Department

Economics for Management StudiesCourse Code

MGEA06H3Professor

Iris AuLecture

14This

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Ch 10 - SOLUTIONS to Supplemental Notes and Exercises #1

1) Keynesian Cross & the Construction of the IS Curve

a) Suppose C

0

= 100, C

1

= 0.6, I

0

= 500, I

1

= 50, and T

0

= G

0

= 500. What is the planned

expenditure function? If r = 4 (percent) what is the resulting equilibrium output? If r = 6

(percent) what is the resulting equilibrium output (in the goods and services market)?

E = [ C

0

+ I

0

+ G

0

- C

1

T

0

] + C

1

Y - I

1

r Planned Expenditure Function (given r)

E = 800 + 0.6Y - 50r

When r = 4 (percent), E = 600 + 0.6Y

Equilibrium, Y = E, implies Y = 600 + 0.6Y

Y* = (600/0.4) = 1,500

When r = 6 (percent), E = 500 + 0.6Y

Equilibrium, Y = E, implies Y = 500 + 0.6Y

Y* = (500/0.4) = 1,250

If C

1

is decreased to C

1

= 0.5, then

E = 850 - 50r + 0.5Y

When r = 4 (percent), E = 650 + 0.5Y

Equilibrium, Y = E, implies Y = 650 + 0.5Y

Y* = (650/0.5) = 1,300

Note, here the IS Multipliers are: δY/δG = 1/[1 - C

1

] = 2.5 = δY/δC

0

= δY/δI

0

δY/δT = - C

1

/[1 - C

1

] = - 1.5

δY/δG = 1 (when δT = δG, balanced budget change)

These expressions are for the case when C

1

= 0.6. Note, when C

1

= 0.5 the last three

expressions above become 2, -1 and 1 respectively.

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2

b) Suppose r = 4 percent solve for the equilibrium level of aggregate real output when

T = tY, where 0 < t <1 is the income tax rate. What is the slope of the IS curve and the

multipliers? i.e., T is endogenous now (determined by/dependent on Y) and all of

the other inputs are the SAME as in part (a)

E = [ C

0

+ I

0

+ G

0

] - C

1

tY + C

1

Y - I

1

r Planned Exp Function (when r is given)

Equilibrium, Y = E, Y = [ C

0

+ I

0

+ G

0

] + C

1

(1-t)Y - I

1

r

Y( 1 - C

1

(1-t) ) = [ C

0

+ I

0

+ G

0

] - I

1

r

So, Y* = (1/[ 1 - C

1

(1-t) ])

[ C

0

+ I

0

+ G

0

] - (I

1

/[ 1 - C

1

(1-t) ])

r

IS Curve

Suppose that t = 0.4 and as before r = 4 (percent), C

0

= 100, C

1

= 0.6, I

0

= 500, I

1

= 50, and G

0

= 500.

Now the IS Curve slope = - [ 1 - C

1

(1-t) ]/(I

1

) = - 0.0128 (in part (a) it was - 0.008)

i.e. the IS Curve has become steeper

r = 4 (percent)

E = 900 + 0.36Y Planned Exp Function

Equilibrium, Y = E, Y = 900 + 0.36Y

Y* = (900/0.64) = 1,406.25

i.e. the equilibrium level of Y has fallen

IS Multipliers: δY/δG = (1/[ 1 - C

1

(1-t) ]) = 1.5625 = δY/δI

0

= δY/δC

0

δY/δt = (- C

1

/[ ( 1 - C

1

(1-t) )

2

])[ C

0

+ I

0

+ G

0

] - (- C

1

I

1

/[ ( 1 - C

1

(1-t) )

2

])r

= (-0.6/(0.64

2

))( 900) + 120/(0.64

2

) = - 1,025.39

(i.e. effect of 100% tax increase)

c) If C = C

0

+ C

1

(Y - T) - C

2

r, what is the IS Curve, its slope and the multipliers? Again all

other inputs are the SAME as in part (a).

E = [ C

0

+ I

0

+ G

0

- C

1

T

0

] + C

1

Y - C

2

r - I

1

r Planned Exp Function (when given r)

Equilibrium, Y = E,

Y = (1/[1 - C

1

])

[ C

0

+ I

0

+ G

0

- C

1

T

0

] - ({I

1

+ C

2

}/[1 - C

1

])

r

IS’ Curve

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3

Note: (1) New slope = - ( [1 - C

1

]/{I

1

+ C

2

}) i.e. flatter IS Curve (as C

2

> 0)

(2) Same Multipliers as in part (a)

Flatter IS curve

Implies a lower Y in part (c) vs. part (a) (i.e.

for any given r and the same values of the

other parameters)

d) If NX = NX

0

- NX

1

Y, what is the IS Curve, its slope and the multipliers? (Later you can

let NX

1

=0.25). Again all other inputs are the same as in part (a), for example same

parameter values and NX

0

= 0 and NX

1

= 0.25.

E = [ C

0

+ I

0

+ G

0

+ NX

0

- C

1

T

0

] + C

1

Y - NX

1

Y - I

1

r Planned Exp Function

Y = (1/[1 + NX

1

- C

1

])

[ C

0

+ I

0

+ G

0

+ NX

0

- C

1

T

0

] - ( I

1

/[1 + NX

1

- C

1

])

r

IS’ Curve

New IS curve slope = - ( {1 + NX

1

- C

1

}/I

1

) = - 0.013 IS steeper than in part (a)

Government Spending Multiplier = δY/δG = (1/[1 + NX

1

- C

1

]) = 1.5385 lower than

part (a)

You will note that this is the SAME result that is referred to in Chapter 11 of the textbook

(i.e. smaller multipliers in the open economy). In the text, the point is that the full

multiplier in equilibrium is smaller now than in the past due to increased globalization

(i.e. more openness and greater economic integration internationally).

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