FUNCTION IN ECONOMICS
(Course 2)
JURUSAN AGRIBISNIS
FAKULTAS PERTANIAN
UNIVERSITAS RIAU
SYAIFUL HADI
DJAIMI BAKCE
Objectives of mathematics for
Objectives of mathematics for
economists
economists
To understand mathematical economics
problems by being able to state the unknowns,
the data and the conditions
To plan solutions to these problems by finding
a connection between the data and the
unknown
To carry out your plans for solving
mathematical economics problems
To examine the solutions to mathematical
economics problems for general insights into
current and future problems
Endogenous & Exogenous
Endogenous & Exogenous
Variables, constants,
Variables, constants,
parameters
parameters
= TR – TC (identity)
Q
d
= Q
s(equilibrium condition)
Y = a + bX
0
(behavioral equation)
Y: endogenous variable
X
0
:
exogenous variable
a: constant
b: parameter / the coefficient of
Functions and Relations
Functions and Relations
Function:
a set or ordered
pairs with the property that for
(x, y) any x value uniquely
determines a single y value
Relation:
ordered pairs with
General Functions
General Functions
Y = f (X)
Y is value or dependent variable
(vertical axis)
f is the function or a rule for
mapping X into a unique Y
X is argument or the independent
Specific Functions
Specific Functions
Algebraic functions
Y = a0 (constant: fixed costs)
Y = a0+ a1 X (linear: S&D)
Y = a0 + a1X + a2X2 (quadratic: prod.)
Y = a0 + a1X + a2X2 + a
3X3 (cubic: t. cost)
Y = a/X (hyperbolic: indiff.)
Y = aXb (power: prod. fn)
lnY = ln(a) + b ln(X) (logarithmic: easier)
Transcendental functions
Y = aX (exponential: interest)
(Chiang & Wainwright, p. 22, Fig. 2.8)
TOTAL AND AVERAGE REVENUE
Total revenue (TR) is price (P) multiplied by
quantity (Q)
TR = P . Q
Average revenue (AR) per unit of output is TR + Q
= P
AR = TR/Q
A market demand curve is assumed to be
If average revenue is given by: P = 72 – 3Q
Sketch this function and also, on a separate graph, the total revenue function.
The average revenue function has P on the vertical axis and Q on the horizontal axis. The general form of linier function is y = a + bx. Comparing our average revenue function we see that it take this linier form with y = P, a = 72, b = -3 and x = Q. We therefore need find only two points on our function to sketch the line and can the extend it as required. For simplicity we choose Q = 0 and Q = 10. The corresponding P values are listed, the two points are plotted and the line is extended to the horizontal axis.
Chosen value Q = 0 and Q = 10
substituting in P = 72 – 3(0) = 72 and P = 72 – 3(10)= 42
0 10 20 30 40 50 60 70 80 Q
P AR =
We next find and expression for TR:
TR = P . Q = (72 – 3Q) . Q = 72Q – 3Q2
so,
Q 0 2 4 6 8 10 12 14 16 18 20 22 24
72Q 0 144 288 432 576 720 864 1,008 1,152 1,296 1,440 1,584 1,728 3Q^2 0 12 48 108 192 300 432 588 768 972 1,200 1,452 1,728 TR 0 132 240 324 384 420 432 420 384 324 240 132
TOTAL AND AVERAGE
COST
A firm’s total cost of production (TC) depends on its
output (Q).
The TC function may include a constant term, which
represent fixed cost (FC).
The part of total cost that varies with Q is called variable
cost (VC).
We have, then, that TC = FC + VC
Remember: FC is the constant term in TC
For a firm with total cost given by: TC = 120 + 45Q – Q2 + 0.4Q3
Identify it AC, FC, VC and AVC functions. List some values of TC and AC, correct to the nearest integer. Sketch the total cost
function and on a separate graph the AC function. TC = 120 + 45Q - Q2 + 0.4Q3
AC = TC/Q = 120/Q + 45 – Q + 0.4Q2
FC = 120 (the constant term in TC)
VC = TC – FC = (120 + 45Q - Q2 + 0.4Q3) – (120) = 45Q - Q2 +
0.4Q3
TC = 120 + 45Q – Q2 +
0.4Q3
PROFIT
Profit is difference between a firm’s total revenue and its
total costs.
Using the symbol as the variable name for profit we have
= TR – TC
A firm has the total cost function: TC = 120 + 45Q – Q2 +
0.4Q3
And faces a demand curve given by: P = 240 – 20Q What is its profit function ?
TR = P . Q = (240 – 20Q) . Q = 240Q – 20Q2
= TR – TC
= (240Q – 20Q2) – (120 + 45Q – Q2 + 0.4Q3)
PRODUCTION FUNCTIONS, ISOQUANTS
AND THE AVERAGE PRODUCTS OF
LABOUR
The long run production function shows that a firm’s
output (Q), depends on the amount of factors it employs (always assuming that whatever factor are employed are used efficiently)
If a production process involves the use of labour (L)
and capital (K), we write Q = f (L, K)
The dependent variable Q is function of two independent variables, L and K.
A firm has the production function Q = 25 (L . K)2 – 0.4(L . K)3. If
K = 1, find the value of Q for L = 2, 3, 4, 6, 12, 14 and 16.
Sketch this short run production function putting L and Q on the axes of your graph. Next suppose the value of K is increased to 2. On the same graph sketch the new short run production
function for the same values of L. Add one further production function to your sketch, corresponding to K = 3, using the same L values again.
For the short run production function with K = 3, find and plot
the average product of labour function.
K\L 2 3 4 6 12 14 16
0 200 400 600 800 1000 1200 1400
2 3 4 6 8 10 12 16
L
A
P
L
The average product of labour function -2,000.0 4,000.0 6,000.0 8,000.0 10,000.0 12,000.0 14,000.0 16,000.0
2 3 4 6 12 14 16
L
Q
For K = 3, we have: Q = 25(3L)2 – 0.4(3L)3
= 225L2 – 10.8L3
APL = Q/L = 225L – 10.8L2
L 2 3 4 6 8 10 12 16
APL 406.8 577.8 727.2 961.2 1108.8 1170 1144.8 835.2
Quis I
1. Sketch the total cost function: TC = 300 + 40Q – 10Q2
+ Q3, write expressions for AC, FC, VC and AVC !
2. If the firm in question 1 faces the demand curve P = 100 – 0.5Q
Find an expression for the firm’s profit function and sketch
the curve !
