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

Answer: In the presence of inferior inputs, the effect of an increase in output price on the demand for capital and labor inputs is ambiguous, as the relationship between output and input demand is no longer straightforward. An increase in output price might increase or decrease the demand for a particular input, depending on whether it is inferior or not.

Step-by-step solution

Part (a): Graphical Explanation #

Let's assume that a firm produces output using two inputs: capital (K) and labor (L). The production function can be represented by the isoquant curve. Isoquants represent combinations of inputs that produce the same level of output. When the output price, \(P\), increases, the firm seeks to maximize profits by producing more output. To explain graphically, we can draw an isoquant map showing isoquants for different levels of output. However, since higher isoquants represent higher outputs, as the output price increases and firms seek to produce more, they will move to a higher isoquant. If neither input is inferior, which means that as output increases, the firm will need to use more of both K and L, the firm will move towards a higher isoquant that requires more capital and labor. Consequently, a rise in \(P\) must not reduce the demand for either factor.

Part (b): Cobb-Douglas Input Demand Functions #

Suppose the firm produces output according to the Cobb-Douglas production function: \[F(K,L) = A K^\alpha L^\beta\] where A, \(\alpha\) and \(\beta\) are constant parameters. In the Cobb-Douglas case, the input demand functions for capital and labor are given by: \[K^* = \alpha \frac{w}{r}L^*\] \[L^* = \beta \frac{r}{w}K^*\] where \(K^*\) and \(L^*\) are the optimal demand for capital and labor, and w and r represent the cost of labor and capital, respectively. Notice that both \(K^*\) and \(L^*\) are increasing functions with respect to P. Thus, the graphical presumption from part (a) is demonstrated by the input demand functions in the Cobb-Douglas case.

Part (c): Profit Function and Inferior Inputs #

Let's assume the profit function is given by: \[\pi = PF(K,L) - wL - rK\] where \(\pi\) represents profit, P is the output price, F(K,L) is the production function, w and r are the costs of labor and capital as before. Taking partial derivatives of the profit function with respect to K and L, we get the following conditions for profit maximization: \[\frac{\partial \pi}{\partial K} = P\frac{\partial F(K,L)}{\partial K} - r = 0\] \[\frac{\partial \pi}{\partial L} = P\frac{\partial F(K,L)}{\partial L} - w = 0\] When neither input is inferior, an increase in P leads to an increase in output, which in turn increases the demand for both K and L. However, if one of the inputs is inferior, an increase in output may reduce the demand for that particular input. In that case, we may observe that \(\frac{\partial F(K,L)}{\partial K}\) or \(\frac{\partial F(K,L)}{\partial L}\) is negative. The presence of inferior inputs creates ambiguity in the effect of \(P\) on input demand because the relationship between output and input demand is no longer straightforward. An increase in \(P\) might increase or decrease the demand for a particular input, depending on whether it is inferior or not. Thus, in the presence of inferior inputs, conclusions drawn from graphical analysis in part (a) and Cobb-Douglas input-demand functions in part (b) may not hold.

Key concepts

Production Function

In economics, a production function is a mathematical equation that describes the relationship between input resources and the maximum output that can be produced. It is an essential concept because it helps understand how different inputs, like labor and capital, affect output levels. A typical production function requires inputs to transform into goods or services. In simple terms, it is about how businesses or firms convert resources into products.

Production functions are vital for making decisions about resource allocation and understanding how changes in input quantities impact the quantity of output.
They allow firms to analyze the efficiency of their production process. A well-known form of production function is the Cobb-Douglas, which will be explained further in this article. This type of function provides insights into the scalability of production and how varying input levels can result in different production outputs.

Isoquants

Isoquants are curves that represent combinations of various inputs that produce the same quantity of output. They are similar to indifference curves in consumer theory but are used in production analysis for firms.

Visualizing Isoquants:

• **Higher Isoquants:** Represent higher levels of output as they indicate more significant quantities of production using different combinations of inputs.
• **Slope Interpretation:** The slope of an isoquant provides information about the rate at which one input can be substituted for another while maintaining the same level of output.

Isoquants help firms decide on the right mix of input resources to achieve efficient production. They are essential in analyzing how firms can adjust inputs in response to changes in external factors such as price changes.

Cobb-Douglas

The Cobb-Douglas production function is a particular form of production function that depicts how two or more inputs interact to produce output. It is expressed as:
\[F(K, L) = A K^\alpha L^\beta\]

In this function:

• **A** represents total factor productivity, indicating shifts in overall production efficiency.
• **K** and **L** are capital and labor, the primary inputs, with **α** and **β** as their respective output elasticities.
• **α** and **β** indicate the proportionate contributions of capital and labor to the production process.

This model is frequently used because it provides a straightforward way to measure the efficiency and impact of input factors on output. The elasticities **α** and **β** are important as they help determine returns to scale, indicating whether doubling input results in double output or not.

Profit Maximization

Profit maximization is the process through which firms decide the best level of output and input combinations to achieve the highest possible profit. The profit function is given by:
\[\pi = PF(K, L) - wL - rK\]

In this function:

• **π** is the profit.
• **P** is the price of the output.
• **F(K, L)** is the production function.
• **w** and **r** are the wages for labor and the rent for capital, respectively.

Firms maximize profits by adjusting their input levels such that the marginal cost of inputs is equal to the marginal product multiplied by the price of the output.

When neither input is inferior, an increase in output price generally increases the demand for both inputs as generating more output becomes profitable. However, if an input is inferior, an increase in output price might not increase its demand due to the negative marginal product, creating ambiguity in expected profits.