Bahan Mata Kuliah Matematika Ekonomi Gratis Terbaru - Kosngosan Situs Anak Kost Mahasiswa Pelajar Course 2
FUNCTION IN ECONOMICS
(Course 2)
SYAIFUL HADI
DJAIMI BAKCE
NOVIA DEWI
JURUSAN AGRIBISNIS
FAKULTAS PERTANIAN
UNIVERSITAS RIAU
Objectives of mathematics for
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
(Polya, G. How to Solve It, 2nd ed, 1975)
Endogenous & Exogenous
Variables, constants,
= TR – TC
(identity)
parameters
Qd = Qs
Y = a + bX0 (behavioral equation)
(equilibrium condition)
Y: endogenous variable
X0: exogenous variable
a: constant
b: parameter / the coefficient of
exogenous variable X0
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
the property that for (x, y) any
x value determines more than
one value of y
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
variable (horizontal axis)
Specific Functions
Algebraic functions
Y = a0
(constant: fixed costs)
Y = a0 + a 1 X
(linear: S&D)
Y = a0 + a1X + a2X2
(quadratic: prod.)
Y = a0 + a1X + a2X2 + a3X3
(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)
6
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
downward sloping.
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
80
70
60
50
P
AR = 723Q
40
30
20
10
0
Q
We next find and expression for TR:
TR = P . Q = (72 – 3Q) . Q = 72Q – 3Q 2
so,
Q
72Q
3Q^2
TR
0
2
0 144
0 12
0 132
4
288
48
240
6
432
108
324
8
576
192
384
10
720
300
420
12
14
864 1,008
432
588
432
420
16
1,152
768
384
18
1,296
972
324
20
22
1,440 1,584
1,200 1,452
240
132
TR = 72Q – 3Q2
24
1,728
1,728
-
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
VC = TC – FC
AC = TC/Q
AVC = VC/Q
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
AVC= VC/Q = (45Q - Q2 + 0.4Q3) / Q = 45 – Q + 0.4Q2
TC = 120 + 45Q – Q2 +
0.4Q3
AC = 120/Q + 45 – Q + 0.4
Q2
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 – Q 2 +
0.4Q3
And faces a demand curve given by: P = 240 – 20Q
What is its profit function ?
TR = P . Q = (240 – 20Q) . Q = 240Q – 20Q 2
= TR – TC
= (240Q – 20Q2) – (120 + 45Q – Q2 + 0.4Q3)
= -120 + 195Q – 19Q2 – 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.
Average product of labour (APL) = Q + L
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
1
2
3
2
96.8
374.4
813.6
3
214.2
813.6
1,733.4
4
374.4
1,395.2
2,908.8
6
813.6
2,908.8
5,767.2
12
2,908.8
8,870.4
13,737.6
14
3,802.4
10,819.2
14,464.8
16
4,761.6
12,492.8
13,363.2
16,000.0
14,000.0
12,000.0
Q
10,000.0
8,000.0
6,000.0
4,000.0
2,000.0
2
3
4
6
12
14
16
L
For K = 3, we have:
1400
Q = 25(3L) – 0.4(3L)
2
1200
3
1000
APL
= 225L2 – 10.8L3
APL=225L – 10.8L2
800
600
400
200
APL = Q/L = 225L – 10.8L2
0
2
3
4
6
8
10
12
L
L
APL
2
406.8
3
577.8
4
727.2
6
961.2
8
1108.8
10
1170
The average product of labour
12
16
function
1144.8
835.2
16
Quis I
1.
2.
3.
Sketch the total cost function: TC = 300 + 40Q – 10Q2
+ Q3, write expressions for AC, FC, VC and AVC !
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 !
A firm in perfect competition sells it output at a price
Rp 12. Plot it total revenue function (TR) = 12Q !
(Course 2)
SYAIFUL HADI
DJAIMI BAKCE
NOVIA DEWI
JURUSAN AGRIBISNIS
FAKULTAS PERTANIAN
UNIVERSITAS RIAU
Objectives of mathematics for
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
(Polya, G. How to Solve It, 2nd ed, 1975)
Endogenous & Exogenous
Variables, constants,
= TR – TC
(identity)
parameters
Qd = Qs
Y = a + bX0 (behavioral equation)
(equilibrium condition)
Y: endogenous variable
X0: exogenous variable
a: constant
b: parameter / the coefficient of
exogenous variable X0
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
the property that for (x, y) any
x value determines more than
one value of y
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
variable (horizontal axis)
Specific Functions
Algebraic functions
Y = a0
(constant: fixed costs)
Y = a0 + a 1 X
(linear: S&D)
Y = a0 + a1X + a2X2
(quadratic: prod.)
Y = a0 + a1X + a2X2 + a3X3
(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)
6
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
downward sloping.
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
80
70
60
50
P
AR = 723Q
40
30
20
10
0
Q
We next find and expression for TR:
TR = P . Q = (72 – 3Q) . Q = 72Q – 3Q 2
so,
Q
72Q
3Q^2
TR
0
2
0 144
0 12
0 132
4
288
48
240
6
432
108
324
8
576
192
384
10
720
300
420
12
14
864 1,008
432
588
432
420
16
1,152
768
384
18
1,296
972
324
20
22
1,440 1,584
1,200 1,452
240
132
TR = 72Q – 3Q2
24
1,728
1,728
-
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
VC = TC – FC
AC = TC/Q
AVC = VC/Q
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
AVC= VC/Q = (45Q - Q2 + 0.4Q3) / Q = 45 – Q + 0.4Q2
TC = 120 + 45Q – Q2 +
0.4Q3
AC = 120/Q + 45 – Q + 0.4
Q2
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 – Q 2 +
0.4Q3
And faces a demand curve given by: P = 240 – 20Q
What is its profit function ?
TR = P . Q = (240 – 20Q) . Q = 240Q – 20Q 2
= TR – TC
= (240Q – 20Q2) – (120 + 45Q – Q2 + 0.4Q3)
= -120 + 195Q – 19Q2 – 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.
Average product of labour (APL) = Q + L
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
1
2
3
2
96.8
374.4
813.6
3
214.2
813.6
1,733.4
4
374.4
1,395.2
2,908.8
6
813.6
2,908.8
5,767.2
12
2,908.8
8,870.4
13,737.6
14
3,802.4
10,819.2
14,464.8
16
4,761.6
12,492.8
13,363.2
16,000.0
14,000.0
12,000.0
Q
10,000.0
8,000.0
6,000.0
4,000.0
2,000.0
2
3
4
6
12
14
16
L
For K = 3, we have:
1400
Q = 25(3L) – 0.4(3L)
2
1200
3
1000
APL
= 225L2 – 10.8L3
APL=225L – 10.8L2
800
600
400
200
APL = Q/L = 225L – 10.8L2
0
2
3
4
6
8
10
12
L
L
APL
2
406.8
3
577.8
4
727.2
6
961.2
8
1108.8
10
1170
The average product of labour
12
16
function
1144.8
835.2
16
Quis I
1.
2.
3.
Sketch the total cost function: TC = 300 + 40Q – 10Q2
+ Q3, write expressions for AC, FC, VC and AVC !
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 !
A firm in perfect competition sells it output at a price
Rp 12. Plot it total revenue function (TR) = 12Q !