Mathematics for Economics and Business 9e by Jacques

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Mathematics for Economics and Business 9e by Jacques is the 9th edition of the textbook authored by Ian Jacques and published by Pearson Education Limited, Harlow, United Kingdom in 2018.

  • Absolute value. The positive value or magnitude of a number.
  • 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 ax2 + bx + c or dx + e.
  • Annual percentage rate (APR). The equivalent annual interest paid for a loan, taking into account the compounding over a variety of time periods.
  • 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 consecutive terms; the nth term takes the form a + bn.
  • Autonomous consumption. The level of consumption 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: ∂Y/∂b.
  • Autonomous savings. The withdrawals from savings when there is no income.
  • Average cost. Total cost per unit of output: AC = TC/Q.
  • Average product of labour (labour productivity). Output per worker: APL = Q/L.
  • Average revenue. Total revenue per unit of output: AR = TR/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, ∂Y/∂G*, assuming that this change is financed entirely by a change in taxation.
  • Capital. Man-made assets used in the production of goods and services.
  • Chord. A straight line joining two points on a curve.
  • Closed interval. The set of all real numbers between and including two given numbers: a ≤ x ≤ b.
  • Cobb–Douglas production function. A production function of the form Q = AKa Lb.
  • Coefficient. A numerical multiplier of the variables in an algebraic term, such as the numbers 4 and 7 in the expression 4x + 7yz2.
  • Cofactor. The cofactor of the element aij 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 Yt = bYt−1 + c when the constant c is replaced by zero.
  • Complementary function of a differential equation. The solution of the differential equation dy dt 5 my 1 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 which is added on to the initial investment, so that this will itself gain interest in subsequent time periods.
  • Concave. Graph bends downwards when f″(x) < 0.
  • 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(lK, lL) = lf(K, L).
  • 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 function. The relationship between national income and consumption.
  • Continuous compounding. The limiting value when interest is compounded with ever-increasing frequency.
  • Continuous. The name given to a function which can be drawn without taking a pen off the paper. More formally, when lim f(x) 5 f(a) x a at all points in the domain.
  • Convex. Graph bends upwards when f″(x) > 0.
  • Coordinates. A set of numbers that 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 determinants. The ith variable xi can be computed using det(Ai)/det(A) where Ai 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).
  • Decision variable. The unknowns in a linear programming problem which can be controlled.
  • 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) = λn f(K, L) where 0 < n < 1.
  • Definite integral. The number #a b f(x)dx which represents the area under the graph of f(x) between x = a and x = b.
  • 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.
  • Degree of homogeneity. The number n in the relation f(lK, lL) = ln f(K, L).
  • Degree. The highest power in a polynomial.
  • 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. 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.
  • Difference of two squares. The algebraic result which states that a2 − b2 = (a + b)(a − b).
  • Differential equation. An equation connecting derivatives of an unknown function.
  • Differentials. Limiting values of incremental changes. In the limit the approximation Dz > −z −x 3 Dx becomes dz 5 −z −x 3 dx where dz and dx are the differentials.
  • Differentiation. The process or operation of determining the first derivative of a function.
  • Discontinuous. The name given to a function which is not continuous everywhere. The graph of the function has jumps or gaps.
  • 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 b2 − 4ac, which is used to indicate the number of solutions of the quadratic equation ax2 + bx + c = 0.
  • Disposable income. Household income after the deduction of taxes and the addition of benefits.
  • Distributive law. The law of arithmetic which states that a(b + c) = ab + ac, for any numbers a, b and c.
  • Domain. The numbers which are used as inputs to a function.
  • Dynamics. Analysis of how equilibrium values vary over time.
  • Economic ordering quantity. The quantity of a product that should be ordered so as to minimise the total cost that includes ordering costs and holding costs.
  • 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 simultaneous 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.
  • Equation. Equality of two algebraic expressions which is true only for certain values of the variable.
  • 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 Yn 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.
  • Equivalent fractions. Fractions which may appear different but which have the same numerical value.
  • 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 expression, 2x5.
  • Exponential form. A representation of a number which is written using powers. For example, 25 is the exponential form of the number 32.
  • Exponential function. The function f(x) = ex; an exponential function in which the base is the number e = 2.718 281 . . .
  • Factor. Part of an expression which, when multiplied by all the other factors, gives the complete expression.
  • Factorisation. The process of writing an expression as a product of simpler expressions using brackets.
  • Factors of production. The inputs into 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.
  • First-order derivative. The rate of change of a function with respect to its independent variable. It is the same as the 'derivative' of a function, y = f(x), and is written as f′(x) or dy/dx.
  • Fixed costs. Total costs that are independent of output.
  • Flow chart. A diagram consisting of boxes of instructions indicating a sequence of operations and their order.
  • Function. 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 arbitrary 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 contains an arbitrary constant. It is the sum of the complementary function and a particular solution.
  • Geometric progression. A sequence of numbers with a constant ratio between consecutive terms; the nth term takes the form, arn−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, education, 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.
