Jacques I. Mathematics for Economics and Business
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with solution
Y
t
=
t
Y
0
If α=0.1 and β=1.4 then Y
t
= (1.08)
t
Y
0
.
As t increases, Y
t
diverges uniformly, so unstable.
12 (a) CF = A(0.1)
t
;
(b) PS = 6(0.6)
t
;
(c) Y
t
= A(0.1)
t
+ 6(0.6)
t
,
Y
t
= 3(0.1)
t
+ 6(0.6)
t
;
(d) stable.
13 (a) CF = A(0.2)
t
;
(b) PS = t + 6;
(c) Y
t
= A(0.2)
t
+ t + 6,
Y
t
= 4(0.2)
t
+ t + 6;
(d) unstable.
14 (a) Y
t
= 0.6Y
0.8
t−1
+ 160
(b) 163.8, 195.4, 200.8, 201.7, 201.9, 201.9, 201.9,
201.9; stable.
(c) No.
15 (a) k
t+1
= 0.99k
t
+ 0.2k
t
0.6
.
(b) 1789; uniform; same behaviour.
Section 9.2
1 (a) 5e
5t
; (b) −6e
−3t
; (c) mAe
mt
.
2 (a) The function that differentiates to 4 times itself is
y = Ae
4t
. The condition
y(0) = 6 gives A = 6, so the solution is y = 6e
4t
.
(b) The function that differentiates to −5 times itself is
y = Ae
−5t
. The condition
y(0) = 2 gives A = 2, so the solution is y = 2e
−5t
.
3 The complementary function is the solution of
= 3y
and is given by
CF = Ae
3t
For a particular solution we try a constant function
y(t) = D
Substituting this into the original equation,
= 3y − 60
gives
0 = 3D − 60
dy
dt
dy
dt
D
F
β
β − α
A
C
y(t) = Ae
mt
+ D (A ≠ 0)
when m > 0.
4 Substituting the expressions for C and I into
= 0.1(C + I − Y )
gives
= 0.1(0.9Y + 100 + 300 − Y )
=−0.01Y + 40
The complementary function is given by
CF = Ae
−0.01t
and for a particular solution we try a constant function
Y(t) = D
Substituting this into the differential equation gives
0 =−0.01D + 40
which has solution D = 4000. The general solution is
therefore
Y(t) = Ae
−0.01t
+ 4000
dY
dt
dY
dt
Solutions to Problems
660
which has solution D = 20. The general solution is
therefore
y(t) = Ae
3t
+ 20
Finally, substituting t = 0 gives
y(0) = A + 20 = 30
and so A is 10. Hence
y(t) = 10e
3t
+ 20
A graph of y against t is sketched in Figure S9.5, which
indicates that y(t) rapidly diverges. We would expect
divergence to occur for any solution
Figure S9.5
MFE_Z02.qxd 16/12/2005 10:51 Page 660
The initial condition, Y(0) = 2000, gives
A + 4000 = 2000
and so A is −2000. Hence
Y(t) =−2000e
−0.01t
+ 4000
This system is stable because the complementary
function is a negative exponential and so Y(t) converges
to its equilibrium value of 4000 as t increases.
5 Substituting the expressions for Q
S
and Q
D
into
=
1
/3(Q
D
− Q
S
)
gives
=
1
/
3[(−P + 4) − (2P − 2)] =−P + 2
The complementary function is given by
CF = Ae
−t
and for a particular solution we try a constant function
P(t) = D
Substituting this into the differential equation gives
0 =−D + 2
which has solution D = 2. The general solution is
therefore
P(t) = Ae
−t
+ 2
The initial condition, P(0) = 1, gives
A + 2 = 1
and so A is −1. Hence
P(t) =−e
−t
+ 2
From the supply and demand equations
Q
S
(t) = 2P − 2 = 2(−e
−t
+ 2) − 2 =−2e
−t
+ 2
Q
D
(t) =−P + 4 =−(−e
−t
+ 2) + 4 = e
−t
+ 2
All three functions involve a negative exponential, so
the system is stable.
6 (a) t
2
+ 7; (b) −
1
/3e
−3t
+
1
/3; (c)
1
/3t
3
+
3
/2t
2
− 5t + 1.
7 (a) −20e
−3t
+ 60; starting at 40, y(t) increases
uniformly to 60.
(b) 20e
−3t
+ 60; starting at 80, y(t) decreases uniformly
to 60.
