Algebra, Equations and
Models
"I would advise you Sir, to study algebra, if you are not already an adept in it:
your head would be less muddy, and you will leave off tormenting your
neighbours about paper and packthread, while we all live together in a world that
is bursting with sin and sorrow."
Overview
At the heart of algebra is the equation - the technical term used to symbolize a relationship between variables. It is
an abstract, but essentially simple concept that you MUST understand if you are to become skilled at "pushing"
those numbers around. As we look more closely at algebra keep in mind that the equations are similar to graphs in
that they are both shorthand ways of conveying information concerning a relationship between phenomena. In the
case of graphs we use a visual means to communicate the information, while with equations we use mathematical
symbols. For example, consider the problem of conveying information on sales at Max's Market, a local retail
outlet. Max needs to put together some numbers and prepare a financial statement before he meets with his banker.
Although we can count on Max to make a full accounting of his situation, we will only look at the revenue side of
the books. We want to compare two revenue projections. The first (Sales A) is based on the assumption Max's
sales are currently $200,000 per month and will increase $2,000 per month for each of the next 120 months (10
years). The second (Sales 2) is that sales will increase 1 percent a month from the same existing base of $200,000.
The task here is to determine how best to capture these mathematical relationships – graphs, tables, or equations –
and below you will see all three. The graph provides a valuable visual representation of the sales projections, while
the table allows us to determine sales for each time period.
Sales Projections Table
Months
Sales A
Sales B
0
200,000
200,000
24
248,000
253,947
48
296,000
322,445
72
344,000
409,420
96
392,000
519,855
120
440,000
660,077
The third representation is algebraic and the two equations below represent the algebraic version of the two
relationships between Max's sales (S) and months (M).
S = 200,000 + 2,000*M
S = 200,000*(1+.01)M
If we want to know exactly what sales will be in 6 months, we simply plug in 6 for the value of M and we get
212,000. This precision is the primary advantage of the algebra, which is why algebraic equations are behind the
majority of the forecasts you hear each month. The disadvantage of the algebraic representation is not as many
people can interpret the language of math necessary to translate the equation back into a picture (the graph) or a
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story (the text). Have you ever listened to a conversation in a foreign language and marveled at the speed with
which they talked? If you have, then you probably have a good idea of how many people feel when they listen to
stories using math. They simply do not know the basis of the language of math that would allow them to
understand what they are seeing / hearing.
Language of math
To help you with the translation we will spend a little time learning some of the basics of the language of math. As
a starting point you should recognize mathematics as a language and think of equations as sentences and the
characters that appear in the equations as the words. You should also recall some of the grammar, the rules that
specify what you can and cannot say. For example you may remember the rule of exponents that specifies (xa)*(ya)
= (xy)a. We'll have a brief refresher of the rules and then in a later section we will talk about models that are the
paragraphs.
So let's start with the words. Rather than talk about verbs, adjectives, or nouns, we will talk about variables and
parameters. Let's return to our Max's market problem. The variables are the terms used to represent the phenomena
being studied. In this equation there are two variables, S that we use to represent the volume of sales, and M that
represents the number of months. We can add more precision to our analysis if we recognize there is causality
implied in the relationship - once we know the month we will know the value for sale. The variable that is the
cause, M, is an exogenous variable or independent variable. These variables tell us something about the external
environment.
In addition to the variables, we also see the numbers 200,000 and 2,000. These are called parameters, constants
that specify the exact nature of the relationship between the variables. If we change one of these numbers it would
change both the picture and the story. Once we know the value of M the parameters in the equation allow us to
determine the value of S, the endogenous variable or dependent variable. Where M was the cause, S is the effect.
When we combine the words with a variety of mathematical symbols (ex. +, -, /, and *) we have equations
providing the reader with some information regarding the nature of the relationship. Once you have the values for
the parameters and exogenous variables, you will be able to calculate the value of the dependent variable. In our
simple example, once you know the value of M (10 months from now), you can determine the value for S
(220,000).
If you return to your English grammar books, you will note there are many types of sentences. The same is true in
mathematics where we can use a number of classification schemes for sorting equations. One way to classify them
is by their mathematical structure. We will talk about linear and some nonlinear relationships that are popular with
economists.
What do we do with our equations? Often times we will combine them to build a model that allows us to specify
the interrelationship between a number of variables, and at the end of the unit we will look at a simple model.
Before we look at these models, however, we need to look at some of the basics of algebra beginning with
equations including some specific types of equations - linear equations, nonlinear equations, and equations of
exchange including present value. When we have completed our analysis of equations, we will look at models.
