Question

How would you expect an increase in output price, \(P\), to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in \(P\) must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb-Douglas case. c. Use the profit function to show how the presence of inferior inputs would lead to ambiguity in the effect of \(P\) on input demand.

Short answer

In summary, an increase in output price must not reduce the demand for capital and labor inputs if neither input is inferior, as the firm will aim to produce more output using higher levels of inputs in response to a higher price. The Cobb-Douglas production function supported this claim, demonstrating that an increase in output price will lead to higher input demands while maintaining their relative proportions. However, the presence of inferior inputs introduces ambiguity in the effect of output price changes on input demand, as some inputs might exhibit a negative substitution effect.

Step-by-step solution

a. Graphically explaining the relationship between output price and input demand

To understand why an increase in output price, \(P\), must not reduce the demand for capital and labor inputs when neither input is inferior, we need to look at the production function and the product transformation curve (PTC) that maps the relationship between the two inputs. When output price increases, the firm's objective is to maximize profit, given by the difference between total revenue and total cost. The total cost consists of the cost of labor and the cost of capital. Assuming both are non-inferior inputs and given a constant production technology, an increase in output price will shift the PTC outward, expanding the production possibilities frontier. As a result, with a higher output price, the firm can afford to demand more capital and labor inputs to produce more output and increase its profits. This relationship is graphically shown by a movement along, and an outward shift of, the PTC as the price of output increases.

b. Cobb-Douglas input demand functions and output price

In the Cobb-Douglas production function, we have: \(Y = AK^{\alpha}L^{\beta}\), where \(Y\) represents output, \(A\) is the total factor productivity, \(K\) is the capital input, \(L\) is the labor input, and \(\alpha\) and \(\beta\) are the output elasticities with respect to capital and labor, respectively. The input demand functions for capital and labor in the Cobb-Douglas case can be derived by solving the cost minimization problem, which involves minimizing the cost function: \(C = wL + rK\), subject to the Cobb-Douglas production function, where \(w\) is the wage rate and \(r\) is the rental rate of capital. Using the Cobb-Douglas production function and the input demand functions, we can show that: \(\frac{K}{Y} = \frac{\alpha}{r}\) and \(\frac{L}{Y} = \frac{\beta}{w}\). Now, let's find the optimal capital and labor utilization as a function of output price: \(K(P) = \frac{\alpha P}{r}Y(P)\) and \(L(P) = \frac{\beta P}{w}Y(P)\). As we can see from these equations, when the output price \(P\) increases, the demand for capital and labor inputs will increase as well, keeping the same proportion between them as established by the production function. This demonstrates the graphical presumption from part (a).

c. The effect of inferior inputs on input demand

To analyze the presence of inferior inputs, we will use the profit function: \(\pi(P) = PY - wL - rK\). For inferior inputs, as their price decreases, the demand for that input decreases as well. This implies a negative substitution effect of an output price increase in input demand. If we differentiate the profit function with respect to output price \(P\) and input prices \(w\) and \(r\), we can find the relationship between the changes in input prices and output price: \(\left( \frac{\partial K}{\partial P}, \frac{\partial L}{\partial P} \right) = -\frac{\partial^2 \pi}{\partial P \partial w}, -\frac{\partial^2 \pi}{\partial P \partial r}\). In the case of inferior inputs, the cross-partial derivatives can be positive or negative, leading to ambiguity in the effect of output price changes on input demand. The presence of inferior inputs breaks the clear relationship between output price and input demand established in parts (a) and (b), creating ambiguity in the response of input demand to changes in output price.

Key concepts

Understanding the Cobb-Douglas Production Function

The Cobb-Douglas production function is a mathematical representation of the relationship between the quantity of output produced and the quantities of capital and labor used in the production process. It is given by the equation:
\[ Y = AK^{\alpha}L^{\beta} \
where \(Y\)\) is the output, \(A\)\) is a constant that represents total factor productivity, \(K\)\) is the capital input, \(L\)\) is the labor input, and \(\alpha\)\) and \(\beta\)\) are the output elasticities with respect to capital and labor, respectively.

This function suggests that the increase in any of the inputs, while holding the other constant, will lead to a proportional increase in the output. It's widely used because it offers both flexibility in the substitution of inputs and mathematical simplicity in deriving input demand functions for optimal input utilization. In the context of the exercise, understanding the interaction between these variables is crucial when analyzing how changes in the output price, \(P\)\), affect the demand for inputs.

Short sentences and a clear, understandable explanation are critical when distinguishing the roles of \(K\)\) and \(L\)\) in production. Remember that when neither input is inferior, the demand for them naturally increases with a rise in output price, given the positive relationship between input usage and output production depicted by the Cobb-Douglas function.

Output Price and Input Demand Correlation

Moving on to output price and input demand, it's important to grasp how a change in the output price \(P\)\) affects the demand for capital \(K\)\) and labor \(L\)\). In a scenario without inferior inputs, an increase in the output price encourages a company to produce more to maximize profits. Since the total revenue a firm can make is the output price multiplied by the quantity of goods sold, a higher price means higher potential revenue.

The firm responds by demanding more of both capital and labor, assuming the price of these inputs remains the same. This response stems from the desire to increase production and take advantage of the higher output price, illustrating a direct positive relationship between output price and input demand.

Furthermore, the graphical representation in exercises helps to visually demonstrate this relationship. When price increases, the expansion of the production possibilities illustrates that more resources can be efficiently allocated toward production, explaining the upward movement and outward shift of the product transformation curve (PTC).

Inferior Inputs in Microeconomics

Lastly, let's delve into the role of inferior inputs in microeconomics. Inferior inputs are inputs for which demand decreases as income or output price increases, contrary to the normal inputs we've discussed above. In the context of an individual's consumption, an inferior good could be something like public transportation; as income rises, people often prefer to drive personal vehicles instead. In production, an inferior input might be a cheaper, less efficient technology or raw material that firms move away from as they grow wealthier.

When considering the profit function \(\pi(P) = PY - wL - rK\)\), if an input is inferior, the effects of an output price increase become ambiguous. This complexity is due to the negative substitution effect, where a decrease in input price doesn’t necessarily increase the quantity demanded for that input. As such, increasing output price can either lead to an increase, decrease, or no change in demand for inferior inputs. This creates a situation where typical correlations between input demand and output price do not apply, and it requires a case-by-case analysis to predict how demand will respond to changes in output price.

Equipping students with the mathematical tools and conceptual understanding of inferior inputs allows them to analyze real-world economic situations more accurately. Emphasizing the unique nature of inferior inputs juxtaposed with the simplicity of the relationships in normal input scenarios is essential for a nuanced understanding of microeconomics.