Agents are assumed to have smooth preferences with natural boundary conditions. For large regular economies, satisfying an indeconposability conditions, it is shown that core allocations and competitive allocations converge to each other with a rate inversely proportional to the number of agents m. To the extent that the indecomposability condition is harmless, 1/m can be regarded as the normal rate of convergence. However, if indifference surfaces are allowed to have kinks, 1/m cannot be regarded as normal. This is treated in Part II [6].