Equations
Algebra is an extremely powerful tool, and one you should make an effort to master. First, a working knowledge
of it is essential if you are to realize the power of spreadsheet software, one of the most popular software programs
on the market and one that employers are increasingly looking for mastery of when they hire college grads. It is
also a valuable tool in economics, which should not be a surprise given our main concern is with the study of
relationships between observed quantities. Algebra simply offers us a way to express the existence of such
relationships. Algebra can be thought of as an alternative to the graphs that fill nearly all economics texts.
At the heart of algebra is the function - the technical term used to symbolize a relationship between variables. It is
an abstract but essentially simple concept. A function is merely a mathematical statement that two or more
variables are related. For example, if we believe two variables, x and y, are related to one another, we may write:
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y = f(x). This expression is read, 'y' is a function of 'x', or 'y' is dependent upon 'x'. The choice of the letter f was
purely arbitrary. An alternative approach would be to repeat the letter used on the left hand side of the equation
(ex. y = y(x)). How we specify the relationship is fairly unimportant. What is important is this notation y = h(x),
should be interpreted as shorthand for "the value of y depends on the value of x."
Oftentimes in economics, however, we are called upon to specify the exact nature of the relationship between the
variables. When we do this we are specifying the relationship in the form of an equation. Some examples of
equations are presented below. These equations are the inputs into models, collections of equations that describe
the interrelationship between a set of variables.
1.
3.
5.
7.
9.
y = 4x + 3
y = 4x1/2 + 3
y = 4x - 2z +3
y = 10/x
y = mx + b
2.
4.
6.
8.
y =1.1x
lny = 4lnx
y = 3 + 4x2
lnQ = -2lnP +1.5lnY
At this point do not be concerned by the fact many of these equations look 'new' to you. Most of them you will
never have to deal with, but you should know they are all just simply ways of expressing relationships between the
variables y and x. We are going to restrict discussion to those you are likely to run into, beginning with a
discussion of linear equations and then moving to some nonlinear equations and then finishing with a discussion of
some important equations of change.
Linear Equations
The first equation is by far the simplest. We can begin by building a table to represent the relationship. In the first
column we put the numbers 1 through 40 and the next column we put in a number 4 times the number in column 1
plus 3. This would produce the table below. This table signifies a relationship between x and y such that every
time x increases by 1, y increases by 4. If we were looking at a graph, the relationship between changes in x and
changes in y show up in the slope that would be 4. This is the distinctive feature of linear relationships - their
graphs are straight lines. In the equation form, the slope (rate of change) appears as the coefficient of x. The other
parameter in the equation is 3 that tells us when x = 0, y is equal to 3. In a graph, this would be the y
intercept. When x = 0, then y equals 3.
The Table
x
0
1
2
3
...
30
31
y
3
7
11
15
...
123
127
The graph
The Equation
y = 4x + 3
If we compare equations 1 and 7 we find there is only one difference. In equation 1 the variables y and x are raised
to the first power (1), while in equation 7, x is raised to the second power (2). This is a significant difference. At
this point you should be able to recognize equations where the variables are raised only to the first power. These
are called linear equations and possess certain desirable properties. The most obvious property is the fact the rate
at which y changes for any change in x is independent of the value of x. In the linear equation, if we are at x = 1
and move to x = 2, y increases by 4. The rate of change Δy/Δx = (11-7)/(2-1) = 4. If we begin at x = 30, however,
and move to x = 31, the rate of change (Δy/Δx) = (127-123)/(31-30) = 4/1 = 4.
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One can conclude from this simple example that when you have a linear equation the rate of change is a
constant. In the above example, every time x increases by one unit, y increases by 4. We can generalize for the
generic equation (9) you should recall from high school.
y = mx + b
In this generic example, the slope equals the rate of change.
(1) m = Δy/Δx
The rate at which x causes y to change is equal to the coefficient of x. What we have here is a measure of the
sensitivity (or responsiveness) of y to changes in x. If we want to demonstrate algebraically that y is more sensitive
to changes in X, then this would show up in an increase in the absolute value of the parameter "m." Graphically, it
would show up as a steeper slope (if the vertical axis is y).
As we will see many times in this section, we can interpret this relationship as one equation with three unknowns
(D y, D x, m). Using the 'laws' of algebra, we can rewrite this equation in the following two ways - each providing
us with a different variable on the left hand side of the = sign.
