Answers to Questions in the Scenario

Y = 104.8 + .6(Y - 599) + 450 + 657

Y = 104.8 + .6Y - 359.4 + 450 + 657

Y - .6Y = 104.8 - 359.4 + 450 + 657

(1 - .6)Y = 852.4

.4Y = 852.4 Y = 852.4/.4 = 2131

Or, you could begin with S + T = I + G, derive the savings function
from C = 104.8 + .6Yd, substitute and come up with the same answer. i.e.,

S + T = I + G

-104.8 + (1 - .6)(Y - 599) + 599 = 450 + 657

.4Y - .4(599) = 450 + 657 + 104.8 - 599

.4Y = 450 + 657 + 104.8 - 599 + 239.6

Y = 852.4/.4 = 2131

And of course, after having done this one way, you should do it the other way to check/verify your answer.

Y = 104.8 + .6(Y - 539.1) + 450 + 657

and recalculated Y to get 2221.

Y = 104.8 + .6Y - 323.46 + 450 + 657

.4Y = 888.34

Y = 888.34/.4 = 2220.85 or 2221

Or, you could do the same thing with S + T = I + G,

Or, you could use the tax multiplier, dY/dT which is easy in this simple model, it will be -b/(1 - b) = -.6/(1 - .6) = 1.5, so that the drop in T of 59.9 would result in an increase in Y of 89.85, so that 2131 + 89.85 = 2220.85 or 2221

C = 104.8 + 1332.6 - 323.46

C = 1113.94

If MPC were .75 instead of .6 then there are two ways of answering the question: 1. You could go back to the beginning and recalculate all the values up to this last one of C (which would of course make all the figures in the scenario wrong), or 2. You could just sub .75 for .6 in the consumption function and calculate accordingly.

You know:

Y = 2922

I = 5.6

Ms = 234

I = 456

Ia = 322

G = 790

X = 285

M = 95

X-M = 190

t = .5

h = .1

n = .05

p = .08

C = 104.8 + .6Yd or C = 104.8 + .6(Y - T)

An investment of the sort: I = g + hY - ji

(You know that the investment function must include the interest rate because you are also asked to find a money demand function and hence the interest rate, and the consumption function doesn't include an interest rate variable so there must be one in the investment function, otherwise there would be no point in having a money demand function.)

An import function of the sort : M = m + nY

A money demand function of the sort: i = k - pMd

Putting the value of the parameters and variables that you know together, you see:

T = To + .5Y, or

T = To + .5(2922)

C = 104.8 + .06(Y - To -tY), or

C = 104.8 + .06[2922 - To - .5(2922)]

M = m + .05Y, or

M = m + .05(2922), or

95 = m + .05(2922)

I = g + hY - ji

456 = 322 + .1(2922) - j(5.6)

i = k - pMd

5.6 = k - .08Md

5.6 = k - .08(234) (234 because Ms must = Md, and we know Ms = 234)

So, the first thing to do is find the obvious: m in the import function, j in the investment function, and k in the money demand function:

95 = m + .05(2922)

95 = m + 146.1

-51.1 = m

so the import function is:

M = -51.1 + .05Y

456 = 322 + .1(2922) - j(5.6)

456 = 322 + 292.2 - j(5.6)

-158.2 = - j(5.6)

-158.2/5.6 = -j

28.25 = j

so the investment function is:

I = 322 + .1Y - 28.25i

5.6 = k - .08(234)

5.6 = k - 18.72

5.6 + 18.72 = k

24.32 = k

so the money demand function is:

i = 24.32 - .08Md

So now, the only thing we don't know is the value of To. To find this we can substitute all our known values into the equation:

Y = C + I + G + X - M

And solve for the only remaining unknown, To

2922 = 104.8 + .06(2922) - .06To - .06[.5(2922)] + 456 + 790 - 95

2922 = 104.8 + 175.32 - .06To - 87.66 + 456 + 790 - 95

1293.54 = -.06To

-21559 = To

So, T = -21559 + .5Y

Therefore your model consists of the following:

C = 104.8 + .6Yd, where Yd = Y - T

T = -21559 + .5Y

I = 322 + .1Y - 28.25i

i = 24.32 - .08Md

M = -51.1 + .05Y

You can, and should, check your math, by combining all the equations, the current parameter values and solving for equilibrium Y:

Y = C + I + G + X - M

Y = 104.8 + .06[Y - (-21559 + .5Y)] + 322 + .1Y - 28.25(5.6) + 790 + 285 - (-51.1 + .05Y)

Y = 104.8 + .06Y + .06(-21559) - .06(.5Y) + 322 + .1Y - 158.2 + 790 + 285 + 51.1 - .05Y

Y - .06Y + .03Y - .1Y + .03Y = 104.8 - 1293.54 + 322 - 158.2 + 790 + 285 + 51.1

Y(1 - .06 + .03 - .1 + .05) = 2688.24

.92Y = 2688.24

Y = 2922

It all checks, so you could now use this model to solve for the kind of policies you are asked about in the first question.