ECON 350 - CLASSICAL ECONOMICS Law of Diminishing Returns Question Bank Set 4

ECON 350 - CLASSICAL

ECONOMICS - Law of Diminishing

Returns

Question Bank - Set 4

Liberty University

Question 1

Question

A toy manufacturing company has a production line where they produce stuffed

animals. Initially, the company had four workers and was producing 200 stuffed

animals per day. As they hired more workers, the production increased. How-

ever, after adding the tenth worker, they noticed that the increase in production

was not as significant as before. Explain this phenomenon using the Law of Di-

minishing Returns in the context of this production line.

Solution

The Law of Diminishing Returns states that as additional units of a variable

input (e.g., labor) are added to fixed inputs (e.g., capital), the marginal product

of the variable input will eventually decrease. This means that each additional

unit of input will yield smaller and smaller increases in output.

Step 1: Initially, the production line had four workers and was produc-

ing 200 stuffed animals per day. As they hired more workers, the production

increased. This is because at the beginning, hiring more workers allowed for

better division of labor and specialization, resulting in increased efficiency and

higher production.

Step 2: However, after adding the tenth worker, the company noticed that

the increase in production was not as significant as before. This is because

at this point, the fixed inputs (such as the equipment and workspace) became

limiting factors. The production line became crowded, leading to congestion,

inefficiency, and decreased productivity per worker.

Step 3: As a result, the marginal product of each additional worker started

to diminish. This means that each new worker added did not contribute as

much to the overall production. Eventually, adding more workers may even

lead to negative returns, where the additional workers actually decrease the

total output due to inefficiency and crowding.

In conclusion, the Law of Diminishing Returns explains why, after adding

the tenth worker, the company experienced a diminishing increase in production

on their stuffed animal production line.

Question 2

Question

An agricultural farm initially employs 20 workers to cultivate a wheat field. As

more workers are hired, the total output of wheat increases. However, after

a certain point, the additional output from each additional worker diminishes.

Define the Law of Diminishing Returns in this context and explain how it man-

ifests in the production process.

Solution

The Law of Diminishing Returns, also known as the Law of Diminishing Marginal

Returns, states that as additional units of a variable input (such as labor) are

added to a fixed input (such as land), the marginal product of the variable input

will eventually decrease. This occurs because the fixed input becomes a limiting

factor in the production process.

Step 1: Initially, with 20 workers, the farm sees a significant increase in

wheat output as each worker specializes in a particular task (e.g., planting,

weeding, harvesting). This is known as the stage of increasing returns, where

the marginal product of labor is rising.

Step 2: As more workers are hired beyond the optimal level, the farm

enters the stage of diminishing returns. Each additional worker adds less to the

total wheat output compared to the workers hired before. This is because the

fixed input (land, machinery) becomes a constraint, leading to inefficiencies and

wastage of resources.

Step 3: Eventually, if the farm continues to add more workers, it reaches

the stage of negative returns. In this phase, the total wheat output begins to

decrease, indicating that the additional workers are now causing more harm

than good to the production process. Overcrowding, lack of resources, and

coordination issues may lead to this decline in productivity.

Therefore, the Law of Diminishing Returns highlights the importance of

optimizing input levels in the production process to achieve maximum efficiency

and output levels.

Question 3

Question

A company has a production function described by Q= 2L0.5K0.5, where Qis

the total output, Lis the amount of labor, and Kis the amount of capital. If

the company currently has 2 units of capital, how many units of labor should

they hire in order to maximize output, according to the Law of Diminishing

Returns?

Solution

1. The marginal product of labor (MPL) is given by:

MP L =∂Q

∂L = 0.5×2×L−0.5K0.5=K0.5

2. The marginal product of capital (MPK) is given by:

MP K =∂Q

∂K = 0.5×2×L0.5K−0.5=L0.5

3. The company’s production is subject to the Law of Diminishing Returns

when MPL and MPK decrease as more labor or capital is added, respectively.

At the point of diminishing returns, MPL = MPK for maximizing output. 4.

Setting MPL = MPK:

K0.5=L0.5

5. Given that the company has 2 units of capital, we substitute K= 2:

20.5=L0.5

L= 22= 4

6. Therefore, the company should hire 4 units of labor to maximize output

according to the Law of Diminishing Returns.

