. The simple model in chapter 3 assumes that taxes are fixed. This is not realis

Question

. The simple model in chapter 3 assumes that taxes are fixed. This is not realistic, because taxes usually depend on income. Suppose taxes are T t, +t,Y (with t, between 0 and 1). (A) Would an increase in autonomous spending of 100 have a larger or smaller impact on Y than in the simpler model? (B) What is the formula for the multiplier in this model? HINT: Use a numerical example and calculate the impact when tl 0 and when t1>0. You should also be able to answer other questions such as the ones below. You do not need to answer them on this HW, but I might ask them on an exam: i. Can you explain why the multiplier changes? ii. If consumers spend a portion of their income on imports, how would the multiplier change?

Explanation / Answer

Solution: Simple model for real income or GDP :Y = C + I + G + NX, C is consumption, I is investment, G is government expenditure and NX is net exports (i.e, Exports, X - Imports, M). Taxes are a part of consumption, as in they reduce the disposable income, and thus, the consumption.

C = C(bar) + MPC*(Y - T)

C = C(bar) + MPC*Y - MPC*T, C(bar) is autonomous consumption (that is consumption when income is 0), Marginal propensity to consume ,MPC is share of disposale income consumed, T is the lumpsum tax, here which is fixed and independent of income. Assuming rest of the variables (G, I, X, M) are independent of Y, for such consumption function, multiplier = 1/(1-MPC)

Now, if T = t0 + t1*Y, our consumption function becomes, C = C(bar) + MPC*(Y- t0 - t1*Y) = C(bar) + MPC*(1 - t1)*Y - MPC*t0. So, now when t1 = 0, this new consumption function is same as the old one, with lumpsum tax, T = t0, so multiplier is 1/(1-MPC) in this case.

When t1 > 1, i.e, taxes are no more independent of Y, multiplier = 1/(1- MPC*(1-t1)); this is the formula for multiplier in this model. Since, clearly, now the denominator is bigger:

Mathematically, both MPC and t1 lie between 0 and 1, thus, MPC > MPC*(1-t1)

So, 1- MPC < 1 - MPC(1-t1), again implying 1/(1-MPC) > 1/ (1 - MPC(1-t1)), so now increase in autonomous spending of 100 will have smaller impact on Y, than in the simple model.

Numerical example: Say autonomous spending = A(bar) which includes all autonomous expenditures of model, i.e A(bar) = C(bar) + I(bar) + G(bar) + X(bar) - M(bar)

With increase by 100, new autonomous spending = A(bar) + 100, or change is 100

(Note: even in numerical example, we haven't alloted any value to A(bar), because evaluating in our multiplier effect, we don't require A(bar), but only change in A(bar wich we are already given as 100). Say, MPC = 0.5 and t1 = 0 in simple model, and t1 = 0.2 in new model.

Than, using multiplier effect, Change in Y = multiplier*change in A(bar)

For simple model, change in Y = [1/(1-0.5)] * 100 = 2*100 = 200

For new model, change in Y = [1/(1 - 0.5(1-0.2))] * 100 = 1.667*100 = 166.7. Hence, we have shown that in our new model, impact on Y will be lower (166.7 < 200) than in our simple model.

1) Why the multiplier changes: Since, now the taxes depend on Y, even a small increase in income implies consumer will have to pay more taxes. In case of lumpsum taxing, no matter what consumer earns, all had to pay a fixed amount of tax. But such tax rate as in our new model, not only affects consumption spending directly (same as in lumpsum tax case), but also the income directly. Since, consumers always have to pay a part of their income as tax and such tax to be paid increases with the income earned, consumers have lower incentive to earn higher income, and as a result of which, the consumption in economy decreases. So, increase in autonomous spending generates relatively, lower increase in income.

2) Now our model becomes: Y = C + I + G + NX with C function being of simple model, i.e, C = C(bar) + MPC*(Y - T) and NX = X - m*Y, where m is the portion of income spent on imports, so M = m*Y

Now, there are two factors impacting multiplier, MPC and m

Y = C(bar) + MPC*Y - MPC*T + I(bar) + G(bar) + X(bar) - m*Y

So, multiplier in this case becomes = 1/(1 - MPC + m)

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