Deriving AD from the Aggregate Expenditures Model

In earlier notes, we derived the formula for equilibrium GDP as the solution to the following equation: Y = C + Ig + G + Xn where C = a(Wn, E) + b(Y -T), Ig = Ig(i, AMOC, BT, TC, K, E), and Xn = Xn(Yf, t, P$, P/Pf). In words, consumption is assumed to be a linear function of disposable income, but the intercept is a function of net wealth and consumer expectations. Investment is a function of the real rate of interest, acquisition, maintenance, and operating cost, business taxes, technological change, the stock of capital, and expectations. Net exports is a function of foreign income, tariff rates, the exchange rate, and the domestic price level relative to foreign prices.

Here we can note that many of these spending determinants are themselves dependent on the domestic price level. Specifically, net wealth, the real rate of interest, and the domestic/foreign price ratio are all functions of the domestic price level: Wn = Wn(P); i = i(P); P/Pf = P/Pf(P). A higher price level reduces autonomous consumption by reducing net wealth. Likewise, it reduces intended investment by increasing the real rate of interest. Finally, it reduces net exports by raising domestic prices relative to foreign prices.

If we hold constant all non-price-level determinants of autonomous spending, we can simplify our expression of GDP to the following: Y = a(P) + b(Y -T) + Ig(P) + G + Xn(P), or even more compactly, Y = A(P) + bY where A(P) = a(P) -bT + Ig(P) + G + Xn(P) is all autonomous (independent of the level of GDP) spending. Since a higher price level reduces autonomous consumption, net investment, and net exports, we know that dA/dP is negative: A’(P) = a’(P) + Ig’(P) + Xn’(P) < 0. The first term is the "real balances effect," the second is the "interest-rate effect," and the final term is the "foreign purchases effect."

Solving our equation for Y we obtain Y = x A(P). This relationship between equilibrium GDP and the price level is the aggregate demand curve, or AD. We see that it is downward sloping: dY/dP = x A’(P) < 0. Further, we note that the AD curve is an "equilibrium" curve of sorts. Every combination of Y and P reflects the expenditures equilibrium-total production equals planned purchases.