Government

We can add a public sector to our economic model by assuming a fixed level of government expenditures, G, and lump-sum taxes, T. The addition of taxes to the model introduces another complication: disposable income no longer is identical to GDP. Rather, DI = Y -T. Here we make use of the convention that "Y" stands for GDP.

Equilibrium still requires that total production equals total purchases, or in this case, Y_e = C + I_g + X_n + G. As before, consumption is assumed to be a linear function of disposable income: C = a + bDI. Substituting Y -T for DI and inserting into our equilibrium relationship, we find Y_e = a + b(Y_e -T) + I_g + X_n + G. Finally, we solve for Y_e to find that Y_e = x (a -bT + I_g + X_n + G).

Following the usual procedure, it is apparent that = and that = . That is, the change in equilibrium GDP from a one dollar change in government expenditures is equal to the standard multiplier, while a one dollar increase in lump-sum taxes decreases equilibrium GDP by b (the MPC) times the standard multiplier. Since the MPC is between zero and one by assumption, it is clear that – < . That is, the impact on equilibrium GDP of a change in government spending exceeds the impact of an equal (but in the opposite direction) change in taxes.

Suppose for example that C = 97.5 + .75DI, I_g = 20, X_n = 0, G = 20, and T = 20. Following the formula, Y_e = x (97.5 - .75x(20) + 20 + 0 +20) = 4 x 122.5 = $490. A $10 increase in government spending will increase equilibrium GDP by x10 = $40, while a $10 increase in lump-sum taxes will reduce equilibrium GDP by x10 = $30.