A firm's productivity depends on two variables, x and y....

A firm's productivity depends on two variables, $x$ and $y$. Currently, $x=a$ and $y=b$, and the firm's productivity is 4,000 units. Productivity is increasing at a rate of 400 units per unit decrease in $x$, and is decreasing at a rate of 300 units per unit increase in $y$. What does all of this information tell you about the firm's productivity function $g(x, y)$ ?

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Key Concepts

Partial Derivatives

Partial derivatives provide the rate at which a multivariable function changes with respect to one variable while holding the others constant. In this context, they indicate how the firm's productivity changes when only one input, either x or y, is varied slightly, with the other kept fixed.

Gradient Vector

The gradient is a vector containing the partial derivatives of a function with respect to its inputs, and it points in the direction of the greatest rate of increase. Here, the gradient of the productivity function at (a, b) would reveal the combined effect on productivity when both x and y are changed infinitesimally.

Marginal Productivity

Marginal productivity refers to the additional output produced by a small change in one of the inputs, reflecting the sensitivity of the production output. The rates given (an increase of 400 units per unit decrease in x and a decrease of 300 units per unit increase in y) inform the corresponding marginal effects in the productivity function.

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