## Proof that Marginal Product = Average Product at the Maximum Point

In economics, Total Product Quantity (Q) is a function of two variables, labour (Z1) and capital (Z2). The function states that the total product quantity (Q) is equal to the product of labour (Z1) and capital (Z2). Marginal product is defined as the increase in total product (Q) when one additional unit of labour (or capital) is employed. Average product is defined as the average amount of total product (Q) produced per unit of labour (or capital). The proof of the statement that marginal product equals average product at the maximum point can be demonstrated mathematically.

## Calculating the Marginal Product

The marginal product of labour (MPZ1) is calculated by taking the first derivative of the total product quantity function with respect to labour (Z1). The formula for the marginal product of labour (MPZ1) is: MPZ1 = dQ/dZ1.

## Calculating the Average Product

The average product of labour (APZ1) is calculated by taking the second derivative of the total product quantity function with respect to labour (Z1). The formula for the average product of labour (APZ1) is: APZ1 = (2Q/Z1) – (d2Q/dZ12).

## Calculating the Maximum Point

The maximum point is determined by finding the point at which the marginal product of labour (MPZ1) is equal to zero. The formula for finding the maximum point is: MPZ1 = 0. Using this formula, the maximum point can be calculated.

## Proving that Marginal Product = Average Product at the Maximum Point

Once the maximum point is calculated, the equation for the marginal product of labour (MPZ1) can be substituted into the equation for the average product of labour (APZ1). This will produce the equation: (2Q/Z1) – (d2Q/dZ12) = 0. Therefore, it is proven that the marginal product of labour (MPZ1) is equal to the average product of labour (APZ1) at the maximum point.

## Related Questions

• What is Total Product Quantity (Q)?
• What is the formula for calculating the marginal product of labour (MPZ1)?
• What is the formula for calculating the average product of labour (APZ1)?
• What is the formula for finding the maximum point?
• What is the proof that marginal product equals average product at the maximum point?
• What is the relationship between marginal product and average product?
• How can marginal product be used to maximize profits?
• What is the difference between marginal product and total product?
• What is the difference between average product and total product?
• What factors affect the marginal and average product?