Economics 151AProfessor David NeumarkProblem Set 1—Regression and Labor Demand - SolutionsDue Tuesday, Jan. 24

1. In the bivariate regression model , the error term ε captures influences on Y that are not included in the model.

a. What has to be true about the relationship between ε and X for the estimate of β to be unbiased?

Answer: E(ε|X) = 0. In other words, ε and X should be uncorrelated.

If a variable that is correlated with both the dependent and independent is not included in the regression when β is estimated, then the unaccounted for variable ends up in the error term. To see this, consider the “true” model for Y.

and the correlation between Xi and Zi is not zero.

If we estimate,

it must mean that , and since Zi is correlated with Xi the estimate of

will be biased and the effect of Z on Y will be attributed to X.

b. Suppose a new pizza shop owner is trying to decide how much to charge, by collecting data on prices of pizzas (X) and quantities sold (Y) from local pizza shops. Give an example of something that might be included in ε and bias the estimate of β. Give an example of something that might be included in ε that would probably not bias the estimate of β.

Answer: In order for an omitted variable to bias the estimate of β, the variable must have a relationship with both the dependent (quantity) and independent (prices) variable. An example of such a variable is the quality of the pizza. If one pizza shop spends more money on quality ingredients (mozzarella from Italy, organic flour and tomatoes, brick oven, etc.) then the prices of the pizza will be higher than in the shop that doesn’t focus on having organic or imported ingredients. The quality of the pizza will most likely also affect the number of pizzas sold since individuals generally prefer higher quality goods to lower quality if prices are similar. Another example of a possible omitted variable that affects both quantity and price is the location of the shop. If one pizza shop is located within walking distance of campus and the next closest is four miles away (and doesn’t have delivery), then the closer shop may be

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able to charge a higher price and will more than likely sell more pizzas since those without access to transportation will have to go to it for their pizza. An example of a variable that would be correlated with quantities of pizza sold but probably have no effect on the price is a special event on campus. If the special event brings extra people to campus around the time that people usually eat, then the local shop will probably see an increase the quantities of pizza sold on that day but the pizza shop is unlikely to raise prices during that time.

c. Suggest a multivariate regression that the pizza shop owner should run to get a better answer to the question of how the price she charges is likely to affect quantities sold. (Remember that a multivariate regression can have many included variables, not just two.)

Answer: Using the examples of possible omitted variables above, the multivariate regression should include variables measuring the quality of the ingredients the shop uses (denoted Wi) and the location of the shop (Zi).

2. Between 1996 and 2004, California raised its minimum wage from $4.75 to $6.75. In that period, total employment grew from 14.3 million to 16.5 million. A policymaker arguing in favor of a minimum wage increase argues that evidence demonstrates that raising the minimum wage does not reduce employment, but rather increases it. How would you evaluate this argument, and how would you propose using data to study this question?

Answer: This is not a strong argument for two reasons. First, the minimum wage does not impact the whole population of workers but only those getting paid minimum wage and around it, i.e. mainly teenage and low-skill workers generally. Hence, to study the impact of minimum wages on employment, we need use data on these population groups and not the total employment changes.

Second, we need to control for other factors like economic conditions in the state during the relevant time period. For example, the period (1996-2004) was a boom period, so employment may have grown even if minimum wages, on their own, had a negative effect on employment. (No one ever said that minimum wages are the only influence on employment.) To account for this, we should probably compare the change in employment in California with changes that occurred in the same period in other comparable states that did not raise their minimum wages. This is the idea behind the “differences-in-differences” approach to evaluating the effects of a policy.

To spell this out more clearly, our goal is to estimate the relationship between minimum wages (MW) and some outcome Y (employment). Suppose we have data on states at a point in time, and estimate the regression

.

