, so the Intermediate Value Theorem implies that the answer to the question is yes: for some value of x between 0 and 1 we have  f (x) = 0.

Circular Flow of Income Diagram

The Circular flow of income diagram models what happens in a very basic economy.

In the very.basic model, we have  two principal components of the economy:

1. Firms. Companies who pay wages to workers and produce output.
2. Households. Individuals who consume goods and receive wages from firms.

This circular flow of income also shows the three different ways that National Income is calculated.

1. National Output. Output  produced by firms.
2. National Income. The total income received by people in the economy. For example, firms have to pay workers to produce the output. Therefore income flows from firms to households.
3. National Expenditure. Total amount spent on goods and services. For example, with wages from work, households can then buy goods produced by firms. Therefore, the spending goes back to firms.

This represents a simple economic model; it is a closed economy without any government intervention.

In the real world it is more complicated. We also add two more components:

1. Government. The government taxes firms and consumers, and then spends money, e.g. health care and education.
2. Foreign sector. We sell exports abroad and buy imports. Therefore, there is a flow of money between one country and the rest of the world.

Withdrawals (W) into Circular Flow of Income

Withdrawals are items that take money out of the circular flow. This includes:

• Savings (S) (money not used to finance consumption, e.g. saved in a bank)
• Imports (M) (money sent abroad to buy foreign goods)
• Taxes (T) (money collected by government, e.g. income tax and VAT)

Injections (J) into Circular Flow of Income

Spending that puts money into the circular flow of income.

• Investment (I). Money invested by firms into purchasing capital stock.
• Exports (X). Money coming from abroad to buy domestically produced goods.
• Government spending (G). Government welfare benefits, spending on infrastructure.
• Circular flow of income 
• Intrepix: http://intrepix.tripod.com/ Dec 2002 
• Copyright © Enoch Lau 2002 
• ···································································································· 
• Page 1 of 2 
• The Circular Flow of Income 
•  
• The five-sector circular flow model describes the operation of the economy and the linkages 
• between the main sectors in the economy. The five-sector model is based on dividing the 
• economy into five sectors. A sector may be defined as a part of the economy where the 
• participants are engaged in a similar type of economy activity. 
• 1. Individuals 
• 2. Businesses 
• 3. Financial institutions 
• 4. Government 
• 5. International Trade 
• Individuals 
• ƒ This sector consists of all individuals in the economy. 
• ƒ These individuals are the owners of productive resources, and the consumers in our 
• economy. 
• ƒ Individuals supply factors of production (inputs) such as labour and enterprise to 
• businesses, which they use to produce goods and services. As a reward for supply 
• resources such as labour and enterprise to firms, individuals receive incomes – rent, 
• wages, interest and profit. 
•  
• Businesses 
• ƒ This sector consists of all the business firms engaged in the production and 
• distribution of goods and services (apart from financial services). 
• ƒ It concerns all their activities involved with buying factors of production and using 
• them to produce and sell goods and services. 
• ƒ Individuals and businesses are interdependent. 
•  
• Financial Institutions ƒ This sector consists of all those institutions that are engaged in the borrowing and 
• lending of money, acting as the intermediaries between those who save, and 
• borrowers of money. 
• ƒ Financial institutions are needed for individuals and firms to be able to undertake 
• saving and investment. They perform the function of mobilising savings for 
• investment. 
• ƒ Savings: leakage; Investment: injection 
• Individuals Businesses 
• Financial Institutions 
• Government
• International Flows 
• Income rewards 
• Expenditure on goods and services 
• Savings Investment
• Taxation Expenditure 
• Imports ExportsCircular flow of income 
• Intrepix: http://intrepix.tripod.com/ Dec 2002 
• Copyright © Enoch Lau 2002 
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• Government 
• ƒ In Australia, this sector consists of the Commonwealth, State and local governments. 
• ƒ It is involved in the satisfaction of collection (community) wants. 
• ƒ It obtains the resources to do this through imposing taxes on the other sectors of the 
• economy. 
• ƒ It uses this tax revenue to undertake various government expenditures.