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Chapter 11: Problem 2

Suppose the production function for widgets is given by \\[ q=K L-.8 K^{2}-. I V \\] where \(q\) represents the annual quantity of widgets produced, \(K\) represents annual capital input, and \(L\) represents annual labor input. a. Suppose \(K-10\); graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(K=10\), graph the \(M P_{L}\) curve. At what level of labor input does \\[ M P_{L}=0 ? \\] c. Suppose capital inputs were increased to \(K-20 .\) How would your answers to parts and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

Short Answer

Expert verified
**Short Answer:** The production function is given as \(q = KL - 0.8K^2 - 0.IV\). When the capital input (K) is 10, the function becomes \(q = 10L - 80\), and the Marginal Product of Labor (MPL) is constant at 10. In this scenario, there is no level of labor input (L) where the MPL equals 0. For a capital input (K) of 20, the MPL is constant at 20. When analyzing the production function for returns to scale, it does not exhibit constant returns to scale. The function has both squared and linear terms, which makes it challenging to determine whether it displays increasing or decreasing returns to scale; it depends on the levels of capital and labor input (K and L).

Step by step solution

01

1. Substitute capital input (K) value into the production function for part (a)#

For part (a), we are given that \(K = 10\). Substitute K into the production function: \[ q = 10L - 0.8(10)^2 - 0. IV \] The production function for part (a) becomes: \[ q = 10L - 80 \]
02

2. Calculate total and average productivity of labor (L) for part (a)#

By rearranging the production function for part (a), we have: \[ Total\, Productivity (q) = 10L - 80 \] The Average Productivity of Labor (APL) can be calculated by dividing total productivity by labor input: \[ APL = \frac {q} {L} = \frac {10L - 80} {L} \]
03

3. Determine the level of labor input (L) that maximizes APL for part (a)#

To find the level of labor input (L) that maximizes APL, we'll take the derivative of APL with respect to L and set it to 0: \[ \frac{d}{dL} \left( \frac {10L - 80} {L} \right) = 0 \] After solving the equation, we find: \[ L = 8 \] Now, plug in the optimal value of L into the production function to find the total productivity (q) at this level: \[ q = 10(8) - 80 = 0 \]
04

4. Calculate the Marginal Product of Labor (MPL) for part (b) and part (c)#

For part (b), the production function is the same as in part (a). Therefore, to find the Marginal Product of Labor (MPL), we can take the derivative of the production function with respect to L: \[ MPL = \frac{dq}{dL} = 10 \] By replacing K with 20 for part (c), the production function becomes: \[ q = 20L - 0.8(20)^{2} - 0.IV \] To find MPL for part (c), we'll take the derivative with respect to L: \[ MPL = \frac{dq}{dL} = 20 \]
05

5. Determine the level of labor input (L) where MPL equals 0 in part (b)#

Since the Marginal Product of Labor (MPL) is constant in part (b), there is no level of labor input (L) where the MPL equals 0.
06

6. Analyze the production function for returns to scale#

To determine whether the production function exhibits constant, increasing, or decreasing returns to scale, we analyze the production function \(q=KL-0.8K^2-0.IV\). For constant returns to scale, the function should exhibit the property: \[ q(tK, tL) = tq(K,L) \] Testing this property, we have: \[ q(tK,tL) = tKL - 0.8(tK)^2 - 0.IV \] The function does not equal \(tq(K,L)\), therefore, the production function does not exhibit constant returns to scale. For increasing returns to scale, \(q(tK, tL) > tq(K,L)\), and for decreasing returns to scale, \(q(tK, tL) < tq(K,L)\). Since the production function has both squared and linear terms, it is not possible to determine whether the function exhibits increasing or decreasing returns to scale generally. The function will exhibit different returns to scale for different levels of capital and labor input (K and L).

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Most popular questions from this chapter

Show that for the constant returns-to-scale CES production function \\[ q=\left[K f+L_{P} Y^{\wedge}\right. \\] a. \(M P_{K}=\left(^{\wedge} | \sim^{P} \text { and } M P_{L}=\left(j-k^{\prime \prime}\right.\right.\) b. \(\quad R T S=[-) \quad\) Use this to show that \(\mathrm{cr}=1 /(1-\mathrm{p})\) \\[ \left.\right|^{K} \boldsymbol{I} \\] c. Determine the output elasticities for Xand \(L\). Show that their sum equals 1 d. Prove that Hence, show Note: The latter equality is useful in empirical work, because in some cases we may approximate \(d q / d L\) by the competitively determined wage rate. Hence, \(a\) can be estimated from a regression of \(\ln (q / L)\) on in \(w\)

The production of barstools \((q)\) is characterized by a production function of the form \\[ q=K^{1 / 2} \cdot U^{2}=V K-L \\] a. What is the average productivity of labor and capital for barstool production \((A P,\) will depend on \(K\), and \(A P_{K}\) will depend on \(L\) ? b. Graph the \(A P_{L}\) curve for \(K=100\) c. For this particular function, show that \(M P_{L}=^{\wedge} A P_{L}\) and \(M P_{K}=\sim A P_{K} .\) Using that infor mation, add a graph of the \(M P\), function to the graph calculated in part (b) (again for \(K=100) .\) What is unusual about this curve? d. Sketch the \(q=10\) isoquant for this production function. e. Using the results from part (c), what is the \(R T S\) on the \(^{\wedge}=10\) isoquant at the points: \(K=L=10 ; L=25, K=4 ;\) and \(K=4, L=25 ?\) Does this function exhibit a diminishing \(R T S ?\)

Constant returns-to-scale production functions are sometimes called homogeneous of degree 1 More generally, as we showed in footnote 1 of Chapter \(5,\) a production function would be said to be homogeneous of degree \(k\) if \\[ f(t K, t L)=t y f(K, L) \\] a. Show that if a production function is homogeneous of degree \(k\), its marginal productiv ity functions are homogeneous of degree \(k-1\) b. Use the result from part (a) to show that marginal productivities for any constant returns-to-scale production function depend only on the ratio \(K / L\) c. Use the result from part (b) to show that the \(R T S\) for a constant returns-to-scale pro duction function depends only on the ratio \(K / L\) d. More generally, show that the \(R T S\) for any homogencous function is independent of the scale of operation - all isoquants are radial expansions of the unit isoquant. Hence, such a function is homothetic. e. Show that the results from part (d) apply to any monotonic transformation of a homo geneous function. That is, show that any such transformation of a homogeneous func tion is homothetic.

As in Problem \(11.8,\) again use Euler's theorem to prove that for a constant returns-to-scale production function with only two inputs \((K \text { and } L), /^{\wedge}\) must be positive. Interpret this result.

Digging clams by hand in Sunset Bay requires only labor input. The total number of clams obtained per hour \((q)\) is given by \\[ q=100 \vee Z \\] where \(L\) is labor input per hour. a. Graph the relationship between \(q\) and \(L\) b. What is the average productivity of labor in Sunset Bay? Graph this relationship and show that \(A P_{L}\) diminishes for increases in labor input. c. Show that the marginal productivity of labor in Sunset Bay is given by \(\boldsymbol{M P}_{L}=50 / V \boldsymbol{Z} .\) Graph this relationship and show that \(M P,

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