Now suppose we take a sample of states that raised their minimum wage, and measure Y before and after the increase in minimum wage (from t-k to t). For simplicity, let us assume

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that every state increases minimum wage by the same amount (i.e. the same level). Then we could estimate a regression

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In this model, β is identified by the changes in Y in the states that increased their minimum wage. To interpret this estimate as a causal effect of minimum wage increase, we also use data on states that did not raise minimum wage, comparing changes for states that raised minimum wage to changes for those that did not. In simplest form we can estimate the following regression:

where Di is a dummy variable for states that raised their minimum wages. In this case the intercept controls for the common change in all states, and captures the effect unique to states that raised their minimum wage. Because we estimate from the difference between the “before and after” difference in states increasing and not increasing minimum wage, we call this technique “differences-in-differences.”

3. Suppose a firm uses labor L and capital K to product output y, with the production function Suppose the firm sells its output in a competitive labor market at price p, and

buys labor in a competitive market at price w, and assume the firm maximizes profits.

a. Let the level of capital be fixed at in the short-run. Provide an expression for the demand curve for labor.

Answer: Profit maximization problem is:

In the short run, firms maximize profits when the marginal revenue product of labor is equal to the marginal cost of labor.

MRPL = MCL (1)

Since the firm is operating in a competitive market, the marginal revenue is equal to price and the marginal revenue product of labor is equal to price times the marginal product of labor:

MRPL = P MPL (2)

Total cost is wage times labor plus rent times capital, which means that the marginal cost of labor is equal to wage:

TC = wL + r (3)MCL = w (4)

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Take the derivative of the production function with respect to labor to find the marginal product of labor and hence MRPL:

(5)

Substituting (5) and (4) into (1) gives the profit maximizing condition for this problem:

(6)

Solving (6) for L provides the labor demand:

b. Using your answer for part a, how does L change with w, p, and ? Explain why each of these changes makes sense.

Answer: Differentiating the expression for labor demand w.r.t. w, p, and respectively, we get,

.

Labor demand decreases with w because the profit maximization condition in the short run states that to be at the profit maximizing point, the marginal product of labor times price must equal the wage, and since wage has increased, to keep this equality the firm must increase the marginal product of labor. To increase the marginal product of labor the firm must reduce labor. The intuition behind this is when wages increase, firms cannot produce at the same output level without increasing costs and so the firm must make adjustments in the input that is flexible in the short run in order to continue to maximize profits.

Labor demand increases with price because in order to meet the profit maximizing condition of the price times the marginal product of labor equal to wage the firm must lower the marginal product of labor. The firm accomplishes this by increasing labor. The intuition is that the firm will want to produce more as prices increase which would mean increasing the flexible input (labor).

Labor demand increases with higher levels of . Higher levels of capital makes each unit of labor more productive and so to continue to meet the profit maximizing condition of the price times the marginal product of labor equal to wage the firm must lower the marginal product of labor. The firm accomplishes this by increasing labor.

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4. Suppose a firm uses only one input (L) to produce output y, with the production function. Suppose the firm sells its output in a competitive market at price p, and buys labor

in a competitive market at price w.

a. Write an expression for the profits of the firm as a function of w, p, and L.

Answer:

b. What is the marginal cost of hiring an additional unit of labor? Graph the marginal cost of labor curve.

Answer:

where TC = wL. So .

c. What is the marginal revenue from hiring an additional unit of labor? Graph the marginal revenue curve on the same graph as in part b.

Answer:The marginal revenue from hiring an additional unit of labor is the change in total revenue from one unit change in the amount of labor. Or if we think about really small changes, then we can use calculus. Then, the marginal revenue of labor is

So,

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Note in this expression that the firm is in a competitive output market, so the marginal revenue is equal to the price. This curve is shown on the same graph as in part (b).

d. Assume the firm maximizes profits. How much labor should it hire as a function of the real wage w/p? Find the solution in terms of L, and also display it on the graph.

Answer:The profit-maximizing firm will increase labor hired until its marginal revenue

product equals its marginal cost. Then, . Solving for labor, we have

.

e. Does the firm hire more or less labor as p increases? Why?

Answer:

The firm hires more labor as p increases. This is because labor demand must increase in order for the marginal product of labor to decrease. By lowering MPL, it

allows the optimizing equation p MPL = w to balance out the increase in price.