(c) 60; y(t) remains at the equilibrium level of 60 for
all time.
8 Y(t) = 5000e
−0.05t
+ 10 000; stable.
9 Y(t) = 2000e
0.15t
− 1800; unstable.
10 P(t) =−e
−2.5t
+ 2; Q
S
(t) =−3e
−2.5t
+ 5;
Q
D
(t) = 2e
−2.5t
+ 5; stable.
dP
dt
dP
dt
11 Substitute assumptions (1) and (3) into (2) to get
β=αY
which rearranges as
= Y
with solution
Y(t) = Y (0)e
(α/β)t
This is unstable because α/β>0.
12 The right-hand side is
my + c = mAe
mt
−+c
= Ame
mt
− c + c = Ame
mt
which we recognize as the derivative of y(t).
13 (a) Ae
−2t
; (b) e
3t
; (c) Ae
−2t
+ e
3t
;6e
−2t
+ e
3t
;
(d) unstable.
14 (a) Ae
−t
; (b) 4t − 7;
(c) y = Ae
−t
+ 4t − 7; y = 8e
−t
+ 4t −7; (d) unstable.
15 y(t) =√(t
2
+ 1). For large t, the solution is well
approximated by a straight line with equation
y(t) = t.
16 (a) y(t) = 0.5
(b) y(t) = 0.5
(c) In (a) the fund decreases and approaches the
equilibrium value of $500 000; in (b) the fund
decreases and is completely exhausted after
2 years.
17 K(t) =
(a) Capital remains constant, at the value c, for all
time.
(b) Capital grows without bound.
(c) Capital decreases to zero.
Appendix 1
1 Slope of chord =
=
= 10 +∆x
Slope of tangent = lim
∆ x →0
(10 +∆x) = 10
25 + 10∆x + (∆x)
2
− 25
∆x
(5 +∆x)
2
− 5
2
∆x
(ac − b)e
a
2
/2
+ b
a
D
F
t − 2
t − 4
A
C
D
F
t + 4
t + 2
A
C
D
F
c
m
A
C
α
β
dY
dt
dY
dt
Solutions to Problems
661
MFE_Z02.qxd 16/12/2005 10:51 Page 661
2 (a) Slope of chord
=
= 8x + 4∆x − 9
Slope of tangent = lim
∆ x →0
(8x + 4∆x − 9) = 8x − 9
(b) f (x +∆x) − f (x) =−
=
=
Slope of chord =
Slope of tangent = lim
∆ x →0
=
3 (a) (a + b)(a + b)
2
= (a + b)(a
2
+ 2ab + b
2
)
= a
3
+ 3a
2
b + 3ab
2
+ b
3
(b) Slope of chord =
= 3x
2
+ 3x(∆x) + (∆x)
2
Slope of tangent
= lim
∆ x →0
(3x
2
+ 3x∆x + (∆x)
2
)
= 3x
2
Appendix 2
1 (a) 2 × 1
2
+ 3 × 2
2
= 2 + 12 = 14
(b) 4x + 6y = 0
6y =−4x
==−
2x
3y
−4x
6y
dy
dx
dy
dx
dy
dx
(x +∆x)
3
+ x
3
∆x
−2
x
3
D
F
−2x −∆x
(x +∆x)
2
x
2
A
C
−2x −∆x
(x +∆x)
2
x
2
−2x∆x − (∆x)
2
(x +∆x)
2
x
2
x
2
− (x +∆x)
2
(x +∆x)
2
x
2
1
x
2
1
(x +∆x)
2
4(x +∆x)
2
− 9(x +∆x) + 1] − (4x
2
− 9x + 1)
∆x
so when x = 1, y = 2,
=−
2 (a) − ; (b) − ; (c) − ;
(d) ; (e) ; (f) −(1 + x + y).
Appendix 3
1 H =
The first principal minor is 2 > 0.
The second principal minor is 2 × 2 − 0 × 0 = 4 > 0.
Hence the stationary point is a minimum.
2 H =
Principal minors are −4 < 0 and 4 > 0, respectively, so
maximum.
3 H =
assuming that L is chosen as the first variable.
At the stationary point,
H =
Principal minors are −1/16 < 0 and 1/4608 > 0,
respectively, so maximum.
4 H
¯
= has determinant 8 > 0, so maximum.
5 H
¯
= has determinant −6 < 0, so minimum.