(2) Δ y = Δ x*m
(3) Δ x = Δ y/m
Equation 2 can be used to answer a question such as: how much will y increase if x increases by 3? In the case
where the coefficient was 4, x would increase by 12 when y increased by 3 [4*3=12]. If on the other hand you
happened to know that y increased by 12, and you wanted to know how much x must have increased, then you
would use Equation 3 because you want the unknown on the left side of the equal sign. If you plug in the values
for Δ y/a and you get 12/4 = 3. If y increased by 12, then x must have increased by 3.
Before we leave our discussion of linear equations, let's return to the list of equations and see if there are any other
linear equations. The secret is the exponent for the variables must be equal to 1. The fourth equation looks
promising, but you will note that X is in the denominator so we could rewrite it as y = 10*X-1 so its exponent is
really -1. Equation 3, meanwhile, does satisfy our condition for a linear relationship, the only difference is that y
depends on two exogenous variables, x and z. The nature of the relationship is identical to what we discussed for
the first equation. In the linear equation, if we are at x = 1 and z = 2 and move to x = 2, then y increases by 4. The
rate of change Δy/Δx = (7-3)/(2-1) = 4. If we begin at x = 30 and move to x = 31, the rate of change (Δy/Δx) =
(123-119)/(31-30) = 4/1 = 4. Once again the coefficient of the variable gives us the slope of a graph and the rate of
change between y and x.
The Table
x
0
1
2
3
31
33
z=0
0
0
0
0
0
0
y
3
7
11
15
123
127
x
0
1
2
3
31
33
z=2
-2
-2
-2
-2
-2
-2
y
-1
3
7
11
119
123
Similarly, the coefficient of z indicates the rate at which y changes when z changes. In equation 3, y will change -2
units for every one-unit change in z. In fact, you can see this in the table. You note that as you move across one
row you are comparing the values of y, for a given value of x, as z changes from 0 to 2. For example, if x = 3 and z
= 0, then y = 15. If you hold x constant at 3 and allow z to increase to 2, the value of y decreases to 11. In this
situation Δy/Δz = (11-15)/(2-0) = -4/2 = -2, which is the coefficient of z. Now let's move on to a discussion of
nonlinear relationships.
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Before moving on, let’s look at an common relationship in economic – demand. For example, consider the
problem of constructing / estimating an equation for the demand for SUVs (Q). What are the factors that affect
demand for the SUVs? The most obvious variable would be the price of SUVs (P), and we could have the
following equation for demand.
Q = a + b*P
In the equation the parameter b would be ΔQ/ΔP and it would tell us how sensitive demand was to price. Our
economics theory, meanwhile, would tell us to expect the parameter to be negative because the theory of demand
tells us to expect a rise in the price to decrease demand. If we were estimating a demand curve, we would expect to
find that b <0, so the following equation would make sense.
Q = 560 – 2*P
Based on this equation, demand for SUVs would fall by 2 units for every one-unit increase in the price. If the price
were 25, then the demand for SUVs would be 510 units.
Q (25) = 560 – 2*25 = 510
In economics, however, things are often not quite so simple so we end up with equations with many variables.
Now let’s look at the multivariate linear equation.
Multivariate Linear Equations
. In this case of demand for SUVs, there are certainly other factors that affect demand. On the short list of factors
would be the price of gas (Pg), family income (Y), and the interest rate (i), and below is the linear equation for the
demand for SUVs.
Q = a + b1*P + b2*Pg + b3*Y + b4*i
What are the expected values / restrictions on the parameters based on economic theory?
• As the price of gas increases we would expect demand to fall, so b2 should be negative.
• As income rises we should expect demand to increase, so b3 should be positive.
•
As interest rates rise we should expect demand to decrease so b4 should be negative.
This would be the specification of a demand equation, and if we conducted the appropriate statistical analysis (we
discuss it later), then we could estimate the parameter values. Below is a possibility based on the economic theory.
Q = 200 - 34*P - 2*Pg + 4.8*Y – 3.25*i
In this case we see that an increase in the price of gas by 1 unit (the coefficient depends on whether the price is in
dollars or cents) will decrease demand by 2 units (maybe thousands). What we have here is the following.
m = ΔQ/ΔP = -34
We could now use this equation to explain what will happen if the price of gas rises by 10.
ΔQ = m /ΔP = -34/10 = -340
If the price rises by 10 units, demand will fall by 340 units.