  • Homogeneous function. A function with the property that when all of the inputs are multiplied by a constant, l, the output is multiplied by ln where n is the degree of homogeneity.
  • Identity. Equality of two algebraic expressions which is true for all values of the variable.
  • 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.
  • Improper integral. An definite integral representing the area of an unbounded region.
  • 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.
  • 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(lK, lL) = ln 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 Y0 (or y(0)) which needs to be specified to obtain a unique solution of a difference (or differential) equation.
  • 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 #a b 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. The points where a graph crosses one of the co-ordinate axes.
  • Internal rate of return (IRR). The interest rate for which the net present value is zero.
  • Interval. The set of all real numbers between (and possibly including) two given numbers.
  • 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.
  • 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.
  • 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: ∂Y/∂I*.
  • Isocost curve. A line showing all combinations of two factors which 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 which 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) 5 a 1 b x, which bends roughly like the letter L.
  • Labour. All forms of human input to the production process.
  • Labour productivity. Average output per worker: Q/L
  • Lagrange multiplier. The number l which is used in the Lagrangian function. In economics this gives the approximate 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) + l[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 associated constrained optimisation problem.
  • Laspeyre index. An index number for groups of data which 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: d2 Q/dL2 < 0 for sufficiently large L.
  • Law of diminishing marginal utility. The law which states that the increase in utility due to the consumption of an additional good will eventually decline: ∂2 U/∂xi 2 < 0 for sufficiently large xi.
  • Like terms. Multiples of the same combination of algebraic symbols.
  • Limited growth. Used to describe an economic variable which increases over time but which tends to a fixed quantity.
  • Limits of integration. The numbers a and b which appear in the definite integral, #a b f x( )dx.
  • Linear equation. An equation of the form y = dx + f.
  • LM schedule. The equation relating national income and interest rate based on the assumption of equilibrium in the money market.
  • Logarithm. The power to which a base must be raised to yield a particular number.
  • Lower limit. The number which appears at the bottom of the sigma notation to indicate the first term in a summation.
  • Marginal cost. The cost of producing 1 more unit of output: MC = d(TC)/dQ.
  • Marginal product of capital. The additional output produced by a 1-unit increase in capital: MPK = ∂Q/∂K.
  • Marginal product of labour. The additional output produced by a 1-unit increase in labour: MPL = ∂Q/∂L.
  • Marginal propensity to consume. The fraction of a rise in national income which goes into consumption: 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: ∂Y/∂a.
  • Marginal propensity to save. The fraction of a rise in national income which goes into savings: 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/∂x1 ÷ ∂U/∂x2.
  • 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 = MPL /MPK.
  • 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/∂xi.
  • Matrix. A rectangular array of numbers, set out in rows and columns, surrounded by a pair of brackets. (Plural matrices.)
  • Maximum (local) point. A point on a curve which has the highest function value in comparison with other values in its neighbourhood; at such a point the firstorder 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 surface looks like the top of a mountain.
  • Method of substitution. The method of solving constrained optimisation problems whereby the constraint is used to eliminate one of the variables in the objective function.
  • Minimum (local) point. A point on a curve which has the lowest function value in comparison with other values in its neighbourhood; at such a point the firstorder 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 surface looks like the bottom of a valley or bowl.
  • Minor. The name given to the cofactor before the '±' pattern is imposed.
  • Modelling. The creation of a piece of mathematical theory which represents (a simplification of) some aspect of practical economics.
  • Modulus. The positive value or magnitude of a number.
  • 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 = en 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.
  • Normal good. A good whose demand increases as income increases.
  • 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 is optimised in a linear programming problem.
  • Open interval. The set of all real numbers between but excluding two given numbers: a < x < b.
  • Optimisation. 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 ax2 + 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 Yt = Bt−1 + c.
  • Particular solution of a differential equation. Any one solution of a differential equation such as dy dt 5 my 1 c.
  • Perfect competition. A situation in which there are no barriers to entry in an industry where there are many firms selling an identical product at the market price.
  • Point elasticity. Elasticity measured at a particular point on a curve: E 5 3 P dQ Q dP.
  • Polynomial. A function of the form anxn + an−1xn−1 + . . . + a0.
  • Power. Another word for exponent. If this is a positive integer, then it gives the number of times a number is multiplied by itself.
  • Precautionary demand for money. Money held in reserve by individuals or firms to fund unforeseen future expenditure.
  • Present value. The amount that is invested initially to produce a specified future value after a given period of time.
  • Price elasticity of demand. A measure of the responsiveness of the change in demand due to a change in price: (percentage change in demand) ÷ (percentage change in price).
  • Price elasticity of supply. A measure of the responsiveness of the change in supply due to a change in price: (percentage change in supply) ÷ (percentage change in price).
  • Primitive. An alternative word for an anti-derivative.
  • Principal. The value of the original sum invested.
  • Producer's surplus. The excess revenue that a producer has actually received over and above the lower revenue that it was prepared to accept for the supply of its goods.
  • Production function. The relationship between the output of a good and the inputs used to produce it.
  • Profit. Total revenue minus total cost: π = TR − TC.
