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Question
B.   Answer the following questions using the aggregate expenditures model of the economy described below.
C=80+.6Yd
T=40+.2Y
Ia=28
Ga=64
Xa=76
M=.18Y

i.   What is the marginal propensity to consume, the marginal tax rate, and the marginal propensity to import?

ii.   What is the saving function?  What is the marginal propensity to save?

iii.   What is the aggregate expenditure function?  What is the autonomous expenditure?  What is the marginal propensity to withdraw?

iv.   What is the equilibrium level of real GDP?

v.   What is the size of the multiplier?

vi.   Suppose the full employment level of real GDP is $340.  Does a recessionary gap or inflationary gap exist?  How can the government eliminate the gap by altering government expenditures?

I cannot get this question, every time I think that I am close to the solution I get stuck.  Can you help me?

Answer
Hi Lauren,

Thank you for seeking my help in respect of the question on aggregate expenditure model. This is a little above intermediate level. There is very little mathematics involved, but a good conception is presupposed based on a notion of government finance and clear concept about extensions of Keynesian macroeconomics. I don’t know about your background, but from the question you have posed it appears you are dealing with a pretty advanced level of comparative statics of macroeconomics. I assume you have some grounding in school-level algebra and some exposure to elementary differential calculus. You may just lay your hand on any standard elementary calculus textbook, or maybe any book on mathematical economics, if you need some brush up.

I have tried to make the explanation as simple as possible, using minimum possible mathematics without which the problem may not lend to any level of intelligible analysis. I hope I will be able to walk you through the following simple steps and make you across to whoever you need to address the problem. So let’s get down to the problem under the following heads.

WE ARE GIVEN THE FOLLOWING SET OF EQUATIONS:
C=80+0.6Yd
T=40+0.2Y
Ia=28Ga=64
Xa=76
M=0.18Y

Remember here we have consumption C as a function of disposable income Yd. It has an intercept 80 –that means consumption is 80 at zero level of income (people do consume even if they don’t earn anything, and this is the vertical intercept in a diagram of positive quadrant with C up the vertical axis and Yd along the horizontal axis. Now as Yd increases, C increases linearly by a multiple of 0.6.

To get at marginal propensity to consume, you have to consider C as a function of Y, not of Yd (because Yd is just left after you pay the taxes). So consider the consumption function.
Your consumption function is given linearly by C=80+0.6Yd

C=80+0.6Yd   and since Yd= Y-T and T=40+0.2Y, we have
C=80+0.6(Y-T) = 80+0.6(Y-(40+0.2Y)) = 80+0.6Y-0.6x40+(0.6x0.2)Y = 80+0.6Y-24-0.12Y
C= 80+0.6Y-24-0.12Y =56+0.48Y

Now MPC is dC/dY= [d/dY](56+0.48Y) = 0.48
So MARGINAL PROPENSITY TO CONSUME is 0.48

Second, Yd = disposable income = income after taxes. However tax itself is a linear function of income, with AUTONOMOUS TAX as the tax-intercept = To=40, and tax increasing by rate 0.2. So if you differentiate your tax function T=40+0.2Y, you get dT/dY=marginal tax rate; and the answer to your second question is:
Marginal tax rate = 0.2

Third, your import function has no intercept, i.e., the function rises exactly from the origin in a positive quadrant with M up the vertical axis and Y along the horizontal axis. Since in differentiation constant doesn’t matter, so never bother about intercept here. Just differentiate M w.r.t. Y and you get dM/dY or marginal propensity to import; and your next answer is:
Marginal propensity to import=0.18

The next question, SAVING FUNCTION, is a trifle tricky. First you have to get the consumption function. Then you derive the saving function. We have already derived the consumption function from C=80+0.6Yd as a function of Y as:

C= 56+0.48Y

You know that Y=C+S, that is consumption and saving equal income, given other things constant.
S= Y-C =Y-(56+0.48Y)
S = Y-56-0.48Y
S=-56+Y-0.48Y = -56+(1-0.48)Y= -56+0.52Y
You get marginal propensity to save by differentiating the savings function wrt Y or dS/dY=MPS. Hence

Marginal propensity to save is 0.52 [So you linear savings function starts with a negative intercept down the vertical axis, a mirror image of the consumption intercept, this being so because consumption a zero level of income means “dissaving” or either dipping into the past savings or borrowing, hence negative.  Note also that MPC+MPS=1.]

I just skip the simple part at iii (aggregate expenditure function) which you will be able to derive after you are done with this explanation. So come straight to iv.

WHAT IS THE EQUILIBRIUM LEVEL OF INCOME (gdp)?
Given
Y=C+Ia+Ga+Xa – M          T=40+0.2Y
C= 80+ 0.6Yd          Yd= Y–T
         M=0.18Y

From that we get

Y= 80+0.6(Y–T) + Ia+Ga +Xa–0.18Y
Y= 80+0.6[Y – (40+0.2Y)] + Ia+Ga+Xa –0.18Y          
Y = 56+0.48Y–0.18Y+ Ia+Ga+Xa
Y = 56+0.3Y+ Ia+Ga+Xa
Y–0.3Y= 56 + Ia+Ga+Xa
Y(1–0.3) = 56 + Ia+Ga+Xa = 56+28+64+76 =224
Y = 224/0.7 = 320

EQUILIBRIUM INCOME or Ye = 320

Now, given whatever values you have for autonomous investment Ia, autonomous exports, and autonomous government expenditure Ga, you can calculate you equilibrium level of income.

v. WHAT IS THE SIZE OF THE MULTIPLIER?

