The basic hypothesis of producer behaviour is that firms seek to maximize profit (the difference between total revenue and total cost). Given this objective, a firm will choose its input combination so that the total product of the last dollar spent on each input is equal.
Why will it do this? Since the marginal product per dollar of land is given by MPT/r, where r represents the (rental) price of land, and the marginal product per dollar of labour is given by MPL/w, where w represents the price of lab our (the wage rate), what must be demonstrated is that, at the profit-maximizing input combination (T*, L*),
Without loss of generality, suppose that
at (T*, L*). If the firm were to reallocate $1 of its expenditures from labour to land, its total product per dollar would change by MPT/r - MPL/w. This number is positive since MPT/r > MPL/w. Thus total revenue can be increased (as a result of higher output an a fixed output price) without changing total cost. This contradicts the statement that (T*, L*) is the profit-maximizing input combination.
Returns to Scale
An important question that might be asked about a production function is how output responds to an increase in all inputs together. For example, suppose all inputs were doubled: Would output double or would the relationship not be quite so simple? This is a question of returns to scale exhibited by the production function.
The production function F exhibits increasing returns to scale (IRS) if
Example: The production function Q = T × L exhibits IRS because
The production function F exhibits decreasing returns to scale (DRS) if
Example: The production function Q = L/T exhibits DRS because
The production function F exhibits constant returns to scale (CRS) if
Example: The production function Q = T + L exhibits CRS because
CRS production functions occupy an important place in economic theory. This is due to the fact that there is a sound economic rationale for expecting an industry's production function to exhibit constant returns. If all output in the industry is produced in plants of an optimal size then doubling all inputs can most effectively be accomplished by doubling the number of these plants. This would double output since there are now exactly twice as many plants. Hence the industry would have a CRS production function.