1 - 31 ECON 2310 Intermediate Microeconomics Algebra Rev iew This course does not require any calculus or “high level” mathematical techniques. However, a comprehensive understanding of basic, elementary algebra and complementary graphs is required. Although most of the mathematics used in this course will have been covered in highschool (grades 9, 10 and 11), it is important that you take this review and preparati on of algebra seriously. Many students never fully understand the models used in economics simply because they lack a full understanding of the simple mathematics underlying them. If you haven’t recently taken a course with a substantial amount of algebra in it, you may have forgotten some of the basics. So check outthis algebra review. Do the Pre-Testto determine your current level of understanding of relevant algebra and graphs. If it isn’t complete ly solid s pend some time on the learning modules. Then re-do the Pre-Test. If y ou are still having some difficulty with the material you should get a textbook on elementary algebra or perhaps an appropriate Schaum’s Outlines on Elementary Algebra and spend some time becoming comfortable with the required mathematics. A review of relevant arithmetic and algebra is also provided in the book Mathematics for Eco nomics Student’s Solutions Manual, by M. Hoy, J. Livernois, C. McKenna, R. Rees, and T. Stengos, Addison Wesley Publishers, 1996, Chapter 1 Algebra and Arithmetic Reviews and Self-tests(pages 1 to 26).
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This course does not require any calculus or “high level” mathematical techniques.
However, a comprehensive understanding of basic, elementary algebra and
complementary graphs is required. Although most of the mathematics used in this coursewill have been covered in highschool (grades 9, 10 and 11), it is important that you take
this review and preparation of algebra seriously. Many students never fully understand
the models used in economics simply because they lack a full understanding of the
simple mathematics underlying them. If you haven’t recently taken a course with a
substantial amount of algebra in it, you may have forgotten some of the basics.
So check out this algebra review. Do the Pre-Test to determine your current
level of understanding of relevant algebra and graphs. If it isn’t completely solid spend
some time on the learning modules. Then re-do the Pre-Test. If you are still having
some difficulty with the material you should get a textbook on elementary algebra or
perhaps an appropriate Schaum’s Outlines on Elementary Algebra and spend some timebecoming comfortable with the required mathematics. A review of relevant arithmetic
and algebra is also provided in the book Mathematics for Economics Student’s
Solutions Manual , by M. Hoy, J. Livernois, C. McKenna, R. Rees, and T. Stengos,
Addison Wesley Publishers, 1996, Chapter 1 Algebra and Arithmetic Reviews and
Self-tests (pages 1 to 26).
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Note that these learning modules are not comprehensive reviews of the
mathematical topics addressed but rather are intended to be short reviews to help those
who have forgotten some of the basics that are essential to this course. When possible,
we use applications from economics that you have already learned in introductory
economics or ideas for which no economic’s background is required. If this reviewmaterial proves not to be sufficient for you, then you will need to take a more aggressive
approach to reviewing the mathematical background required for this course.
Module 1: Basic Arithmetic Operations, Including Powers
In this, and following modules, letters such as a, b, c, d, x, y, z are used to
represent real number values. Rules for manipulating expressions are generally given in
terms of “unspecified” numbers but we will usually provide specific examples with
specific numbers as well. These rules are used to manipulate and simplify or solve for
expressions.
Commutative Law of Addition: a+b = b+a
This law simply indicates that adding two numbers together gives the same answer
regardless of the order in which they are added. So, for example, 5+2=7 and also 2+5=7.
This law can be applied iteratively and so the addition of any number of terms is also
independent of the order of addition. Thus, a+b+c = a+c+b = c+a+b = c+b+a, etc. So,
for example, 8+6+(-4) = 10 as is 6+(-4)+8 = 10.
Commutative Law of Multiplication: ab = ba
This law simply indicates that multiplying two numbers together gives the same answer
regardless of the order in which they are multiplied. So, for example, 3×6=18 and also
6×3=18. As with the case for addition, this law can be applied iteratively and so the
multiplication of any number of terms is also independent of the order of multiplication.
Thus, a×b×c = a×c×b = c×a×b = c×b×a, etc. So, for example, 3×7×2 = 42 as is 2×3×7 =
42.
Associative Law: (a+b)(c+d)=ac+ad+bc+bd
This law can be illustrated in two steps:
Step 1 (a+b)(c+d) = a(c+d) + b(c+d)
Step 2 a(c+d) + b(c+d) = ac+ad+bc+bd
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If there are more than two terms to be added in one of the brackets then the associative
law continues to apply, just iteratively. Thus, (a+b)(c+d+e) = (a+b)((c+d)+e) =
(a+b)(c+d)+(a+b)e = ac+ad+bc+bd+ae+be = ac+ad+ae+bc+bd+be. In words, to expand
this expression multiply the first element in the left hand bracket (a) by each of the
elements in the right hand bracket (c,d,e), adding all the results together (ac+ad+ae), andthen do the same with the second element in the left hand bracket (b) to get (bc+bd+be),
and then add all terms together.
