A note on the three-sector Cobb–Douglas GDP function

The GDP function chooses the most efficient point on the production possibility frontier with curved surface. We derive a three-sector Cobb–Douglas GDP function including three goods (agriculture, manufacturing, and services) and three factors (capital, labor, and land). It is very difficult to derive the Stolper–Samuelson theorem in the three-sector version. But we derive it numerically. By applying the empirical factor income share across sectors, we show that an increase in the relative agricultural goods price in terms of manufacturing goods decreases the real interest rate and the wage rate while it increases the land rent. On the other hand, an increase in the relative service price in terms of the manufacturing goods decreases the real interest rate while it increases the wage rate and the land rent. The three-sector Cobb–Douglas GDP function with three factors must be useful for empirical studies in structural change.