If we look at the other coefficients we see that a one unit increase in the price of SUVs will reduce demand by 34
units, a one unit increase in income will increase demand by 4.8 units, and a one unit increase in interest rates will
decrease demand by 3.25.
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Now for one last question: What will the equation look like if we estimate it in the UK and the results indicate that
demand is more sensitive to the price of gas in the UK? Because we are talking about the sensitivity of demand to
the price of gas we are looking at the coefficient of Pg. In the US the number is -2, so if demand is more sensitive
in the UK then this number must get larger (in terms of absolute value), so in the UK the coefficient might be -4.
In the US every one unit increase in the price of gas reduces SUV demand by 2, while in the UK every one-unit
increase in the price of gas reduces demand by 4.
Nonlinear relationships
Not all relationships can be captured by linear equations, but you probably know that from your algebra and
trigonometry courses. You probably also remember that nonlinear equations were often more difficult to deal with,
and that has not changed. You can, however, make your life a bit easier if you realize that as long as all of the
parameters have specified values, you can use tables to help you visualize the nature of the nonlinearities. For
example, let's look at the seventh equation, a simple quadratic.
The Table
x
0
1
2
3
...
30
31
The Graph
y
3
7
19
39
...
3603
3847
The Equation
y = 4x2 + 3
We can set up a simple table in which we put in values for x extending from 1 to 31 and calculate the value for y
appearing in the second column by plugging the values of X into the equation. In this nonlinear case the rate of
change varies with the initial value of x. If we are at x = 1 and move to x = 2, y increases by 12. The rate of
change Δy/Δx = (19-7)/(2-1) = 12. If we begin at x = 30 and move to x = 31, the rate of change (Δy/Δx) = (38873603)/(31-30) = 284/1.It should come as no surprise that if we plot these points we have what looks like a
parabola. As X increases the slope increases.
Equation 6 ]lny = 4lnx] is also a nonlinear equation. It is called a log linear equation and economists like this
specification of a relationship between two variables. Fortunately, you do not need to know anything about logs at
this time. If you ever have a question where you run into logs, you can use excel, which has logarithmic functions,
but more on that later. What is important now is the interpretation of the coefficient. In the linear equation the
coefficient of x was equal to Δy/Δx. In the log linear case the coefficient of lnx equals %Δy/%Δx. In this equation,
y increases 4 percent for every 1 percent increase in x.
(1’) 4 = %Δy/%Δx
As we saw with the linear equations, we can interpret this relationship as one equation with three unknowns. Using
the 'laws' of algebra, we can rewrite this equation in the following two ways - each providing us with a different
variable on the left hand side of the = sign. If you know the value of the coefficient, which is 4 in equation 6, then
you rewrite the equation two ways:
(2') %Δy = %Δx *4
(3') %Δx = %Δy /4
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Equation 2' can be used to answer questions such as: how much will y increase if x increases by 3 percent? If x
decreases by 4 percent, what will happen to y? In the first case y would increase by 12 percent when x increased
by 3 percent [ 3*4 = 12]. In the second case y would decrease by 16 percent when x decreased by 4 percent [ -4*4
= -16]. If on the other hand you happened to know that y increased by 12 percent, and you wanted to know how
much x must have increased, then you would use Equation 3’ because you want the unknown on the left-side of
the equal sign. If you plug in the values for %Δy/a you get 12/4 = 3. If y increased by 12 percent, then x must
have increased by 3 percent.
The reason why this formulation is so important is the ratio of percentage changes has a special significance in
economics. From your microeconomics you will recall the concept of elasticity. The price elasticity of demand
(ep) is defined as the percentage change in demand (Qd) divided by the percentage change in price (p) [ep =
%ΔQd/%DP]. Similarly the income elasticity of demand is defined as the percentage change in demand divided
by the percentage change in income (Y) (ey = %ΔQd/%ΔY). If we wrote the demand equation as ln(Qd) = a*ln(P)
+ b*ln(Y) then a would be the price-elasticity of demand and b would be the income elasticity of demand.
Equation 5 (y = 1.1x) is also an equation you are likely to run into repeatedly. This is an exponential function,
which is at the heart of present and future value analysis, which in turn, is at the center of the large majority of
financial decisions. The graph for the values of x from 1 to 31 is presented below and it looks very much like the
parabola we looked at earlier.
Now let's look at a couple of important equations that specify change.