You can check the associative law with a specific numerical example, such as
There are several useful rules for simplifying expressions involving powers
(exponents) and roots. Although you would profit from a more thorough review of
powers (see Mathematics for Economics Student’s Solutions Manual , by Hoy, et al.,pages 4 to 6) the following condensed review contains most of the operations you will
require.
The square of a number is obtained by multiplying the number by itself (e.g., the
square of 5 is just 5×5 = 25). The square root is the inverse operation - that is, finding
the square root of a number, z (often written z), is finding that number which, when
multiplied by itself, gives the result z (e.g., the square root of 9 is 3, written 9 = 3).
Since finding the square root of a number, z, is the inverse operation of finding the
square of a number, we also write
z as z1/2. The rationale for this notation is that ½ is
the inverse of the number 2. If we multiply a number by itself 3 times we indicate this as
z×z×z = z
3
and refer to the result as the cube of z. So, for example, the cube of 4 is 64and we can write 43 = 4×4×4 = 64. The number, which when multiplied by itself three
times gives the value of some original number, say y, is called the cube root of y. So, for
example, the cube root of 64 is 4. Since finding the cube root is the inverse operation of
finding the cube of a number we write the cube root of a number y as y1/3. Thus, we say
641/3 = 4.
The above operations of multiplying a number by itself two or three times can be
extended to any number of times, n, and so we use the notation zn to represent the result
of multiplying the number z by itself n times and y1/n represents the result of finding the
number which, when multiplied by itself n times will give the number y. That is:
z×z×z×...z×z = zn
and y1/n
×y1/n
×y1/n
×...×y1/n
×y1/n
= y , where in each case there are “nmultiplications.
The following rules of powers follow directly from these definitions:
Power Rule for Inversions: If ya = x, then x = y1/a (e.g., if y2 = x, then x =
y1/2)
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This result is not generally noted explicitly as a rule for powers since it is really just a
restatement of the inverse operation of taking powers (e.g., the relationship between
finding the square of a number or the inverse operation of finding the square root of a
number). However, it is so useful it is included here explicitly.
Power Rule for Multiplication: (ya)(yb) = ya+b
For example, (25 )(22) = 25+2 = 27 since (2×2×2×2×2)×(2×2) gives the result of 2 being
multiplied by itself 7 times.
Power Rule for Division: ya /yb = ya-b
For example, 25 /22 = 25-2 = 23 since (2×2×2×2×2)/(2×2) = 2×2×2.
The power rule for division applies even if a-b is a negative number with theconvention that y-n represents 1/yn. These rules also apply for non-integer values of the
exponents. Another useful convention is y0 = 1 for any number y. This makes sense
once you note that using the power rule for division implies ya /ya = ya-a = y0 = 1 for any
numbers y and a, y not zero.
An example using the power rules.
In Macroeconomics we study the topic of economic growth. A typical problem that the
student encounters is to determine the steady state per capita capital stock of an economy
given an exogenous savings rate, capital depreciation rate, and a production function.
The fundamental relationship used to find the steady state capital stock is given by
equating savings to depreciation or , where s is the savings rate, f(k) is the per capitals f (k )
k
production function, k the per capita capital stock, and is the capital depreciation rate. If
savings (addition to the capital stock as savings equals investment) equals depreciation (the
reduction in the capital stock) the net gain in capital is zero and we say the economy is in
a steady state (capital is unchanging).
Suppose the per capita production function is given by . When savings equalsak 1/2
depreciation the steady state capital stock can be found by solving for k in the equation
Using the power rule for division this equation can be rewritten as
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common denominator 4, with and so we have = = = . Once one1
2
2
4
1
2
1
4
2
4
1
4
(2
1)
4
3
4
takes into consideration more complicated expressions appearing in the denominator one
needs to be a bit more methodical about this. For example, consider the following
simplification:
Notice that to compute expressions such as the above first find the common
denominator by multiplying together the denominators of each fraction in the series
(2×7×8) and then for each fraction we multiply the numerator by the values of the
denominators of the other fractions. The rule is easiest to see when using fractions which
are ratios of variable names, such as the following:
So, you can use this simple rule to obtain simplifications of expressions such as
To see how to obtain this result note that the common denominator for this expression is(x+y)(x-y) and so this expression can be expanded and then simplified as shown below.
