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 \(, \partial K / \partial w \cdot w / K) ?\)

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.

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: \(\varepsilon_L = \varepsilon_p = -2\).

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%.

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%.

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.