In the labor market, we can also talk about responsiveness or elasticity with respect to wages

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In the labor market, we can also talk about responsiveness— or elasticity — with respect to wages (and other prices) on both the demand and supply sides.
A: For each of the following statements, indicate whether you think the statement is true or false (and why):
(a) The wage elasticity of labor supply must be positive if leisure and consumption are normal goods.
(b) In end-of-chapter exercise 9.5, we indicated that labor supply curves are often “back ward bending”. In such cases, the wage elasticity of labor supply is positive at low wages and negative at high wages.
(c) The wage elasticity of labor demand is always negative.
(d) In absolute value, the wage elasticity of labor demand is at least as large in the long run as it is in the short run.
(e) (The compensated labor supply curve, which we will cover more explicitly in Chapter 19, is the labor supply curve that would emerge if we always insured you reached the same indifference curve regardless of the wage rate.) The wage elasticity of compensated labor supply must always be negative.
(f) The (long run) rental rate (of capital) elasticity of labor demand (which is a cross-price elasticity) is always positive.
(g) The output price elasticity of labor demand is positive and increases from the short to the long run.
B. Suppose first that tastes over consumption and leisure are Cobb-Douglas.
(a) Derive the functional form of the labor supply function.
(b)What is the wage elasticity of labor supply in this case? Explain how this relates to the implicit elasticity of substitution in Cobb-Douglas tastes.
(c) Next, suppose that the decreasing returns to scale production process takes labor and capital as inputs and is also Cobb-Douglas. Derive the long run wage elasticity of labor demand.1
(d) Derive the rental rate elasticity of labor demand. Is it positive or negative?
(e) Derive the long run output price elasticity of labor demand. Is it positive or negative?
(f) In the short run, capital is fixed. Can you derive the short run wage elasticity of labor demand and relate it to the to long run elasticity you calculated in part (c)?
(g) Can you derive the short run output price elasticity of labor demand and compare it to the long run elasticity you calculated in part (e)?
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