NON LINEAR FUNCTION

Definition

The power of the variable in the function is not 1 (it can be more or less than 1)

There are 4 types of non-linear functions that used to be discussed in the economic analysis. Those are:a. Quadratic Functionb. Cubic Functionc. Exponential Functiond. Logarithmic Function

Find by your self the explanation of it. Give example for each function (do it as homework)

Economics Application

1. Demand, Supply and Market Equilibrium. The analysis is almost the same with the

analysis in the linear function Market equilibrium still shown by Qd = Qs The influence of Tax and Subsidy (changing

the selling price offered by the producer) Supply func.change market equilibrium change.

Tax leads to the increasing of equilibrium price & the decreasing of equilibrium quantity

Subsidy leads to the decreasing of eq. price & the increasing of eq. quantity

Example

a. Suppose Demand & Supply function of a product: Qd = 19 – P2

Qs = -8 + 2P2

Determine the Equilibrium P and Q.Answer: Qd = Qs

19 – P2 = -8 + 2P2

27 = 3P2

P2 = 9 P = 3 Q = 19 – P2 = 19 – (32) = 10 P = 3 ; Q = 10

Example

b. If for that product, specific tax is imposed at the amount of Rp. 1/unit, then:

Answer. Q’s = -8 + 2(P-1)2

= -8 + 2(P2-2P+1)= -8 + 2P2 – 4P + 2= -6 + 2P2 – 4P

The new market equilibrium:Qd = Q’s

Example

19 – P2 = -6 – 4P + 2P2

3P2 – 4P – 25 = 0 Using ABC formula, we find: P1 = 3,63 and

P2 = -2,30 (irrational)ABC Formula:

P1,2 = (-b +/- √b2 – 4ac)/2a

P = 3,63 Q’s = 5,82 In the imposition of TAX: P’e = 3,63 & Q’e =

5,82

Example

The tax that must be afforded by consumer per product unit:

Tc = P’e – Pe = 3,63 – 3 = Rp. 0,63 The tax that must be afforded by

producer per product unit:Tp = t – tk = 1 – 0,63 = Rp. 0,37

The tax that is received by the government:

T = Q’e x t = 5,82 x 1 = Rp. 5,82

Example

2. If we know that:Qd = 40 – P2

Qs = -60 + 3P2

Calculate the equilibrium P & Q. If to that product, government imposes tax

at the amount of Rp.2/unit:a. Find the new equilibrium P & Qb. How much the portion of tax that afforded by consumer & producer?c. How much the total tax that received by government?

Example

a. Qs = -60 + 3P2

3P2 = Qs + 60P = (√1/3Qs + 20)

After t = 2;P = (√1/3Qs + 20) + 2P – 2= (√1/3Qs + 20) (P – 2)2 = 1/3Qs + 20P2 – 4P + 4 = 1/3Qs + 201/3Qs = P2 – 4P - 16Q’s = 3P2 – 12P - 48

Example

Using ABC formula:P1,2 = (3 +/- (√9 + 88))/2

= (3 +/- 9,8)/2

P1 = 6,4P2 = -3,4 (irrational)

Cost Function

There are some specific terms in Cost function, i.e:a. Fixed cost : FC = k (k: Constant)b. Variable cost : VC = f(Q)c. Total cost : TC = FC + VC = k + f(Q)d. Avr. Fix. Cost : AFC = FC/Qe. Avr. Var. Cost : AVC = VC/Qf. Avr. Cost : AC = TC/Q = AFC + AVCg. Marginal Cost: MC = ∆C/∆Q

Example

a. The Total Cost of a firm can be illustrated as:TC = 2Q2 – 24Q + 102- At how many production unit, the total cost of that firm is stated as minimum?- In that production unit, Calculate the minimum TC- Calculate also the FC, VC, AFC & AVC- If there is an increasing in the production at the amount of 1 unit, how much the marginal cost?

Example

Using the formula of parabola extreme point, TC is minimum when: Q = -b/2a = 24/4 = 6 units.

TC min = 2Q2 – 24Q + 102= 2(6)2 – 24(6) + 102= 30

When Q = 6 and TC = 2Q2 -24Q + 102, then:FC = 102VC = 2Q2 -24Q = 2(6)2 – 24(6) = -72AC = TC/Q = 30/6 = 5AFC = FC/Q = 102/6 =17AVC = VC/Q = -72/6 = -12

Example

If Q increases by 1 unit, then Q = 7 and TC = 2(7)2 – 24(7) + 102 = 32MC = ∆C/∆Q = (32 – 30)/(7-6) = 2

So, in order to increase production from 6 to 7, that firm needs the marginal cost of 2.

Example

b. The Total Cost of a firm can be illustrated as:TC = 5Q2 – 1000Q + 85000- How much the TC if firm produce 90 units of outputs?- At how many production unit, the total cost of that firm is stated as minimum?- In that production unit, Calculate the minimum TC- Calculate also the FC, VC, AFC & AVC- How much the marginal cost?