Equations of Exchange
You will often find yourself in a situation where you are concerned with change. How much will your paycheck
increase if the government increases payroll taxes by $5.00 a pay period? What time will we get in if we are taking
a 5-hour flight that leaves at 9 AM? What will the balance be in my saving account in five years if interest
accumulates at 5 percent per year? These are just a sampling of the questions that involve change, and behind the
answers to each is an equation.
Two popular measures of change are the simple change and the rate of change, what you may have heard of as
percentage change. They may sound very much the same, but they are not, and if you think they are, you are ripe
to get ripped off. The difference is evident in the two graphs of population growth reproduced below. The first
graph contains data for the change in population, while the second contains percentage change data.
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In the first graph Asia appears to be the center of population growth, while in the second Asia trails significantly
behind Africa. The differences become clear when we look at the equations that generated the graphs.
In the discussions of change between any two dates, we will use the notation (PV) to designate the original point
(present value) and (FV) to designate the end point (future value).
Let's begin with a measure of change - what is represented in the first graph of population change.
(1) Change = Δ = (FV - PV)
When calculating change we have one equation containing three variables - Change (Δ), New (O), and Old
(O). Given there is only one equation, we can solve for only one unknown which means we will need information
on the other two unknowns. If you know where you started and where you end up, you can certainly calculate the
change. Similarly, if you know where you started and the change, you could easily calculate the new. Finally, you
could determine where you started if you knew where you ended up and how much you changed to get there.
To answer each of these questions you can use the logic of algebra to generate three equations, each one isolating
the unknown variable on the left side of the = sign. The three formulations would be:
(1a) Δ = FV - PV
(1b) FV = PV + Δ
(1c) PV = FV - Δ
You would use the first equation to solve a question such as the following: What was the change in our bank
balance between 1980 and 1995 if it increased from $10,000 to $14,500? The second equation would be
appropriate when we needed to know our bank account balance next year if it is currently $2,000 and it will
increase $200 this year. Finally, if we needed to know the balance in a checking account at the beginning of the
period if we knew it is now $1,200 and it had grown $200 during the period, we would use the third specification.
A second popular measure of change is percentage change or rate of change, a concept that has caused trouble for
many students over the years. When we are measuring percentage change (%D) we will use the following formula:
(2) Percentage Change = % Δ = (FV - PV) / PV
Once again we have one equation with three unknowns (%Δ, N, and O) so we have three formulations. In addition
to the formula for percentage change above, we can rewrite the equation as follows:
(2a) % Δ = (FV - PV) / PV
(2b) FV = PV*(1+% Δ)
(2c) PV = FV / (1 + % Δ)
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You would use the first equation to solve for the growth rate in sales if sales rose from $120 to $150 million. The
second equation would be used to calculate the world's population be in ten years if it increases by 5 percent from
its current level of 5 billion, while equation 3 would be used to determine how much we must we put in the bank
today if we want to have $20,000 in the bank next year if we expect the interest rate to be 6 percent?i
You are now ready to turn to the present value analysis, an important use of the equation of exchange concept present value analysis.
Present value
You all know that $1 today is not worth $1 a year from now because you would not trade me the $1 today to get
$1 a year from now, but how much will you be willing to spend today to get $1 in a year? Present value analysis
is the technique for translating $s from one time period to $ in another time period. We will begin with an equation
that gives us the relationship between an original value, an ending value, a growth rate (rate of change), and the
number of time periods. In the equation above (2’) we can calculate the ending value (FV) if we know its original
value (PV) and the growth rate (g). Now all we need to do is allow this process to go on for T periods. The
equation describing this growth is:
(3) FV = PV*(1 + g)T
If we call the old value the present value (PV) and the new value the future value (FV), the equation that describes
the mathematical structure of the relationship between these four variables will be the formula for compound
growth you will find in your economics and finance courses. As you would expect by now, we can think of this as
an equation with 4 unknowns. You can use it to solve for any one of the variables when you have information on
the other three variables. To solve for the unknown all that you need to do is rewrite the equation so the unknown
is on the left hand side of the = sign. To make it easy for you I have done the algebra to help you solve the three
most common problems.
Computation of Future Value: givens = initial value, growth rate, time period. You would use this for questions
like, what would the world's population be in ten years if it increases by 5 percent from its current level of 5
billion?
(3a) FV = PV(l+g)T
Computation of Present Value: givens = future value, growth rate, time period. You would use this for questions
like, What must we put in the bank today if we want to have $20,000 in the bank in 10 years if we expect the
interest rate to be 6 percent?
(3b) PV = FV/(l+g)T
Computation of Average Growth Rate: givens = future and present value, time. You would use this for questions
like, What was the growth rate in sales if they went from $120 to $150 million in 5 years?