Normally, multiplier is the reciprocal of MPS or simply 1 divided by 1 minus MPC. So the l;arger the MPC, the larger the multiplier. That you can calculate easily from the data given.

Now the question is what the multiplier is given all those functions. Let’s look at (a) export multiplier, (b) autonomous import multiplier, and (c) autonomous taxation multiplier.

  EXPORT MULTIPLIER
This is dY/dXa = 1/[1–0.6 –0.12+ 0.18] >0

Because 0<b, 0.18<1. A 1-unit increase in exports will have a positive effect on equilibrium income, which is given by the multiplier.

  IMPORT MULTIPLIER
This is dY/dM= –1/[1–0.6 –0.12+ 0.18] <0
An increase in autonomus imports will lead to a decrease in equilibrium income.

  TAXATION MULTIPLIER is given by
dYe/dTo = dYe/d(40) = [0.18 – 0.48]/ [1–0.6 –0.12+ 0.18] <0
Because a country’s marginal propensity to import m=0.18 is usually smaller than its marginal propensity to consume (MPC= 0.48 in our data set). With 0.18 < 0.48, 0.18–0.48 <0. An increase in autonomous taxes will lead to decrease in national income, but the presence of 0.18 in the numerator has a mitigating effect on the decrease in income. As 0.18 is positive, increased taxes will reduce cash outflows for imports and thus reduce the negative effect of increase taxes on the equilibrium level of income.

DOES A RECESSIONARY OR INFLATIONARY GAP EXIST WITH FULL-EMPLOYMENT LEVEL OF INCOME AT $340?
Equilibrium level of income is 320. This falls short of full-employment level of income (340) by 20. As real GDP is less than potential GDP, there is an inflationary gap.

The government will eliminate the gap by some initiative.
The desired level in economic activity is the difference between the full-employment level of income  (320) and the present level (320).

The government can eliminate the gap (20) by ALTERING GOVERNMENT EXPENDITURE (Go):

DY=[dY/dGo]DGo          [D stands for upper-case delta, since Greek fonts are not admissible here]

dY/dGo= 1/[1-0.6+0.6(0.2)] = 1/ 0.52 = 1.92

DY=[dY/dGo]DGo  = 1.92[DGo]
20 =  1.92[DGo]
Hence DGo= 10.42

Therefore, the government, by increasing government expenditure by an amount of 10.42, will set in motion GDP propagation through a multiplier effect, and will achieve the desired level of potential GDP by increasing GDP by 20.

[By the way, the government may also try to achieve the equivalent result by altering autonomous taxes, in which case we will have DY=[dy/dTo]DTo, and that can be calculated easily by similar formula.]

If my explanation has been of some help to get you out of where you say you get stuck in grappling with somewhat tricky macroeconomic topic, I would be quite happy. I wish you best of success in your pursuit of knowledge.

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Eklimur Raza

Expertise

It appears some students in this website are confused about elasticity of demand and the slope of the demand curve when they are trying to figure out why rectangular hyperbola comes up in case of unitary demand curve. First, they don't know that RH can be depicted in a positive quadrant of price,quantity plane. Secondly, they make the mistake that the slope of RH is constant at -1. Two points could help them: first, e=1 at each and every point of the RH, because the tangent at any point shows lower segment=upper segment (another geometric definition of e); yet slopes at different points,dQ/dP, are different; second, e is not slope but [(Slope)(P/Q)]in absolute terms. Caveat: only if we measure (log P) along the horizontal axis and (log Q) up the vertical axis, can we then say slope equals elasticity --in which case RH on P,Q plane is transformed into a straight-line demand curve [with slope= -tan 45 deg] on (log Q),(logP) plane, and e= -d(log Q)/d(log P). [By the way, logs are not used in college textbooks --although that is helpful in econometric estimation of elasticity viewed as an exponent of P, when demand equation is transformed into log-linear form.] I have not found the geometrical explanation I have given in any textbook followed in undergraduate and college classes in Canada (including the book followed in a university where I taught for a short time and in the book followed in George Brown College, Toronto, where I teach.

Experience

About 11 years' teaching economics and business studies, and also English, history and elementary French.Practical experience in a development bank, working with international donor agencies like the World Bank and the ADB. Experience in free-lance journalism, including Canada's "National Post."

Organizations
I teach micro- and macroeconomics at George Brown College (continuing education), Toronto, ON, Canada.

Publications
Many articles and editorials, on different subjects, in English newspapers. Recently an applied Major Research Paper, based on a synthesis of the Solow growth model and the Lewis two-sector model, has be accepted by Ryerson University, Toronto. Professors Thomas Barbiero and Eric Cam, Ryerson University, accepted the paper.

Education/Credentials
Master degree in Interantional Economics and Finance and diploma with honours in Business Administration from Canada.

Awards and Honors
Received First Prize in an inter-university Literary Contest.

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