Please answer part d,e and fand here is the answer of first three parts:part c answer is 1350 **Problem 1**A soap manufacturer has the production function $ q = LK^2 $ and faces a wage rate of $45 and rental rate of capital of $90.(a) **Find the cost-minimizing ratio of capital to labor.** (This is also called the output expansion path.)(b) **Suppose the firm plans to produce $ q = 1,000 $.** Determine the cost-minimizing amount of labor and capital.(c) **How much will it cost to produce $ q = 1,000 $ (long run)?**(d) **Suppose the firm decides to increase its output to $ q = 1331 $, but is unable to increase its quantity of capital from the amount in b.** How much labor will the firm need to employ and what will be the cost of producing $ q = 1331 $?(e) **If the firm continues to produce $ q = 1331 $ over a long period of time, how will it alter its capital and labor from part d?** What will it cost to produce $ q = 1331 $ in the long run?(f) **On a graph, draw the two isoquants from above and the isocost lines corresponding to the LR and SR costs (there should be three).** Identify the optimal L and K for each part on the graph.**Explanation of Graphs and Diagrams:**- **Isoquants:** These curves represent combinations of labor (L) and capital (K) that produce the same output level. In this case, draw isoquants for $ q = 1,000 $ and $ q = 1331 $.- **Isocost Lines:** These lines show combinations of L and K that result in the same total cost. You will draw isocost lines based on prevailing wage and rental rates for long-run and short-run scenarios.- **Intersection Points:** The points where isoquants and isocost lines intersect indicate the optimal combinations of L and K for different production levels and time horizons. **Part A:** The cost-minimizing ratio of capital (K) to labor (L) ratio can be determined by using the following rule:MRTS (marginal rate of technical substitution) = W/R = 45/90 = 0.5.MRTS = $\frac{\partial Q}{\partial L} \big/ \frac{\partial Q}{\partial K} = \frac{K^2}{2LK} = \frac{K}{2L}$MRTS = $\frac{w}{r} = \frac{45}{90} = 0.5$MRTS = $\frac{K}{2L} = 0.5$$\frac{K}{L} = 1$**Part B:** The amount (amt) of L and K that minimizes the cost (C) of producing 1000 units of output (o/p) will be the one that lies on the expansion path also. Therefore, by using the equation $K/L=1$ or $K=L$, we can calculate the cost-minimizing value of K and L:Q = LK$^2 = 1000$L(L)$^2$ = 1000 .............. \{K = L\}L$^3$ = 1000L = K = 10**Part C:** The cost of producing 1000 units is:

A soap manufacturer has the production function q = LK² and faces a wage rate of $45 and rental rate of capital of $90. (a) Find the cost-minimizing ratio of capital to labor. (This is also called the output expansion path). (b) Suppose the firm plans to produce q=1,000. Determine the cost- minimizing amount of labor and capital. (c) How much will it cost to produce q=1,000 (long run)? (d) Suppose the firm decides to increase its output to q = 1331, but is unable to increase its quantity of capital from the amount in b. How much labor will the firm need to employ and what will be the cost of producing q = 1331? (e) If the firm continues to produce q = 1331 over a long period of time, how will it alter its capital and labor from part d.? What will it cost to produce q = 1331 in the long run? On a graph, draw the two isoquants from above and the isocost lines corresponding to the LR and SR costs (there should be three). Identify the optimal L and K for each part on the graph. (f)

A soap manufacturer has the production function q = LK² and faces a wage rate of $45 and rental rate of capital of $90. (a) Find the cost-minimizing ratio of capital to labor. (This is also called the output expansion path). (b) Suppose the firm plans to produce q=1,000. Determine the cost- minimizing amount of labor and capital. (c) How much will it cost to produce q=1,000 (long run)? (d) Suppose the firm decides to increase its output to q = 1331, but is unable to increase its quantity of capital from the amount in b. How much labor will the firm need to employ and what will be the cost of producing q = 1331? (e) If the firm continues to produce q = 1331 over a long period of time, how will it alter its capital and labor from part d.? What will it cost to produce q = 1331 in the long run? On a graph, draw the two isoquants from above and the isocost lines corresponding to the LR and SR costs (there should be three). Identify the optimal L and K for each part on the graph. (f)

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

Please answer part d,e and f

and here is the answer of first three parts:

part c answer is 1350

**Problem 1**

A soap manufacturer has the production function $ q = LK^2 $ and faces a wage rate of $45 and rental rate of capital of $90.

(a) **Find the cost-minimizing ratio of capital to labor.** (This is also called the output expansion path.)

(b) **Suppose the firm plans to produce $ q = 1,000 $.** Determine the cost-minimizing amount of labor and capital.

(c) **How much will it cost to produce $ q = 1,000 $ (long run)?**

(d) **Suppose the firm decides to increase its output to $ q = 1331 $, but is unable to increase its quantity of capital from the amount in b.** How much labor will the firm need to employ and what will be the cost of producing $ q = 1331 $?

(e) **If the firm continues to produce $ q = 1331 $ over a long period of time, how will it alter its capital and labor from part d?** What will it cost to produce $ q = 1331 $ in the long run?

(f) **On a graph, draw the two isoquants from above and the isocost lines corresponding to the LR and SR costs (there should be three).** Identify the optimal L and K for each part on the graph.

**Explanation of Graphs and Diagrams:**
- **Isoquants:** These curves represent combinations of labor (L) and capital (K) that produce the same output level. In this case, draw isoquants for $ q = 1,000 $ and $ q = 1331 $.
- **Isocost Lines:** These lines show combinations of L and K that result in the same total cost. You will draw isocost lines based on prevailing wage and rental rates for long-run and short-run scenarios.
- **Intersection Points:** The points where isoquants and isocost lines intersect indicate the optimal combinations of L and K for different production levels and time horizons.

**Part A:** The cost-minimizing ratio of capital (K) to labor (L) ratio can be determined by using the following rule:

MRTS (marginal rate of technical substitution) = W/R = 45/90 = 0.5.

MRTS = $\frac{\partial Q}{\partial L} \big/ \frac{\partial Q}{\partial K} = \frac{K^2}{2LK} = \frac{K}{2L}$

MRTS = $\frac{w}{r} = \frac{45}{90} = 0.5$

MRTS = $\frac{K}{2L} = 0.5$

$\frac{K}{L} = 1$

**Part B:** The amount (amt) of L and K that minimizes the cost (C) of producing 1000 units of output (o/p) will be the one that lies on the expansion path also. Therefore, by using the equation $K/L=1$ or $K=L$, we can calculate the cost-minimizing value of K and L:

Q = LK$^2 = 1000$

L(L)$^2$ = 1000 .............. \{K = L\}

L$^3$ = 1000

L = K = 10

**Part C:** The cost of producing 1000 units is:

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