J
K
K
L
4 −1 −1
−1 0−1
−1 −10
G
H
H
I
J
K
K
L
−22−1
2 −2 −1
−1 −10
G
H
H
I
J
K
L
−1/16 0
0 −1/288
G
H
I
J
K
L
−4L
−3/2
0
0 −6K
−3/2
G
H
I
J
K
L
−4 −2
−2 −2
G
H
I
J
K
L
20
02
G
H
I
2x + 2y
2
3 − 4xy
(1 − e
x
)y
e
x
− x − 2y
e
x−y
2
1 + 6x
2
3y
x
y
1
3
dy
dx
Solutions to Problems
662
MFE_Z02.qxd 16/12/2005 10:51 Page 662
Adjustment coefficient The constant of proportionality in the simple macroeconomic
model in which the rate of change of national income is assumed to be proportional to excess
expenditure.
Algebraic fraction Ratio of two expressions: p(x)/q(x) where p(x) and q(x) are algebraic
expressions such as ax
2
+ bx + c or dx + e.
Annual percentage rate (APR) This is the equivalent annual interest paid for a loan, taking
into account the compounding over a variety of time periods.
Annual rate of inflation The percentage increase in the level of prices over a 12-month
period.
Annuity A lump-sum investment designed to produce a sequence of equal regular payments
over time.
Anti-derivative A function whose derivative is a given function.
Arbitrary constant A letter representing an unspecified constant in the general solution of a
differential equation.
Arc elasticity Elasticity measured between two points on a curve.
Arithmetic progression A sequence of numbers with a constant difference between consec-
utive terms; the nth term takes the form a + bn.
Autonomous consumption and savings The levels of consumption and savings when there
is no income.
Autonomous consumption multiplier The number by which you multiply the change in
autonomous consumption to deduce the corresponding change in, say, national income.
Average cost Total cost per unit of output: AC = TC/Q.
Average product of labour Output per worker: AP
L
= Q/L.
Average revenue Total revenue divided by quantity: TR/Q = PQ/Q = P.
Balanced budget multiplier The number by which you multiply the change in government
expenditure to deduce the corresponding change in, say, national income, assuming that this
change is financed entirely by a change in taxation.
Bordered Hessian matrix A Hessian matrix augmented by an extra row and column con-
taining partial derivatives formed from the constraint in the method of Lagrange multipliers.
Capital Man-made assets used in the production of goods and services.
Glossary
MFE_Z03.qxd 16/12/2005 10:52 Page 663
Chord A straight line joining two points on a curve.
Cobb–Douglas production function A production function of the form Q = AK
α
L
β
.
Coefficient A numerical multiplier of the variables in an algebraic term, such as the numbers
4 and 7 in the expression, 4x + 7yz
2
.
Cofactor The cofactor of the element a
ij
is the determinant of the matrix left when row i
and column j are deleted, multiplied by +1 or −1, depending on whether i + j is even or odd,
respectively.
Column vector A matrix with one column.
Comparative statics Examination of the effect on equilibrium values due to changes in the
parameters of an economic model.
Complementary function of a difference equation The solution of the difference equation
Y
t
= bY
t−1
+ c when the constant c is replaced by zero.
Complementary function of a differential equation The solution of the differential equation
= my + c when the constant c is replaced by zero.
Complementary goods A pair of goods consumed together. As the price of either goes up,
the demand for both goods goes down.
Compound interest The interest that is added on to the initial investment, so that this will
itself gain interest in subsequent time periods.
Constant of integration The arbitrary constant that appears in an expression when finding
an indefinite integral.
Constant returns to scale Exhibited by a production function when a given percentage
increase in input leads to the same percentage increase in output: f(λK, λL) =λf (K, L).
Constrained optimisation Maximisation (or mimimisation) of an objective function subject
to at least one constraint.
Constraint A restriction on the values of an objective function.
Consumer’s surplus The excess cost that a person would have been prepared to pay for
goods over and above what is actually paid.
Consumption The flow of money from households to firms as payment for goods and
services.
Continuous compounding The limiting value when interest is compounded with ever-
increasing frequency.
Coordinates A set of numbers which determine the position of a point relative to a set of
axes.
Cramer’s rule A method of solving simultaneous equations, Ax = b, by the use of determin-
ants. The ith variable x
i
can be computed using det(A
i
)/det(A) where A
i
is the determinant of
the matrix obtained from A by replacing the ith column by b.
Cross-price elasticity of demand The responsiveness of demand for one good to a change in
the price of another: (percentage change in quantity) ÷ (percentage change in the price of the
alternative good).
