The aggregate spending multiplier crucially depends on the sectoral origin of government purchases. To establish this result, we characterize analytically the response of aggregate output to sector-specific government spending shocks in a tractable production-network economy. The response is larger when government purchases originate in sectors with a relatively small contribution to private final demand and in downstream industries. These findings hinge on the pivotal role of relative-price changes. In the special case where price adjustment is inhibited, the response of aggregate output is invariant to the origin of the spending shock. Using a quantitative multi-sector model, we show that the heterogeneity across U.S. industries translates into significant dispersion in the aggregate spending multiplier associated with sectoral government purchases. Finally, we illustrate how differences in the sectoral composition of purchases across U.S. government levels (i.e., general, federal, and state and local) lead to sizable differences in the spending multiplier.