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In economics we often use linear functions in examples and illustrations because
of their simplicity. Although actual relationships between economic variables often are
not linear, the use of linear functions as approximations is often useful and reasonable.It is important to realize that when using a general functional notation, such as
y=f(x) - where we say y is a function of x - or a specific linear notation, such as y= mx+b,
to describe a relationship between two variables, one must not presume any direction or
notion of causality such as “a change in x causes a change in y” although, in some
circumstances, this may be an appropriate interpretation. For example, if the relationship
is between x, the level of labour used as an input into production, and y, the level of
output produced, then the notion of causality does make sense with changes in the level
of x causing or resulting in changes in the level of y. In particular, if y=5x describes
some linear production function, with x representing the number of person-hours used
and y the number of units of some product produced, the it does make sense to think of “changes in x” causing “changes in y” according to the function y=5x. However, even
when there is a clear notion of causality we can still write the function equivalently as
x=0.2y [or x=(1/5)y] and note that in this form we are expressing the idea that for every y
units of output the firm wishes to produce it needs to employ x=0.2y units of input. This
would allow us to determine the labour costs for producing any given level of output y as
CL = wx = 0.2wy, where w is the cost per person hour of labour, and so we can write
CL(y) = 0.2wy as the “cost of labour” function.
As noted above, one general way to denote a linear function, for y=f(x), is to write
where m and b are constants.
Note that in the inverse form this same relationship can be written
It is often useful to write linear equations in both “direct” form and “inverse” form in
order to easily find the intercept terms which are helpful in plotting the graph of thefunction. For the case above, the y intercept (i.e., the value of y when x is zero) is b
while the x intercept (i.e., the value of x when y is zero) is -b/m. The term m represents
the slope of the linear function and the fact that this slope is the same regardless of the
choice of the value of x is the key property of a linear function. If m is a positive value
(m>0) that means the function has a positive slope while if m is a negative number (m<0)
that means the function has a negative slope. If m=0 then the value of y is the same
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expect that we would use the horizontal axis to measure price, p, and the vertical axis to
measure quantity, q. However, the convention is just the opposite. Thus, if given the
function q = 25-2p it is a good idea to also write it in its inverse form, p = 12.5-0.5q in
order to facilitate graphing the function with q on the horizontal axis and p on the
vertical axis. It is also useful to write out demand and supply functions (and many othersas well) in direct (or original) form and also the inverse form as this makes it very simple
to compute the intercepts and hence more easily draw the graph.
We draw the graph for the above mentioned example below. Note that, since
q = 25-2p or, in inverse form, p = 12.5-0.5q, the q-intercept is 25 while the p-intercept is
12.5. Since the function is linear the graph of the function can be drawn by simply
drawing a straight line which
connects the intercept points
(q,p)=(0,12.5) for the p-intercept
and (q,p) = (25,0) for the q-
intercept. The graph is illustratedbelow.
Since negative values for a price or a quantity are not meaningful, we often do notextend the demand function into the regions where q<0 or p<0. The p-intercept for
demand functions (12.5 in this example) is sometimes referred to as the choke price since
it is the (smallest) price at which quantity demanded becomes zero (i.e., demand is
choked off). The q-intercept value for demand functions (q=25 in this example) is the
quantity that would be demanded if price were zero.
Consider next an example of a linear supply function, also quantity being a
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Module 3: Simple Nonlinear Functions and Their Graphs
Although we often use linear functions for examples and illustrations in
economics, in this course we will also need to use some simple nonlinear functions. The
fact that linear functions have constant slopes is precisely why we cannot always usethem to illustrate important economic properties.
Consider, for example, a simple linear production function, Q=aL, with a>0 a
pararmeter, which describes the relationship between the amount of input labour (L) used
and the level of output produced (Q). Since we are considering variations in the level of
only one input, labour, this function is sometimes called the total product of labour
function and its graph the total product of labour curve. The implication, generally, is
that other inputs - and in particular capital or machinery - is held at some fixed level.
This makes sense in a short-run setting since one can employ existing workers more
intensively, through overtime, or hire new workers in a much shorter time period than
can one set up new production lines and factories. Linearity of the function means thatan increase in one unit of L leads to an increase in output in the amount of “a” units
regardless of the existing level of labour employed. This is illustrated in the following
graph.
