Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The productivity of a South American country is given by the function $$f(x, y)=20 x^{3 / 4} y^{1 / 4}$$ when \(x\) units of labor and \(y\) units of capital are used. a. What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 256 units and 16 units, respectively? b. Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity?

Step-by-Step Solution

Answer

a) The marginal productivity of labor (MPL) is 120, and the marginal productivity of capital (MPK) is 80, when the amounts expended on labor and capital are 256 units and 16 units, respectively. b) The government should focus more on increasing expenditure on labor rather than encouraging capital investment at this time to increase the country's productivity.

Step 1: Compute the partial derivative of productivity function with respect to labor

Take the partial derivative of f(x, y) with respect to x: \[ \frac{\partial f(x, y)}{\partial x} = 20 \cdot \frac{3}{4} x^{(3/4)-1} y^{1/4} = 15 x^{-1/4} y^{1/4} \]

Step 2: Compute the partial derivative of productivity function with respect to capital

Take the partial derivative of f(x, y) with respect to y: \[ \frac{\partial f(x, y)}{\partial y} = 20 \cdot x^{3/4} \cdot \frac{1}{4} y^{(1/4)-1} = 5 x^{3/4} y^{-3/4}\]

Step 3: Calculate marginal productivity of labor and capital

Evaluate the derivatives obtained in Steps 1 and 2 for x=256 and y=16: Marginal productivity of labor (MPL): \[ MPL = \frac{\partial f(x, y)}{\partial x} \Big|_{x=256, y=16} = 15 (256)^{-1/4} (16)^{1/4}= 15(4)(2)= 120\] Marginal productivity of capital (MPK): \[ MPK = \frac{\partial f(x, y)}{\partial y} \Big|_{x=256, y=16} = 5 (256)^{3/4} (16)^{-3/4} = 5(64)(1/4) = 80\] a) The marginal productivity of labor is 120, and the marginal productivity of capital is 80, when the amounts expended on labor and capital are 256 units and 16 units, respectively.

Step 4: Determine whether to encourage capital investment or labor expenditure

Compare MPL and MPK at this point: MPL = 120 MPK = 80 Since the marginal productivity of labor is higher than the marginal productivity of capital, it implies that an additional unit of labor will result in a greater increase in productivity compared to an additional unit of capital at the current levels of labor and capital. b) The government should focus more on increasing expenditure on labor rather than encouraging capital investment at this time to increase the country's productivity.

Key Concepts

Partial Derivative

Understanding partial derivatives is essential when examining how changes in one variable influence a function that depends on more than one variable. For instance, consider you are baking a cake and want to know just how the sweetness changes with the amount of sugar, regardless of other ingredients. Analyzing partial derivatives is doing precisely that but in a mathematical sense.

In our exercise, we assessed the partial derivatives of the productivity function with respect to labor (x) and capital (y). This process gives us the marginal productivity of each. A key point for students is to recognize that taking a partial derivative with respect to one variable treats all other variables as constants. When we calculated the partial derivatives, we focused on how productivity changes with an incremental adjustment in either labor or capital, holding the other constant.

Productivity Function

A productivity function like the one we used, signifies the relationship between inputs (labor and capital) and the output (products or services). This particular function can become complex, but the core idea is rather straightforward. Think of it like a recipe where the inputs are ingredients and the output is the final dish. The higher the quality of the inputs and the better they work together, the better is the dish.

The productivity function in our question uses labor and capital in a certain mathematical form that shows how these inputs combine to create output. It is paramount to understand that changes in these inputs have a direct impact on output, as sought by the partial derivatives representing the marginal increase in productivity per unit increase in labor or capital.

Labor and Capital

The concepts of labor and capital are fundamental to economics and productivity analysis. Labor refers to the human effort put into the production of goods and services. Capital, on the other hand, includes tools, equipment, machinery, and factories used in production. Both are vital and often complement each other.

In the context of our exercise, these two elements are varied to see how productivity is affected. As one of these inputs increases while the other remains constant, we want to know if the 'cook' is becoming more efficient (productivity rises) with each additional 'ingredient' (labor or capital). Understanding the relationship and interplay between labor and capital is critical in decision-making for maximizing productivity.

Capital Investment

Finally, let's delve into the concept of capital investment. In the broadest sense, this is the money spent by a business or economy on acquiring or maintaining fixed assets such as land, buildings, and equipment. When an entity invests in capital, it's like planting a seed for future benefits.

The question of whether to bolster the economy through capital investment or labor expenditure is not just about the numbers. It is also about strategy and long-term growth. In our scenario, even though the marginal productivity of labor is higher, it does not inherently mean that increasing labor is always the best choice. The decision should consider other factors such as the current state of the economy, available technology, and growth goals. However, based solely on our function and the marginal productivity calculated, it appears that additional labor is the current path to increased productivity.