5. Assume everything is the same as in Problem 6, except that the firm is a monopolist, and faces downward sloping demand curve p(y) = a −by.

a. Write an expression for the profits of the firm as a function of w, p, and L.

Answer:The monopolist has to account for the fact that his output choice determines the price of the product, so we get the following expression for profits as a function of hired labor:

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.

b. What is the marginal revenue from hiring an additional unit of labor? Graph the marginal revenue curve and the marginal cost curve (which is the same as in 6.b).

Answer:The marginal revenue of hiring an additional unit of labor is:

c. Assume the firm maximizes profits. How much labor should it hire as a function of a, b, and w? Find the solution in terms of L, and also display it on the graph.

Answer:The profit-maximizing firm, monopolist or otherwise, will increase labor hired until

its marginal revenue product equals its marginal cost. Then, .

Solving for labor, we have .

d. Why is the equation for the profit-maximizing choice for L in this case not a function of p?

Answer:Labor demand is not a function of price because the monopolist’s output choice determines the price of the product.

e. Show that labor demand declines if the wage increases.

Answer:

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A quick look at the labor demand equation shows that an increase in wages leads to a decrease in labor demand. We can get a precise statement of the effect of changes in wages on labor demand by taking the derivative with respect to w. This shows that

.

f. Show that labor demand increases if demand for the firm’s product increases.

Answer:Take another look at the demand curve for the output good: p(y) = a −by. An increase in demand can be seen in this curve as an increase in ‘a’, which would shift the demand curve to the right. Since labor demand is a positive function of ‘a’, an increase in ‘a’ would increase labor demand. Taking the partial derivative, we have

.

6. Suppose a firm’s production function is Q = L + 2∙K.

a. Graph isoquants for Q = 6, 9, and 12.

Answer:

b. Suppose w, the price of labor, and r, the price of capital, both equal 1. Draw the isocost lie for C (the cost) = 6. What combination of inputs L and K does the firm choose?

Answer:The graph from part (a) also includes the answer for this problem. The most productive use of inputs on the isocost line of 6 when w=r=1 (i.e. the isocost line is L+K = 6) is at the point , at which point the firm is producing 12 units of the output good using only capital as an input. This makes sense because no matter what combination of L and K is chosen, capital is twice as productive as labor, but cost the same, so the firm will use only K.

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c. Now suppose w = 1 and r = 3. Draw the isocost line for C = 6. What combination of inputs L and K does the firm choose.

Answer:

Answer:Now that capital has become relatively more expensive, the isocost line which is given by the equation L+3K = 6 becomes more flat, favoring a relative increase in use of labor. Now, the most productive use of inputs at the cost of 6 is at the point

, where the firm uses only labor as an input. This makes sense because no matter what combination of L and K the firm chooses, capital is twice as productive as labor but cost three times more than the labor cost, so the firm will use only L.

d. When r goes from 1 to 3, what is the magnitude of the substitution effect for L? What is the magnitude of the scale effect for L?

Answer:

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Holding the level of output constant the substitution effect represents the switch from using the more expensive input (K) to the less expensive input (L). Initially in part (b), the firm produced 12 units of output with 6 units of K. To produce the same level of output with just labor (and the same costs), the firm will need 12 units of L. Therefore the substitution effect is a 12 unit increase in L. An increase in r also implies that the firm can no longer use the same combination of inputs to produce their output and at the same time, keep their costs the same. Their new combination of inputs requires the same total cost as before the rental rate increase. But since L has a lower marginal product than K, it implies that the firm will be on a lower isoquant (producing fewer units of output). Rather than using 12 units of L to produce 12 units of output, the firm will now only need 6 L to produce 6 units of output. So the scale effect is a decrease of 6 units of labor. Note that here our analysis involved moving to a lower isocost line, but in a profit maximizing framework, we would find the scale effect corresponding to a movement to a lower isoquant.

7. Suppose a firm’s production function is Q = min(L,K). (This means the level of Q produced is the smaller of L and K).

a. Graph some isoquants for this firm.