Decreasing function A function, y = f(x), in which y decreases as x increases.
Decreasing returns to scale Exhibited by a production function when a given percentage
increase in input leads to a smaller percentage increase in output: f(λK, λL) =λ
n
f(K, L) where
0 < n < 1.
Definite integration The process of finding the area under a graph by subtracting the values
obtained when the limits are substituted into the anti-derivative.
dy
dt
Glossary
664
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Degree of homogeneity The number n in the relation f(λK,λL) =λ
n
f(K, L).
Demand function A relationship between the quantity demanded and various factors that
affect demand, including price.
Denominator The number (or expression) on the bottom of a fraction.
Dependent variable A variable whose value is determined by that taken by the independent
variables; in y = f(x), the dependent variable is y.
Derivative The gradient of the tangent to a curve at a point. The derivative at x = a is written
f ′(a).
Derived function The rule, f ′, which gives the gradient of a function, f, at a general point.
Determinant of a matrix A determinant can be expanded as the sum of the products of the
elements in any one row or column and their respective cofactors.
Difference equation An equation that relates consecutive terms of a sequence of numbers.
Differential equation An equation connecting derivatives of an unknown function.
Differentials Limiting values of incremental changes. In the limit the approximation
∆z ×∆x becomes dz =×dx where dz and dx are the differentials.
Differentiation The process or operation of determining the first derivative of a function.
Discount rate The interest rate that is used when going backwards in time to calculate the
present value from a future value.
Discounting The process of working backwards in time to find the present values from a
future value.
Discriminant The number, b
2
− 4ac, which is used to indicate the number of solutions of the
quadratic equation ax
2
+ bx + c = 0.
Disposable income Household income after the deduction of taxes and the addition of
benefits.
Distributive law The rule which states that a(b + c) = ab + ac, for any numbers a, b and c.
Dynamics Analysis of how equilibrium values vary over time.
Elastic demand Where the percentage change in demand is more than the corresponding
change in price: E > 1.
Elements The individual numbers inside a matrix. (Also called entries.)
Elimination method The method in which variables are removed from a system of simultan-
eous equations by adding (or subtracting) a multiple of one equation to (or from) a multiple
of another.
Endogenous variable A variable whose value is determined within a model.
Equilibrium (market) This state occurs when quantity supplied and quantity demanded are
equal.
Equilibrium value of a difference equation A solution of a difference equation that does not
vary over time; it is the limiting value of Y
n
as n tends to infinity.
Equilibrium value of a differential equation A solution of a differential equation that does
not vary over time; it is the limiting value of y(t) as t tends to infinity.
Euler’s theorem If each input is paid the value of its marginal product, the total cost of these
inputs is equal to total output, provided there are constant returns to scale.
Exogenous variable A variable whose value is determined outside a model.
Exponent A superscript attached to a variable; the number 5 is the exponent in the expres-
sion, 2x
5
.
∂z
∂x
∂z
∂x
Glossary
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Exponential form A representation of a number which is written using powers. For example,
2
5
is the exponential form of the number 32.
Exponential function The function f(x) = e
x
; an exponential function in which the base is the
number e = 2.718 28...
External demand Output that is used by households.
Factors of production The inputs to the production of goods and services: land, capital,
labour and raw materials.
Feasible region The set of points which satisfy all of the constraints in a linear programming
problem.
Final demand An alternative to ‘external demand’.
First-order derivative The rate of change of a function with respect to its independent vari-
able. It is the same as the ‘derivative’ of a function, y = f(x), and is written as f ′(x)or dy/dx.
First principal minor The 1 × 1 determinant in the top left-hand corner of a matrix; the
element, a
11
of a matrix, A.
Fixed costs Total costs that are independent of output.
Flow chart A diagram consisting of boxes of instructions indicating the sequence of opera-
tions and their order.
Function (of one variable) A rule that assigns to each incoming number, x, a uniquely
defined outgoing number, y.
Function of two variables A rule which assigns to each pair of incoming numbers, x and y,
a uniquely defined outgoing number, z.
Future value The final value of an investment after one or more time periods.
General solution of a difference equation The solution of a difference equation that contains
an arbitary constant. It is the sum of the complementary function and particular solution.
General solution of a differential equation The solution of a differential equation that con-
tains an arbitary constant. It is the sum of the complementary function and particular solution.
Geometric progression A sequence of numbers with a constant ratio between consecutive
terms; the nth term takes the form, ar
n−1
.
