Question

The mowing of lawns requires only labor (gardeners) and capital (lawn mowers). These factors must be used in the fixed proportions of one worker to one lawn mower, and production exhibits constant returns to scale. Suppose the wage rate of gardeners is \(\$ 2\) per hour, lawn mowers rent for \(\$ 5\) per hour, and the price elasticity of demand for mowed lawns is -2 a. What is the wage elasticity of demand for gardeners (that is, what is \(\partial L / \partial w \cdot w / L\) )? b. What is the elasticity of demand for lawn mowers with respect to their rental rate (that is, \(\partial K / \partial v \cdot v / K) ?\) c. What is the cross elasticity of demand for lawn mowers with respect to the wage rate (that is \(, \partial K / \partial w \cdot w / K) ?\)

Short answer

Answer: The wage elasticity of demand for gardeners is -2, the elasticity of demand for lawn mowers with respect to their rental rate is -2, and the cross elasticity of demand for lawn mowers with respect to the wage rate is 0.

Step-by-step solution

Understanding the production function

In this problem, the production function is fixed, with one worker (gardener) and one lawn mower required for each unit of output. This means the output can be represented as Q = min(L, K), where L is the number of workers and K is the number of lawn mowers.

Calculate the price elasticity of demand for gardeners

We are given the price elasticity of demand for mowed lawns: \(\varepsilon_p = -2\). Since there is a fixed proportion between the labor and capital and the production function exhibits constant returns to scale, we can assume that the elasticity of demand for gardeners and lawn mowers is the same: \(\varepsilon_L = \varepsilon_p = -2\).

Calculate the wage elasticity of demand for gardeners

The wage elasticity of demand for gardeners is defined as \(\partial L / \partial w \cdot w / L\). Using the result from Step 2, we can calculate this as follows: Wage elasticity of demand for gardeners = \(\varepsilon_L = \frac{\partial L}{\partial w} \cdot \frac{w}{L} = -2\) This means that for every 1% increase in the wage rate of a gardener, the demand for gardeners will decrease by 2%.

Calculate the elasticity of demand for lawn mowers with respect to their rental rate

We need to find the elasticity of demand for lawn mowers with respect to their rental rate: \(\partial K / \partial v \cdot v / K\). Using the information from Step 1, we can see that elasticity of demand for lawn mowers is also -2 (since they are used in fixed proportions with constant returns to scale): \(\varepsilon_K = \frac{\partial K}{\partial v} \cdot \frac{v}{K} = -2\) This means that for a 1% increase in the rental rate of lawn mowers, the demand for lawn mowers will decrease by 2%.

Calculate the cross elasticity of demand for lawn mowers with respect to the wage rate

We need to find the cross elasticity of demand for lawn mowers with respect to the wage rate: \( \partial K / \partial w \cdot w / K\). Here, we already know that the elasticity of demand for gardeners and lawn mowers is the same, since they are used in fixed proportions and the production function exhibits constant returns to scale. Thus, the cross elasticity of demand for lawn mowers with respect to the wage rate is: \(\varepsilon_{K,w} = \frac{\partial K}{\partial w} \cdot \frac{w}{K} = 0\) This means that when the wage rate of gardeners changes, the demand for lawn mowers remains unchanged. In summary: a. The wage elasticity of demand for gardeners is -2. b. The elasticity of demand for lawn mowers with respect to their rental rate is -2. c. The cross elasticity of demand for lawn mowers with respect to the wage rate is 0.

Key concepts

Constant Returns to Scale

Constant returns to scale (CRS) in production means that if all inputs are increased by a certain factor, the output increases by the same factor. It implies that the production process is perfectly efficient, with no wasted effort or resources.
For example, in the lawn mowing business scenario, one gardener and one lawn mower produce one mowed lawn. If you double both the gardeners and mowers, the output doubles to two mowed lawns. In mathematical terms, if we double input quantities represented as \(L\) (labor) and \(K\) (capital), then output \(Q\), given as \(Q = min(L, K)\), also doubles.
This feature is crucial, as it simplifies several economic calculations. The behavior of demand elasticities, like wage elasticity and rental rate elasticity, hinges on the assumption of constant returns to scale.

Price Elasticity

Price elasticity of demand measures how much the quantity demanded of a good responds to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price.
In this exercise, the given price elasticity of demand for mowed lawns is \(-2\). This negative value indicates that the quantity demanded is inversely related to price: as price goes up, demand decreases, and vice versa. Specifically, a 1% increase in the price results in a 2% decrease in demand for mowed lawns.
Because the production of the mowed lawns involves a fixed proportion of one gardener and one lawn mower, both factors exhibit the same price elasticity. This fixed relationship ensures the elasticity of demand for gardeners and lawn mowers mirrors the demand for mowed lawns.

Fixed Proportions Production

Fixed proportions production refers to a production process where inputs must be used in a specific ratio to each other to produce output. This situation is described by a Leontief production function often represented as \(Q = min(L, K)\), indicating that production cannot occur unless the inputs are in precise proportions.
In the lawn mowing example, production requires one gardener for each lawn mower. This fixed ratio impacts the elasticity calculations. With fixed proportions and constant returns to scale, elasticity remains consistent across inputs.
Such a production setup simplifies elasticity analysis because if all input prices change, demand for each input responds the same way due to their interconnected usage. However, the cross-elasticity between input factors, such as between gardener's wages and lawn mower rentals, is unaffected. This setup results in a cross elasticity of zero because a change in one input's cost does not alter the need for the other input when used in fixed proportions.