Consider a government that increases spending by $100 billion in the each of the next three years. Economists estimate the spending multiplier (which is the multiple by which income increases for a given injection of spending) to be 1.5 and the impact is immediate and exhausted in each year. They also estimate that the import propensity is 0.2 (meaning that imports rise by 20 cents for every dollar generated in the economy). They also estimate the tax multiplier (impact of tax changes on income) to be equal to 1 and the current tax rate is equal to 30 per cent. So for every extra dollar produced, tax revenue rises by 30 cents. Which of the following statements is correct? The cumulative impact of this fiscal expansion on nominal GDP is
Answer: $450 billion and the private sector saves 24 cents out of every extra dollar generated.
The answer was Option (a) $450 billion and 24 cents.
The question involves two parts: (a) working out what is relevant to the answer; and (b) reverse engineering some of the relevant data to get the marginal propensity to consume (and hence the saving propensity).
To work out the cumulative impact you need to understand the concept of the spending multiplier which is the easier part of the question.
In Year 1, government spending rises by $100 billion, which leads to a total increase in GDP of $150 billion via the spending multiplier. The multiplier process is explained in the following way. Government spending, say, on some equipment or construction, leads to firms in those areas responding by increasing real output. In doing so they pay out extra wages and other payments which then provide the workers (consumers) with extra disposable income (once taxes are paid).
Higher consumption is thus induced by the initial injection of government spending. Some of the higher income is saved and some is lost to the local economy via import spending. So when the workers spend their higher wages (which for some might be the difference between no wage as an unemployed person and a positive wage), broadly throughout the economy, this stimulates further induced spending and so on, with each successive round of spending being smaller than the last because of the leakages to taxation, saving and imports.
Eventually, the process exhausts and the total rise in GDP is the "multiplied" effect of the initial government injection. In this question we adopt the simplifying (and unrealistic) assumption that all induced effects are exhausted within the same year. In reality, multiplier effects of a given injection usually are estimated to go beyond 4 quarters.
So this process goes on for 3 years so the $300 billion cumulative injection leads to a cumulative increase in GDP of $450 billion.
It is true that total tax revenue rises by $135 billion but this is just an automatic stabiliser effect. There was no change in the tax structure (that is, tax rates) posited in the question.
That means that the tax multiplier, whatever value it might have been, is irrelevant to this example.
Some might have decided to subtract the $135 billion from the $450 billion to get answer (c) on the presumption that there was a tax effect. But the automatic stabiliser effect of the tax system is already built into the expenditure multiplier.
So answers (c) and (d) were there to lure you into thinking the tax parameters were important for the first part of the solution.
However, the second part of the question required you to reverse engineer the multiplier. In mathematics the general rule is that you can only solve for unknown parameters if you have as many equations as unknowns. So if you have y = 2x. You cannot solve for y because you don't know what x is. If I tell you x = 2 then you have one equation (y = 2x) and one unknown (y) so it becomes trivial y = 4.
Similar reasoning applies in this question.
The expenditure multiplier is defined as the change in real income that results from a dollar change in exogenous aggregate demand (so one of G, I or X). We could complicate this by having autonomous consumption as well but the principle is not altered.
Consumption and Saving
So the starting point is to define the consumption relationship. The most simple is a proportional relationship to disposable income (Yd). So we might write it as C = c*Yd - where little c is the marginal propensity to consume (MPC) or the fraction of every dollar of disposable income consumed. The marginal propensity to consume is just equal to 1 minus the marginal propensity to save (which is the 24 cents or 28 cents in the dollar that we are seeking in the question).
The * sign denotes multiplication. You can do this example in an spreadsheet if you like.
Taxes
Our tax relationship is already defined above - so T = tY. The little t is the marginal tax rate which in this case is the proportional rate - 0.3 in the question. Note here taxes are taken out of total income (Y) which then defines disposable income.
So Yd = (1-t) times Y or Yd = (1-0.3)*Y = 0.7*Y
Imports
If imports (M) are 20 per cent of total income (Y) then the relationship is M = m*Y where little m is the marginal propensity to import or the economy will increase imports by 20 cents for every real GDP dollar produced.
Multiplier
If you understand all that then the explanation of the multiplier follows logically. Imagine that government spending went up by $100 and the change in real national income is $150. Then the multiplier is the ratio (denoted k) of the
Change in Total Income to the Change in government spending.
Thus k = $150/$100 = 1.50
That is the value assumed in the question. This says that for every dollar the government spends total real GDP will rise by $1.50 after taking into account the leakages from taxation, saving and imports.
When we conduct this thought experiment we are assuming the other autonomous expenditure components (I and X) are unchanged.
But the important point is to understand why the process generates a multiplier value of 1.50.
The formula for the spending multiplier is given as:
k = 1/(1 - c*(1-t) + m)
where c is the MPC, t is the tax rate so c(1-t) is the extra spending per dollar of disposable income and m is the MPM. The * denotes multiplication as before.
This formula is derived as follows:
The national income identity outlined in Question 4 is:
GDP = Y = C + I + G + (X - M)
A simple model of these expenditure components taking the information above is:
GDP = Y = c*Yd + I + G + X - m*Y
Yd = (1 - t)*Y
We consider (in this model for simplicity) that the expenditure components I, G and X are autonomous and do not depend on the level of income (GDP) in any particular period. So we can aggregate them as all autonomous expenditure A.
Thus:
GDP = Y = c*(1- t)*Y -m*Y + A
While I am not trying to test one's ability to do algebra, and in that sense the answer can be worked out conceptually, to get the multiplier formula we re-arrange the previous equation as follows:
Y - c*(1-t)*Y + m*Y - A
Then collect the like terms and simplify:
Y[1- c*(1-t) + m] = A
So a change in A will generate a change in Y according to the this formula:
Change in Y = k = 1/(1 - c*(1-t) + m)*Change in A
or if k = 1/(1 - c*(1-t) + m)
Change in Y = k*Change in A.
So in the question you have one equation (the multiplier) and one unknown (c). This is because of the 3 behaviorial parameters (c, t and m) two are known (t and m) and you also know the value of the left-hand side of the equation (1.5). So in effect you can solve for c:
k = 1/(1 - c*(1-t) + m)
Thus k*[1 - c*(1-t) + m] = 1
Thus k - c*k*(1-t) + k*m = 1
Thus k + k*m -1 = c*k*(1-t)
Thus c = (k + k*m - 1)/(k*(1-t))
Then you plug in the values of the knowns and the result is:
c = (1.5 + 0.3 - 1)/(1.5*0.7)
c = 0.8/1.05 = 0.761905
So the MPS (marginal propensity to save) = (1 - c) = approximately 24 cents.
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