SM��� Mathematical Methods for Economics Fall ���� Uhan

Lesson ��. More Economic Applications of Linear Systems

� Overview

● In this lesson/worksheet, you will solve two types of economicmodels using the techniques for solving systemsof linear equations we covered in Lessons �� and ��.

� Marketmodels

● Consider the following two commoditymarketmodel:

D� = S� (�)D� = �� − �P� + P� (�)S� = −� + �P� (�)

D� = S� (�)D� = �� + P� − P� (�)S� = −� + �P� (�)

where

D� = demand for product � D� = demand for product �S� = supply for product � S� = supply for product �P� = price for product � P� = price for product �

● Look at equations (�) and (�). Explainwhy product � and product � are substitutes. (See Lesson �� for a refresher.)

● We want to �nd the equilibrium prices P� and P� in thismarket.

● First, simplify the system (�)-(�) above. Because of equation (�), you can set the right hand sides of equations (�)and (�) equal to each other. You should obtain an equation with � variables: P� and P�. Simplify the equation bycollecting terms and putting all the P� and P� terms on the le�, and the constant on the right.

(A)

● Do the same with equations (�), (�) and (�):

(B)

�

Pat ⇒ DIT } Therefore , product I and product 2

RT ⇒ Dat are substitutes .

10 - ZP, t Pz = - Z t 3 P,

⇒ - 5P,+ Pa = - 12

15 t P, - Pz = - I t 2Pa

⇒ P,- 3Pa = - 16

● Putting together the equations you found in (A) and (B), you should end up with the following system ofequations: −�P� + P� = −��

P� − �P� = −�� (C)

(Did you get the same equations? Youmay have the same equations, butmultiplied by −�, which is OK.)

● Rewrite the system (C) in matrix form AX = B:

A = X = B =

● Now, use Cramer’s rule to solve this system and �nd the equilibrium prices:

● You should �nd that the equilibrium prices are P� = ��� and P� = ��

� .

� Amodel for national income

● Consider the following national incomemodel:

Y = C + I� +G� (�)C = a + bY (� < b < �) (�)

where

Y = national incomeC = consumer expenditureI� = business expenditure (i.e., investment)G� = government expenditure

(We saw amore complicated version of thismodel back in Lesson �.)

● Equation (�) says that national income equals total expenditure by consumers, business, and government.

�

iii. I l :L t.in

aa:# at *÷

= Fa = 2¥ = Fa -- 4¥

● What does equation (�) say about the relationship between consumer expenditure and total national income?

● Now suppose that I� = �, G� = �, a = �, and b = �� . �en equations (�) and (�) become

Y = C + �� (�)

C = � + ��Y (��)

● We want to solve for the national income Y and consumer expenditures C.

● First, simplify equations (�) and (��) by putting the Y and C terms on the le�, and the constants on the right:

(D)

● Rewrite the system of equations you wrote in (D) in matrix form AX = B:

A = X = B =

● Now, use Cramer’s rule to solve this system and �nd the national income and consumer expenditure:

● You should �nd that the national income Y = ��� and consumer expenditure C = ��

� .

�

Consumer expenditure depends directly on total national income :

as T T,then C T

Y - C = 13

- TT t C = 4

E. it til lil13

* i÷i÷. at'÷i¥÷= ¥ = Iz = 25¥ = Iz

� Exercises

Problem �. Consider the three commoditymarketmodel given by

D� = S� D� = S� D� = S�D� = � − P� + �P� + �P� D� = � + �P� − P� + �P� D� = � − �P� + �P� − P�S� = −� + P� S� = −� + P� S� = −� + P�

Using a similarmethod to the one outlined in Section � of this worksheet, simplify the abovemodel into a systemof � linear equations and � variables P�, P�, and P�. Solve this system to �nd the equilibrium prices P�, P�, and P� byforming the augmentedmatrix of the system and �nding the RREF.

Problem �. Suppose that in a national income model as in Section �, we have I� = ��, G� = �, a = �, b = �� . Use

Cramer’s rule to �nd Y and C.

�

D setting D,-- Si

,etc . :

2 - P, t 213+213=-2+17 - 219 213=-4

5-24 - Pgp!2% I,:P. } ⇒ -24+-22%7213=-6

5- 2P,t Pz B

-LP,t 2Pa - 2B =

- 8

Augmented matrix :

i÷÷÷÷¥l÷÷÷÷¥i:÷÷÷⇒i÷÷¥i: :÷÷÷t÷÷÷,¥3,1 ! ! ! II ) so

, 13=-5,

Pa -- I,13=1

RREF

② Io -

- 12, Go -- 4

,a -- I,

b -- I,

Model : Y = Ct 16

C -- it # y } ⇒Y - c -- 16

- YY t C = I

Using Cramer 's rule :

" t:÷ii÷i¥=¥a- ¥÷iit÷=÷i¥