(3c) g = (FV/PV)(1/T) -1
To see how we might use the compound growth equation, let's look at three simple questions and their
answers before we return to a few of the Future Questions from our introduction.
Q1. What will the population of India be in the year 2020 if the population in 1985 was estimated to be
751 million and the growth rate is expected to remain at 2.5% a year for the entire time period?
A1: This is a future value problem. The initial value is 751, the growth rate is 2.5% (.0251), and the time
horizon is 35 years.
FV = PV*(l+g)T = 751*(1.0251)35 = 751*2.381 = 1,788 million
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Q2. How much should you pay for a piece of paper that guarantees you $1000 three years from now if
you expect the interest rate to be 8% per year for this three-year period? Stated somewhat differently,
how much would I have to invest today to receive $1,000 in three years
A2: This is a present value problem. The end value is $1,000, the time horizon is 3 years, and the growth
rate of money is 8%.
PV = FV/(l+g)T = 1,000/(1+.08)3 = $793.83
Q3. The average weekly wage rate for workers in the US increased from $102 per week in 1970 to $389 a
week in 1989. What was the average yearly rate of increase?
A3. This is an average yearly growth rate problem. The initial value is 102, the end value is 389, and the
time horizon is 19.
g = (FV/PV)1/T -1= (389/102)1/19-1 =3.81370526 -1
= 1.0729 - 1 = .0729 (growth rate = 7.29%)
If you feel confident about your mastery of the compounding formula, you are ready to look at a few of
those opening questions. More specifically, you should check out the problems concerned with
calculating the value of a college education, the future costs of a college degree, and the value of a timeshare unit.
Models
For example, let's look at the model of supply and demand. You remember those supply and demand curves and
the designation of the intersection as the equilibrium point giving us reason to believe price will be W* and output
will be Q*.
Supply and Demand Curves
Just as we saw there was an algebraic skeleton to the revenue graphs, we can now create the algebraic structure of
the supply - demand model of prices. We begin with the information on the behavior of suppliers, where we
specify the quantity supplied (Qs) as being dependent upon price (P). The second piece of information is the
behavior of demanders, where the quantity demanded (Qd) is specified as being dependent upon price (P). The last
bit of information would be the equilibrium condition specifying the conditions under which there would be no
tendency for change in the market - the supply equals demand condition. We can represent this complete model
algebraically with our three-equation, linear model where I have specified the values of the parameters.
Qs = 100 +2*P
Qd = 400 - 4*P
Qs = Qd
If we take these three pieces of information together we have a simple economic model that will explain the
market price (P), quantity supplied (Qs), and quantity demanded (Qd). The graphical analysis would lead us to
look for the intersection of the supply and demand curves, while the algebraic approach would lead us toward
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simultaneously solving the three equations for the equilibrium values. In both cases we would find the equilibrium
values to be, P (50), QS (200), and Qd (200).
What will happen to the price of a commodity if there is an increase in demand? What will happen to the level of
national income if the government increases defense spending? How can we describe the choices open to a student
allocating study time between two courses? How can we explain the slowdown in the growth in output in the US?
Why is tuition increasing so fast in US universities? What will be the rate of interest on government securities
next month?
These are just some questions you may have come across in your economics courses, but how do we approach
them given the answer to each question requires a rather involved logical chain of reasoning? We are going to
look at the first three of these questions in some detail to provide you with an opportunity to see how models are
developed and used. We will begin here with modeling the impact of an increase in demand on price.
Let's begin our work with models by looking at the first question and following through the logic you discussed in
your introductory course. If there is an increase in demand, there will be a shortage in the market. The supplier in
the market will respond to this shortage by raising prices that will lower quantity demanded and raise quantity
supplied. Eventually the fall in demand and the rise in quantity supplied will eliminate the surplus and the market
will reestablish equilibrium at a higher price.
How can we capture this logic? In most courses we do it with some graph similar to what we have below. The
increase in demand (D1 to D2) means that at the original equilibrium price (P1) there is now a shortage (S < D).
Equilibrium is reestablished at P2.