Geometric ratio The constant multiplier in a geometric series.
Geometric series A sum of the consecutive terms of a geometric progression.
Government expenditure The total amount of money spent by government on defence, edu-
cation, health, police, etc.
Gradient The gradient of a line measures steepness and is the vertical change divided by the
horizontal change between any two points on the line. The gradient of a curve at a point is that
of the tangent at that point.
Hessian A matrix whose elements are the second-order partial derivatives of a given function.
Homogeneous function A function with the property that when all of the inputs are multi-
plied by a constant, λ, the output is multiplied by λ
n
where n is the degree of homogeneity.
Identity matrix An n × n matrix, I, in which every element on the main diagonal is 1 and the
other elements are all 0. If A is any n × n matrix then AI = I = IA.
Implicit differentiation The process of obtaining dy/dx where the function is not given
explicitly as an expression for y in terms of x.
Income elasticity of demand The responsiveness of demand for one good to a change in
income: (percentage change in quantity) ÷ (percentage change in income).
Increasing function A function, y = f(x), in which y increases as x increases.
Glossary
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Increasing returns to scale Exhibited by a production function when a given percentage
increase in input leads to a larger percentage increase in output: f(λK, λL) =λ
n
f(K, L) where
n > 1.
Indefinite integration The process of obtaining an anti-derivative.
Independent variable A variable whose value determines that of the dependent variable; in
y = f(x), the independent variable is x.
Index Alternative word for exponent or power.
Index number The scale factor of a variable measured from the base year multiplied by 100.
Indifference curve A curve indicating all combinations of two goods which give the same
level of utility.
Indifference map A diagram showing the graphs of a set of indifference curves. The further
the curve is from the origin, the greater the level of utility.
Inelastic demand Where the percentage change in demand is less than the corresponding
change in price: E < 1.
Inferior good A good whose demand decreases as income increases.
Inflation The percentage increase in the level of prices over a 12-month period.
Initial condition The value of Y
0
(or y(0)) which needs to be specified to obtain a unique
solution of a difference (or differential) equation.
Input–output analysis Examination of how inputs and outputs from various sectors of the
economy are matched to the total resources available.
Integer programming A linear programming problem in which the search for solution is
restricted to points in the feasible region with whole-number coordinates.
Integral The number
b
a
f(x)dx (definite integral) or the function
f(x)dx (indefinite integral).
Integration The generic name for the evaluation of definite or indefinite integrals.
Intercept Points where a graph crosses one of the coordinate axes.
Intermediate output Output from one sector which is used as input by another (or the same)
sector.
Internal rate of return (IRR) The interest rate for which the net present value is zero.
Inverse function A function, written f
−1
, which reverses the effect of a given function, f,so
that x = f
−1
(y) when y = f(x).
Inverse matrix A matrix A
−1
with the property that A
−1
A = I = AA
−1
.
Inverse (operation) The operation that reverses the effect of a given operation and takes you
back to the original. For example, the inverse of halving is doubling.
Investment The creation of output not for immediate consumption.
Investment multiplier The number by which you multiply the change in investment to
deduce the corresponding change in, say, national income
Isocost curve A line showing all combinations of two factors that can be bought for a fixed
cost.
IS schedule The equation relating national income and interest rate based on the assumption
of equilibrium in the goods market.
Isoquant A curve indicating all combinations of two factors that give the same level of output.
L-shaped curve A term used by economists to describe the graph of a function, such as
f(x) = a + , which bends roughly like the letter L.
b
x
Glossary
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Labour All forms of human input to the production process.
Labour productivity Output per worker: AP
L
= Q/L.
Lagrange multiplier The number λ which is used in the Lagrangian function. In economics
this gives the change in the value of the objective function when the value of the constraint is
increased by 1 unit.
Lagrangian The function f(x, y) +λ[M −ϕ(x, y)] , where f(x, y) is the objective function and
φ(x, y) = M is the constraint. The stationary point of this function is the solution of the associ-
ated constrained optimization problem.
Laspeyre index An index number for groups of data that are weighted by the quantities used
in the base year.
Law of diminishing marginal productivity (law of diminishing returns) Once the size of
the workforce exceeds a particular value, the increase in output due to a 1 unit increase in
labour will decline: d
2
Q/dL
2
< 0 for sufficiently large L.
Leontief inverse The inverse of I − A, where A is the matrix of technical coefficients.
