# Balanced budget and change in taxes

Consider a simple Keynesian economy in which the government expenditure ($$G$$) exactly equals its total tax revenue: $$G = tY$$ where $$t$$ is the tax rate and $$Y$$ is the national income. Suppose that the government raises $$t$$. Then what happens to $$Y$$?.

My working: With the balanced budget multiplier in effect $$Y$$ becomes equal to $$\frac I {(1- c) (1 - t)}$$

However without the balanced budget in effect ( when taxes are increased over government expenditure), $$Y$$ becomes equal to $$\frac {I + G} {1 - c(1-t)}$$ Is the working right ?. How should I proceed in order to come to a solution ?.

• All sorts of different and possibly contradictory things depending on the other assumptions of the model. Spell them out clearly in your post, and maybe a meaningful and useful answer can be given. – Alecos Papadopoulos Dec 29 '17 at 15:41
• Alecos this was a question asked for a Econ post grad entrance exam that I'm preparing for. The question posted is exactly what was asked at the exam. Options for the answer were- a. Y decreases b. increases c.remains unchanged d.may increase or decrease. – madhavU Dec 29 '17 at 15:44
• Then I guess the person who drafted this question, puts a lot of emphasis on the assumptions hidden behind the word "Keynesian". So supposedly, the "correct" answer depends totally on how the materials you have to study for this exam discuss/describe the "simple Keynesian Economy". – Alecos Papadopoulos Dec 29 '17 at 15:55
• The syllabus covers the simple Keynesian economy that is taught in under-graduation, nothing further than that. Could you please read my working and suggest a way out if that's not too much to ask ? – madhavU Dec 29 '17 at 15:58
• We examine such questions by taking the total differential on both sides of the fundamental macroeconomic identity, and then keep $dY$ on one side of the equation and everything else on the other, and see what kind of answer we can give. – Alecos Papadopoulos Dec 29 '17 at 20:22

Here's my take, which could be wrong. We assume a closed economy, so net exports, imports and exports themselves are all zero.

We have Y = C + I + G Suppose we have the simple consumption function C= k +c*(Y-T). We are only allowed to consume out of disposable income or have autonomous consumption.

Then Y = k+c*(Y-T) + I + G This goes to

Y*(1-c) = k - c*T + I + G

You take T = t*Y. So this is

Y*(1-c) = k - ctY + I + G

Subtract ctY from both sides. Then we have

Y*(1-c*(1-t)) = k + I + G

The solution in the absence of the balanced budget would be

Y = [k + I + G]/(1 - c*(1-t))

Now have G = t*Y - the balanced budget

Y*(1 - c*(1-t)) = k + I + t*Y or

Y*((1-t) - c*(1-t)) = k+ I or

Y*((1-t)*(1-c)) = k + I

or

Y = [k + I]/((1-t)*(1-c))

So my take is that the multiplier is right, in both cases, but that there is or will have to be some autonomous consumption expenditure in the economy, and that's what's left out. You can say, "I don't need autonomous expenditure!" But just try coming up with a Keynesian cross to settle output in nominal terms without one!

• Thanks a lot for verifying and making that correction Julian. How can we get to an answer from the multipliers figured out though ? – madhavU Dec 30 '17 at 3:39

Let me be careful about multipliers, because they are a controversial topic. Barro has an extremely hostile view of multipliers from spending by the Federal government, which is on display in

https://www.imf.org/external/np/seminars/eng/2013/fiscal/pdf/barro.pdf

I believe that his view is widely shared; economists want to get paid, and sucking up to people who rabidly want to cut back government is a successful strategy these days.

For the Keynesian mulitipliers (a la (G+I + k)/(1-c)), let /\ denote change.

So if we have /\G =40, then, under the Keynesian assumptions, and we let M denote our multiplier, we have /\Y = M*/\G. If the numerical value of M were 2, as used to be assumed, then /\Y = 80.

My own feeling is that the harsh view of multipliers is misleading. Suppose the government builds a road where there was none before. My impression, which could be verified with an intensive search that I can't promise to do now, is that in Barro's view the spending increases from the road in the 1st year ONLY are counted as a response to the government spending. But of course the road promotes spending for a number of years, and all the spending should be discounted and put in to get an honest answer. You keep getting things like this - when West Germany absorbed East Germany to reunify the German state, my recollection is that someone calculated some multiplier involved as 0.2, conveniently overlooking that the West German government paid off the (worthless) debts of the East in legitimate marks.

On the occasions when I pretend to be a "respectable" economist, I do regional analysis. The work usually involves making some use of IMPLAN, one of the few input-output programs of the necessary size around these days. The key assumption behind IMPLAN is that you have constant returns to scale and no bottlenecks, and multipliers wind up from 1.7 to a little over 2 for that sort of work.

To return to Barro, his assumptions fall very easily from his assumptions about aggregate supply. If you assume aggregate supply is nearly perfectly inelastic with regard to any government spending, then you get all his results about crowding out and inflation and the rest. This aggregate supply assumption is of course not Keynesian; it is the opposite, that there really is no slack. It is difficult to develop an unambiguous test for whether there is slack or not.

A good example of standard thought about multipliers now, not that I necessarily agree with it, is in Price Fishback, "How Successful Was the New Deal? The Microeconomic Impact of New Deal Spending and Lending Policies in the 1930s", Journal of Economic Literature, December 2017, Vol. 55, No. 4, 1435 - 1485.