The Supply-Demand Model
In many instances, however, economists are interested in using algebra to express the relationships conveyed in
the graph above. They would create a model that is nothing more than a set of equations. The first step requires us
to make some decisions regarding what kind of model we want to build. To make our life easy we will assume that
the model is linear. It will consist of three equations: (1) the demand curve, (2) the supply curve, and (3) the
equilibrium condition. The first two of these would be called behavioral equations because they describe behavior,
while the last would be called an equilibrium equation because it specifies the condition for equilibrium. In very
general terms we could develop a qualitative model and specify the model as:
(1) Qd = a + bP
(2) Qs = c + dP
(3) Qs = Qd
Before you panic because you have never seen anything like this, remember this is a model you have seen before,
so let's try to make sense of it. The equations are clearly linear, which means the equations would appear as
straight lines on a graph. Given they are linear, the coefficients of price (P) are:
b = ΔQd/ΔP
d = ΔQs/ΔP
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The coefficient b tells us how quantity demand changes when price changes, and from our economics we have
come to expect they would be negatively related. An increase in price will reduce demand so we would expect b
to be a negative number. For supply, meanwhile, price and quantity are positively related, so we would expect d to
be a positive number.
To answer the opening question with this model, we need to solve it for the unknown variables (Qs, Qd, and P).
Given we are looking to solve for P, I suggest we substitute equations (1) and (2) into equation (3) to obtain the
following equation (4):
(4) a + bP = c + dP
Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution for the
equilibrium value of P (P*).
(5) P* = (a-c)/(d-b)
What do we know about the right side? We know from our economics that the supply curve has a positive slope
and the demand curve has a negative slope. This means that b < 0 and d > 0 so we can expect the term (d - b) >
0. We are almost there, but first we need to know how to show an increase in supply. If you return to the supply
equation, c is the shift parameter. An increase in c would represent an increase in supply that would show up as an
outward shift in the supply curve. Returning to equation (5) we can conceptually rewrite it as:
(6) P = [1/(d-b)]a -[1/(d-b)]c
What we have here is a linear equation specifying the relationship between P and c. The red term represents the
intercept and the green term represents the slope of a relationship between P and c. Now we are home. We know
the slope is by definition equal to DP/Dc and equal to the coefficient of c, so DP/Dc = -[1/(d-b)]. If you go back
and plug in the values for the parameters you find the coefficient is a negative number since ( d - b) > 0. Putting
them together we have that an increase in c will result in a decrease in P, but that is what we already found with
our graphs and our narrative.
Before we leave this, let's redo things with actual numbers and build a quantitative model. Our model will be:
(1') Qd = 100 - 2P
(2') Qs = 10 + 1P
(3') Qs = Qd
Following the procedure above, we substitute equations (1') and (2') into equation (3') to obtain the following
equation:
(4') 100 - 2P = 10 + 1P
Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution. The
equilibrium value of P equals 30.
(5') P = 90/3 = 30
Now let us increase supply by 30 so that our supply equation becomes Qs = 40 + 1P. Following the same
procedure we end up with equation (4''):
(4'') 100 - 2P = 40 + 1P
Collecting terms and bringing P to the left side of the =, we get an equation specifying the solution. The new
equilibrium value of P is equal to 20. As we expected, the equilibrium price has fallen.
(5') P = 60/3 = 20
Not as bad as it looked? To see if you really understand it, please try the following exercises.
• What will happen to price if there is an increase in demand?
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•
How will the price change be affected by the sensitivity of demand to price changes? [Hint: a more
responsive demand would be represented by a larger value for the coefficient (absolute value) of price in
the demand curve]
Multiplier example
It's time to look at a macroeconomic model, one that has been very important in the evolution of our thinking on
how the economy works. Here we will look at the simple income determination model of the macro economy.
The idea behind the model is simple enough: what are the factors that determine the size of the economy?
To help you build the model, the following specifications are being supplied. The linear assumption is designed to
make life easy.
Statements (words)
1.
consumption (C) is a linear function of disposable income (D) (spending by households in the economy
depends upon the level of income - taxes.
2. disposable income (D) equals income (Y) less taxes (T)
3. investment (I) spending is exogenously determined (a fixed value)
4. government (G) spending is exogenously determined
5. taxes (T) are exogenously determined
6. aggregate demand (AD) is equal to the sum of consumption, investment, and government spending
7. equilibrium exists when income (Y) equals aggregate demand (AD)
It's now time to turn the story into equations. The translations appear below.
Equations
1. C = a + b*D
2. D = Y - T
3. I = I*
4. G = G*
5. T = T*
6. AD = C + I + G
7. Y = AD
The behavioral equation is the consumption equation. The identities are all of the other except the equilibrium
condition, which is the last equation. The goal is to have an equation specifying income (Y) in terms of exogenous
variables and parameters. Plugging all of the information into the equilibrium equation derives the solution.