Limits of integration The numbers a and b which appear in the definite integral,
b
a
f(x)dx.
Linear equation An equation of the form y = ax + b.
Linear programming Optimization of a linear objective function subject to linear
constraints.
LM schedule The equation relating national income and interest rate based on the assump-
tion of equilibrium in the money market.
Logarithm The power to which a base must be raised to yield a particular number.
Marginal cost The cost of producing 1 more unit of output: MC = d(TC)/dQ.
Marginal product of capital The extra output produced by 1 more unit of capital:
MP
K
=∂Q/∂K.
Marginal product of labour The extra output produced by 1 more unit of labour:
MP
L
=∂Q/∂L.
Marginal propensity to consume The fraction of a rise in national income which goes on
consumption. It is the slope of the consumption function: MPC = dC/dY.
Marginal propensity to consume multiplier The number by which you multiply the change
in MPC to deduce the corresponding change in, say, national income.
Marginal propensity to save The fraction of a rise in national income which goes into sav-
ings. It is the slope of the savings function: MPS = dS/dY.
Marginal rate of commodity substitution (MRCS) The amount by which one input needs
to increase to maintain a constant value of utility when the other input decreases by 1 unit:
MRTS =∂U/∂x
1
÷∂U/∂x
2
.
Marginal rate of technical substitution (MRTS) The amount by which capital needs to rise to
maintain a constant level of output when labour decreases by 1 unit: MRTS = MP
L
/MP
K
.
Marginal revenue The extra revenue gained by selling 1 more unit of a good:
MR = d(TR)/dQ.
Marginal utility The extra satisfaction gained by consuming 1 extra unit of a good: ∂U/∂x
i
.
Matrix A rectangular array of numbers, set out in rows and columns, surrounded by a pair
of brackets. (Plural matrices.)
Matrix of technical coefficients A square matrix in which element a
ij
is the input required
from the ith sector to produce 1 unit of output for the jth sector.
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Maximum (local) point A point on a curve which has the highest function value in com-
parison with other values in its neighbourhood; at such a point the first-order derivative is zero
and the second-order derivative is either zero or negative.
Maximum point (of a function of two variables) A point on a surface which has the highest
function value in comparison with other values in its neighbourhood; at such a point the sur-
face looks like the top of a mountain.
Method of substitution (for constrained optimization problems) The method of solving
constrained optimization problems whereby the constraint is used to eliminate one of the vari-
ables in the objective function.
Minimum (local) point A point on a curve which has the lowest function value in com-
parison with other values in its neighbourhood; at such a point the first-order derivative is zero
and the second-order derivative is either zero or positive
Minimum point (of a function of two variables) A point on a surface which has the lowest
function value in comparison with other values in its neighbourhood; at such a point the sur-
face looks like the bottom of a valley or bowl.
Modelling The creation of a piece of mathematical theory which represents (a simplification
of) some aspect of practical economics.
Money supply The notes and coins in circulation together with money held in bank deposits.
Monopolist The only firm in the industry.
Multiplier The number by which you multiply the change in an independent variable to find
the change in the dependent variable.
National income The flow of money from firms to households.
Natural logarithm A logarithm to base e; if M = e
n
then n is the natural logarithm of M.
Net investment Rate of change of capital stock over time: I = dK/dt.
Net present value (NPV) The present value of a revenue flow minus the original cost.
Nominal data Monetary values prevailing at the time that they were measured.
Non-negativity constraints The constraints x ≥ 0, y ≥ 0, etc.
Non-singular matrix A square matrix with a non-zero determinant.
Number line An infinite line on which the points represent real numbers by their (signed)
distance from the origin.
Numerator The number (or expression) on the top of a fraction.
Objective function A function that one seeks to optimize (usually) subject to constraints.
Optimization The determination of the optimal (usually stationary) points of a function.
Order of a matrix The dimensions of a matrix. A matrix with m rows and n columns has
order m × n.
Origin The point where the coordinate axes intersect.
Paasche index An index number for groups of data which are weighted by the quantities
used in the current year.
Parabola The shape of the graph of a quadratic function.
Parameter A constant whose value affects the specific values but not the general form of a
mathematical expression, such as the constants a, b and c in ax
2
+ bx + c.
Partial derivative The derivative of a function of two or more variables with respect to one
of these variables, the others being regarded as constant.
Particular solution of a difference equation Any one solution of a difference equation such
as Y
t
= bY
t−1
+ c.
Glossary
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