  • Quadratic function. A function of the form f(x) = ax2 + bx + c where a ≠ 0.
  • Range. The numbers which form the set of outputs from a function.
  • Real data. Monetary values adjusted to take inflation into account.
  • Rectangular hyperbola. A term used by mathematicians to describe the graph of a function, such as f(x) 5 a 1b x, which is a hyperbola with horizontal and vertical asymptotes.
  • Recurrence relation. An alternative term for a difference equation. It is an expression for Yn in terms of Yn−1 (and possibly Yn−2, Yn−3, etc.).
  • Reduced form. The final equation obtained when exogenous variables are eliminated in the course of solving a set of structural equations in a macroeconomic model.
  • Reverse flow chart. A flow chart indicating the inverse of the original sequence of operations in reverse order.
  • Row vector. A matrix with one row.
  • Saddle point. A stationary point which is neither a maximum nor a minimum and at which the surface looks like the middle of a horse's saddle.
  • Scale factor. The multiplier that gives the final value in percentage problems.
  • Second-order derivative. The derivative of the firstorder derivative. The expression obtained when the original function, y = f(x), is differentiated twice in succession and is written as f ″(x) or d2 y/dx2.
  • Second-order partial derivative. The partial derivative of a first-order partial derivative. For example, fxy is the second-order partial derivative when f is differentiated first with respect to y and then with respect to x.
  • Shadow price. The change in the optimal value of the objective function due to a 1-unit increase in one of the available resources.
  • Simple interest. The interest which is paid directly to the investor instead of being added to the original amount.
  • Simultaneous linear equations. A set of linear equations in which there are (usually) the same number of equations and unknowns. The solution consists of values of the unknowns which satisfy all of the equations at the same time.
  • Singular matrix. A square matrix with a zero determinant. A singular matrix fails to possess an inverse.
  • Sinking fund. A fixed sum of money saved at regular intervals which is used to fund some future financial commitment.
  • Slope of a line. Also known as the gradient, it is the change in the value of y when x increases by 1 unit.
  • Small increments formula. The result Dz > Dx 1 Dy −z −x −z −y .
  • Speculative demand for money. Money held back by firms or individuals for the purpose of investing in alternative assets, such as government bonds, at some future date.
  • Square matrix. A matrix with the same number of rows as columns.
  • Square root. A number that when multiplied by itself equals a given number; the solutions of the equation x2 = c which are written { x.
  • Stable (unstable) equilibrium. An economic model in which the solution of the associated difference (or differential) equation converges (diverges).
  • Statics. The determination of the equilibrium values of variables in an economic model which do not change over time.
  • Stationary point of inflection. A stationary point that is neither a maximum nor a minimum; at such a point both first- and second-order derivatives are zero.
  • Stationary points (critical points, turning points, extrema). Points on a graph at which the tangent is horizontal; at a stationary point the first-order derivative is zero.
  • Structural equations. A collection of equations that describe the equilibrium conditions of a macroeconomic model.
  • Substitutable goods. A pair of goods that are alternatives to each other. As the price of one good goes up, the demand for the other rises.
  • Superior good. A normal good for which the percentage rise in consumption exceeds the percentage increase in income.
  • Supply function. A relationship between the quantity supplied and various factors that affect supply, including price.
  • Tangent. A line that just touches a curve at a point.
  • Taxation. Money paid to government based on an individual's income and wealth (direct taxation) together with money paid by suppliers of goods or services based on expenditure (indirect taxation).
  • Time series. A sequence of numbers indicating the variation of data over time.
  • Total cost. The sum of the total variable and fixed costs: TC = TVC + FC.
  • Total revenue. A firm's total earnings from the sales of a good: TR = PQ.
  • Transactions demand for money. Money used for everyday transactions of goods and services.
  • Transpose (of a matrix). The matrix obtained from a given matrix by interchanging rows and columns. The transpose of a matrix A is written AT.
  • Transpose a formula. The rearrangement of a formula to make one of the other letters the subject.
  • U-shaped curve. A term used by economists to describe a curve, such as a parabola, which bends upwards, like the letter U.
  • Unbounded region. A feasible region that is not completely enclosed by a polygon. The associated linear programming problem may not have a finite solution.
  • Uniformly convergent sequence. A sequence of numbers which progressively increases (or decreases) to a finite limit.
  • Uniformly divergent sequence. A sequence of numbers which progressively increases (or decreases) without a finite limit.
  • Unit elasticity of demand. Where the percentage change in demand is the same as the percentage change in price: |E| = 1.
  • Unlimited growth. Used to describe an economic variable which increases without bound.
  • Unstable equilibrium. An economic model in which the solution of the associated difference (or differential) equation diverges.
  • Upper limit. The number which appears at the top of the sigma notation to indicate the last term in a summation.
  • Utility. The satisfaction gained from the consumption of a good.
  • Variable costs. Total costs that change according to the amount of output produced.
  • X axis. The horizontal coordinate axis pointing from left to right.
  • Y axis. The vertical coordinate axis pointing upwards.
  • Zero matrix. A matrix in which every element is zero.
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