Since the amount of capital (machinery) is fixed it makes sense that the amount of
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extra output generated by adding more labour will depend on the existing amount of
labour employed. The more labour already using the fixed amount of machinery, the less
might be the added amount of output per additional unit of labour employed. If this is
the case, we would say that labour is subject to the law of diminishing marginal
productivity. Since the slope of a function indicates the rate at which the variable on thevertical axis rises (or falls for a negative sloped function) and a “flatter” function means
this rate is lower, then according to our above description of what might happen when
more labour is used we should expect the slope of the total product of labour curve to fall
as more labour is added. A function with this property is Q=10 L or Q=10L1/2. The
following table gives some values which can be used to plot this function. From the
table, it is also clear that adding an extra unit of labour indicates less and less of an
We generally expect that with a fixed amount of machinery (capital), additional
amounts of labour will at least eventually lead to smaller amounts of additional output.
That is, the law of diminishing marginal productivity of labour will eventually apply.
However, it is certainly possible that at zero or low levels of existing labour beingemployed, additional amounts of labour will not add so much to output. It is quite
plausible that the number of machines along a production process could not be used very
productively until a certain threshold level of labour were employed since with small
numbers of workers there would be a lot of running around from machine to machine in
order to generate any output. So additional output from additional units of labour may
well be created at an increasing rate, rather than a diminishing rate, at least for low levels
of output.
Again, since the slope of a function indicates the rate at which the variable on the
vertical axis rises and a “steeper” function means this rate is higher, then according to our
above description of what might happen when more labour is used we should expect theslope of the total product of labour curve to rise (at least initially) as more labour is
added. A function with this property is Q=L2. The following table gives some values
which can be used to plot this function. From the table, it is also clear that adding an
extra unit of labour indicates less and less of an increase in output.
L 0 1 2 3 4 5 6 7 8 9 10
Q=L2 0 1 4 9 16 25 36 49 64 81 100
Q - 1 3 5 7 9 11 13 15 17 19
We sketch this function,
Q=L2 below.
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The simplest function which has the above shape for its graph is a cubic function.
We won’t investigate further here how to determine the graph of a cubic function as suchspecific exercises are not so important in this course. However, you do need to
understand the meaning of convex and concave shapes of functions in a number of
contexts, most notably the total product of labour concept discussed here.
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The concepts of “lesser than”, “lesser than or equal to”, “greater than”, “greater
than or equal to”, are quite straightforward when applied to (real) numbers. If an
inequality statement includes the “equal to” part, it is called a strict inequality, while if itdoes not include equality, it is referred to as a weak inequality. Four examples, with
graphs relating to the real number line for each, are given below. Note that an open dot,
, means the number on the line is not included in the set, while a closed dot, , is
included.
x
5 (read “x is less
than or equal to 5"),
x < 2 (read “x is less
than 2"),
x -3 (read “x is
greater or less than-3"),
x > 8 (read “x is
greater than 8")
The above are sets of real numbers and these sets are drawn in a single dimension.In economics one important concept involving sets described by an equality relation is
the budget set for consumers. Given a certain income, or budget amount, we need to find
a convenient method of describing this graphically. Although a typical consumer selects
his/her consumption from a wide variety of goods, it turns out that much of the economic
intuition concerning this choice problem can be described by using the budget line and
corresponding budget set for two goods. Let x1 and x2 represent the consumption
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In this course you will need to be able to solve systems of linear equations. These
will involve only cases of two equations and two unknown values which is a relatively
straightforward procedure. In the majority of cases this technique is applied to the caseof finding equilibrium in demand-supply models. We have already discussed how to
graph linear demand and linear supply functions, so we continue with that example.
In the above examples we had:
Demand: q = 25-2p or, in its inverse form, p = 12.5-0.5q
Supply: q=-5+p, or, in its inverse form, p=q+5
There are several techniques for solving systems of linear equations. When there
are just two equations and two unknowns, the simplest method is to write one variable interms of the other variable in one of the two equations and then substitute out for that
variable in the other equation. This provide a solution for one of the two variables. One
can then substitute this solution into either of the equations to solve for the other
variable. This is particularly simple in demand-supply models since generally q is
written explicitly as a function of price in both equations. Thus, equilibrium price can be
solved determined by simply noting that quantity demanded = quantity supplied, which
Another Example: In Macroeconomics the IS-LM diagram is used to find the short run
equilibrium output level and real interest rate. The IS curve represents the interest rateand output pairs that equate planned and actual expenditure in the goods market (or
equivalently, the demand and supply of loanable funds). The LM curve represents the
interest rate and output pairs that equate the demand and supply of real money balances
in the money market.
Suppose that we are given the following information.
C = 250 + 0.8 [Y - T ]
I = 200 - 50r
( M / P)d = Y - 50r
where C is consumption, Y is output, T is taxes, I is investment, r is the real interest rate, and
M/P are real money balances. Let government spending, G, equal 210, and taxes, T , be equal to
200. Let the money supply, M , be 2000 and the price level, P, be 2. Y is income and r is the real
interest rate in percent.
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