1. is alone sufficient to enable a unique maximum output to be obtained from the given capital stock.
2. must hold together with the condition that private returns everywhere equal social returns if a maximum output is to be obtained from any given aggregate capital stock.
3. is irrelevant to maximizing output as long as private returns to investment everywhere equal social returns.
4. holds in every function that can be defined.
Choose the correct option.
The correct answer is option 2. Diminishing returns is a well-known property of production functions (but not all functions). Investors place their capital where the return to them is highest. Diminishing returns is necessary to ensure that greater investment in the high-return than low-return opportunities will drive the returns into equality. Only if this occurs will all types of capital be present in the final equilibrium. For aggregate output to be maximized an additional condition must hold---namely, that the private returns to all investments equal the social returns, ensuring that equality of the social returns to all forms and locations of capital will occur.
Recall that the principle of diminishing returns means that an expansion of one type of capital relative to others will cause the marginal product of the expanding type of capital to fall and the marginal product of the relatively contracting types of capital to rise. The marginal product of any type of capital is the increase in output associated with a small (one-unit in this case) expansion of that type of capital. This means that when investors invest more in high-return capital types and equivalently less in low-return capital types output rises because the increase in output resulting from the expansion of the high-return types of capital will exceed the decrease in output resulting from the contraction of the low-return capital types. And the return to the high-return types of capital will tend to fall relative to the return to the low-return types of capital, driving the their returns into equality. If there were no diminishing returns to drive the returns to all types of capital into equality, only one form of capital, the one with the highest return, would eventually remain.
The private and social returns to every type of investment must be the same because private investors make investment decisions based on the returns to themselves rather than to society. If the private returns are below the social returns to investment in a particular type of capital investors, being insufficiently rewarded, will invest too little in that type of capital.