Answer:

b. Let w = 2, r = 1, and suppose the firm’s expenditures are C = 12. What are the firm’s demands for L and K? What is the share of labor in the cost of output?

Answer:The graph from part (a) also includes the answer for this problem. The trick for solving questions with a production function such as Q = min(L,K) is to notice that

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the optimizing firm will only ever produce at the kink points on its isoquants. Producing using a bundle of inputs located anywhere else on those isoquants will simply waste either capital or labor. In this case, those kink points are associated with input bundles such that K=L. Finding the cost minimizing bundle of inputs is a matter of solving the linear system of K=L and C=wL+rK, the firm’s cost equation.

C = wL+rK => 12 = 2L +1K => 12 - 2L = K equation of isocost lineNow, since K = L, we solve the above equation to get L i.e.12 – 2L = L => L = 4.Therefore, K = L = 4 and Q = min(L,K) = min(4,4) = 4.

Thus we see that given the input costs, the firm reaches the highest isoquant when output is set at 4 units, which is produced using 4 units of labor and 4 units of capital. Labor’s share of total costs is (total labor costs)/(total costs) or

.

c. Now let w rise to 3. What are the firm’s new demands for L and K?

Answer:

If the wage increases to 3 and total costs stay fixed at 12, the maximum level of production decreases to 3 units which are produced with 3 units each of labor and capital. You can get this result algebraically using the same system of equations mentioned above: K=L and C=wL+rK.

d. Now suppose instead that the production function is Q = min(L,2K). Draw some isoquants for this firm.

Answer:

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e. Again, let w = 2 and r = 1, and suppose the firm’s expenditures are C = 12. What are the firm’s demands for L and K? What is the share of labor in the cost of output?

Answer:The graph from part (d) also shows the answer to this question. Consider the advice from part (b). Under a perfect-complements production technology, we set the two arguments in the production function equal to each other. When Q = min(L, K), the optimal combination of labor and capital will always be such that K=L. When Q = min(L,2K), the optimal combination of labor and capital will always be such that 2K=L. To find the optimal basket of inputs, we solve the following system of equations: 2K=L and C=wL+rK i.e.

C = wL+rK => 12 = 2L +1K => 12 - 2L = K equation of isocost lineNow, since 2K = L => K = 0.5L, we solve the above equation to get L i.e.12 – 2L = 0.5L => L = 4.8Therefore, K = 0.5L = 2.4 and Q = min(L,2K) = min(4.8,4.8) = 4.8.

Thus we find that the firm produces 4.8 units of output using 4.8 units of labor and 2.4 units of capital.

f. Now let w rise to 3. What are the firm’s new demands for L and K?

Answer:

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Using the same method above, we find the optimal basket of inputs solving the following system of equations: 2K=L and C=wL+rK.

C = wL+rK => 12 = 3L +1K => 12 - 3L = K equation of isocost lineNow, since 2K = L => K = 0.5L, we solve the above equation to get L i.e.12 – 3L = 0.5L => L = 3.4.Therefore, K = 0.5L = 1.7 and Q = min(L,2K) = min(3.4,3.4) = 3.4.

Thus we find that the firm produces 3.4 units of output using about 3.4 units of labor and 1.7 units of capital.

g. Why does labor demand fall more in part f than in part c? (Hint: Use one of Marshall’s Laws.)

Answer:Marshall: "The demand for anything is likely to be less elastic, the less important is the part played by the cost of that thing in the total cost of some other thing, in the production of which it is employed."

In our answers above, we have found that labor demand is more elastic under the production technology Q = min(L,2K) than the production technology Q = min(L, K). That is, the demand response was greater under the former technology than the latter given the same change in the wage. Under Q = min(L,2K), capital is twice as productive as labor and so labor costs constitute a greater share of total costs than under Q = min(L,K), where capital and labor are equally productive. When the wage increases, the firm with the technology Q = min(L,2K) is hit that much harder in the labor share of costs, and so its labor demand falls by a greater amount.

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