**Q.1: Long-Run Labor Demand and Factor Substitutability**

Suppose there are two inputs in the production function, labor (L) and capital (K), which can be combined to produce Y units of output according to the following production function:

**Y = 30K + 10L**

The firm wants to produce 600 units of output.

1. Draw the isoquant that corresponds to that level of production (600 units) in a graph that has L on the horizontal axis and K on the vertical axis.

2. The shape of the isoquant tells us about the relationship between the two inputs in production. How substitutable are L and K in the production of Y? In particular, how many units of L can be replaced by one unit of K without affecting the level of output?

3. Is this isoquant convex (bowed toward the origin)?

4. In class, we said that isoquants are convex under our "standard assumptions." To see which standard assumption is violated in this case, hold K fixed at some level (for convenience, suppose K is fixed at zero). Graph Y as a function of L for L = 0, ...,

5. By looking at your graph, determine the marginal product of labor (MP_{L)}. That is, what is the change in Y (ΔY ) when L increases by 1 unit (ΔL = 1)?

6. How does the marginal product of labor (MP_{L}) change as L increases? How is this di↵erent from the "standard assumption" about the MP_{L} we made in class?

7. Suppose the firm can choose whatever combination of capital (K) and labor (L) it wants to produce 600 units. Suppose the price of capital is $1,000 per machine per week. What combination of inputs (K and L) will the firm use if the weekly salary of each worker is $400?

8. What if everything is same as in the previous question but the weekly salary of each worker is $300? Now what combination of inputs (K and L) will the firm use to produce its 600 units?

9. (Bonus) What is the (wage) elasticity of labor demand for this firm as the wage falls from $400 to $300?

**Q.2: Own-price elasticity**

Suppose the market labor demand curve is given by L^{D} = 20-(1/2)W and the market labor supply curve is given by L^{S} = 2W.

1. Graph the labor demand curve and the labor supply curve on the same graph (with L on the horizontal axis and W on the vertical axis, as we have done in class).

2. Determine the equilibrium employment (L*) and wage (W*) in this market.

3. Now suppose the government implements a minimum wage (W^{M}) of $10 in thismarket. What will the new level of employment be?

4. Calculate the elasticity of the labor demand curve when the wage changes from its equilibrium level (W*) to the minimum level (W^{M}) set by the government. Is the demand curve elastic or inelastic in this range?

5. Suppose that the wage in some other labor market goes up so that labor supply in this market is now given by L^{S} = 2W -10. Graph the new supply curve on your graph from Part #1.

6. Now that supply has shifted, what will employment and the wage paid to workers be in this market? What is the e↵ect of the minimum wage given in Part #3 on employment now?

7. The government implements a new minimum wage of $14 in this market. What will the new level of employment be? Calculate the elasticity of the labor demand curve when the wage changes from what it is in Part #6 to the new minimum wage of $14. Is the demand curve more or less elastic in this range than it is in Part #4?

**Q3: Cross-price elasticity**

Consider teenage labor and adult labor as separate inputs in production for fast-food restaurants. Suppose the wage of teenage workers increases (but the adult wage remains the same). Analyze the e↵ect of the teenage wage increase on fast-food restaurants' employment of adult labor, given that:

1. Teenage labor costs are a large share of total costs at fast-food restaurants.

2. Adults dislike the tasks teenagers do at fast-food restaurants (i.e. cleaning bathrooms), so it takes big increases in their wages to get them to do this kind of work. Given these 2 facts, are teenage workers and adult workers more likely to be gross substitutes or gross complements in fast-food production, holding all other factors constant? **Explain your answer.**

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