Y=C+I+G
now substitute for each of these
Y = a + bD + I* + G*
now substitute for D
Y = a + b(Y - T*) + I* + G*
now collect all Y terms on the left side
Y - bY = a + I* + G* -bT*
now collect the terms and isolate Y on left
(1-b)*Y = a + I* + G* -bT*
This gives you the following solution for Y.
Ye = 1/(1-b)*(a + I* + G* -bT*)
This equation gives us the solution for the model. Once you have the values for the exogenous variables and
parameters, you will have a solution for equilibrium level of income (Ye). More importantly, you can now use the
model to explain how the equilibrium income is affected by changes in the exogenous variables. The secret is to
realize the equation above is a linear equation in all of the exogenous variables so it is easy to separate out the
individual factors. You could rewrite the equation as:
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Ye = 1/(1-b)*(a) + 1/(1-b)*I + 1/(1-b)*G* -1/(1-b)*(bT)
The common element is the 1/(1-b) term. Once you recognize this as a linear relationship, you can use what you
know about linear relationships - the coefficient is the slope. The impact government spending has on income
(DY/DG) would be determined by the coefficient of G. In macroeconomics you will call this the multiplier. In this
simple model the multiplier is:
ΔY/ΔG = 1/(1-b)
The only parameter that influences the multiplier would be b since it is in the equation for the multiplier. If you
look where we first encountered b, it is the coefficient of income in the consumption equation. The parameter b
gives us a measure of how consumption changes with income (DC/DY) . Keynes called b the marginal propensity
to consume.
Possibility curve example
Consider the plight of a student in ECN201 who has a serious problem as she prepares for finals. She has two
final exams that are going to have a significant impact on her GPA. One of the courses is that infamous
introductory economics course (ECN201) and the other is introductory history (HIS 180). The difficulty she faces
is the allocation of her study time between the two courses. You have to decide what grade possibilities are open to
her. To help you the following information has been collected.
Statements
1. there are twenty hours of study time available
2. in both courses, you get a zero with no time spent studying
3. for each hour spent in the study of economics, your grade increases 4 points
4. for each hour spent in the study of history, your grade increases 3.5 points
With these "facts" we can now begin the process of translating the words into equations and then combine the
equations into a model. We will also translate what we have into graphs to link our current work with the earlier
graphical analysis.
One of the first things we will need to do is introduce some characters/letters to represent the important variables.
We will specify the following characters:
• E = economic grade
• H = history grade
• T = total time
• TE = time spent in economics
• TH = time spent in history
With these characters we can now translate the statements into equations.
Equations
1. T = 20 (this is a constraint)
2. T = TE + TH (the constraint when we build in the assumption that the time will be spent on studying the
two subjects)
3. E = 4*TE (this is a grade production relationship that links the hours spent in economics and the grade in
economics)
4. H = 3.5*TH (this is a grade production relationship that links the hours spent in history and the grade in
history)
You will now need to build in all of the information to find the relationship between the grades in history and
economics. These are the two outputs we are concerned about. What are the grade possibilities open to our
student? The secret with solving any model is to know what you want to end up with and then use the equations to
"get rid" of the other variables. In this problem you will substitute the two behavioral relationships into the
constraint. We do this by substituting these two equations into the time constraint. To do that, we need to rewrite
the two production equations in terms of hours.
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TE = E/ 4
TH = H /3.5
Substituting into the time constraint gives us:
20 = E/8 + H/7
Now we get H on the left side since we graphed it on the vertical axis. The equation is:
H = 70 - 7/8*E
This is the equation for the grade possibility curve. This specifies the relationship between the history and
economic grades. You should recall from the earlier discussion how to interpret this equation. The 70 is the H
intercept and tells you the grade in History if all of the time is spent studying history. The -7/8 is the slope and
tells us that every increase of 1 point in the ECN grade will reduce the HIS grade by 7/8 point.
The picture of the grade possibility curve (the maximum grade possibilities assuming full use of the time) appears
below. It should look familiar to anyone who has taken an ECN201 or ECN202 course.
At this point you should be up to speed on your algebra and modeling, so it's time to move on to spreadsheets to
get some practice at using your new skills.
i
There is one final measure of change that we should briefly mention. Consider the problem of estimating change in a variable
(4) R = P*Q
(5) Δ R = Δ P*Q + P* Δ Q
(6) % Δ R = % Δ P + % Δ Q
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