Question 4

Question

A company operates a factory with fixed capital and a variable number of work-

ers. Initially, hiring more workers increased the output proportionally due to

specialization and division of labor. However, after reaching a certain point,

adding more workers leads to diminishing returns. Explain the concept of the

Law of Diminishing Returns using a production function and provide a graphical

representation of this concept.

Solution

The Law of Diminishing Returns states that if one input factor is increased

while keeping all other factors constant, the marginal output of that input will

eventually decrease. This law arises due to fixed factors of production (such as

capital) and variable factors of production (such as labor).

Let’s consider a production function Q=f(L), where Qis the total output,

Lis the number of workers, and fis the production function.

Step 1: Increasing Marginal Returns Initially, as more workers are

hired, there is an increase in output due to specialization and division of labor.

This leads to increasing marginal returns. Mathematically, this is represented

by a positive slope of the total product curve.

Step 2: Diminishing Marginal Returns After a certain point, adding

more workers causes the fixed factor (capital) to become a constraint, leading

to diminishing marginal returns. Each additional worker contributes less to the

total output, ultimately resulting in a decrease in the marginal product of labor.

Mathematically, this is represented by a decreasing slope of the total product

curve.

Step 3: Graphical Representation We can represent the concept of di-

minishing returns graphically using the total product curve. Initially, the curve

rises at an increasing rate, indicating increasing marginal returns. However, it

eventually flattens out and starts to slope downward, representing diminishing

marginal returns.

Overall, the Law of Diminishing Returns highlights the importance of effi-

ciency in production processes and the optimal allocation of resources to maxi-

mize output.

Question 5

Question

A firm is currently operating in the short run with one variable input and one

fixed input. The total product of the variable input is given by Q= 10L−0.5L2,

where Qis the total output and Lis the quantity of the variable input. Find

the marginal product of labor and determine at what level of labor does the law

of diminishing returns set in.

Solution

Step 1: To find the marginal product of labor, we differentiate the total product

function Q= 10L−0.5L2with respect to L.

dL = 10 −L

Step 2: The marginal product of labor is given by dQ

dL , so the marginal

product of labor is 10 −L.

Step 3: The law of diminishing returns sets in when the marginal product

of labor starts to decrease. Setting 10 −L= 0, we find the level of labor where

the law of diminishing returns begins:

10 −L= 0

L= 10

Therefore, the law of diminishing returns sets in when L= 10 units.

Question 6

Question

Suppose a farmer has a fixed amount of land and hires labor to cultivate the land.

Initially, as more labor is added, the overall output increases at an increasing

rate. However, at some point, adding more labor starts to yield diminishing

returns. Explain the concept of the Law of Diminishing Returns in the context

of this scenario.

Solution

The Law of Diminishing Returns states that when one factor of production is

increased while other factors are held constant, the marginal increase in output

will start to decrease after a certain point. This concept can be understood in

the context of the farmer and labor scenario described above.

Step 1: Initially, when the farmer hires more labor to cultivate the fixed

amount of land, the overall output increases at an increasing rate. This is

because the additional labor helps in utilizing the land more efficiently, resulting

in higher yields per unit of labor input.

Step 2: However, as more and more labor is added to the fixed amount of

land, a point is reached where the marginal increase in output starts to diminish.

This occurs because the fixed amount of land has a limited capacity to support

additional labor effectively. As a result, each additional unit of labor contributes

less to the overall output compared to the previous units of labor.

Step 3: Beyond this point of diminishing returns, adding even more labor

may lead to a situation where the overall output starts to decrease. This is

because the fixed amount of land becomes overworked or overcrowded with

labor, leading to inefficiencies, resource wastage, and ultimately lower yields.

In summary, the Law of Diminishing Returns highlights the fact that increas-

ing one input factor (labor in this case) while holding other factors constant can

eventually lead to diminished marginal returns and may even result in negative

returns if pushed too far.

Question 7

Question

Suppose a farmer is cultivating a piece of land. Initially, he plants one crop

per square meter and the yield is increasing. After a certain point, he decides

to plant a second crop per square meter as well. Describe how the law of

diminishing returns applies to this scenario and why the farmer might experience

diminishing marginal returns.

Solution

The law of diminishing returns states that as one input is increased while other

inputs are held constant, a point will be reached at which the resulting incre-

ments of output will decrease. This is exactly what the farmer would experience

when planting multiple crops per square meter.

Step 1: Initially, when the farmer plants only one crop per square meter,

he may experience increasing yields as more crops are added to the land. This

is due to factors such as efficient resource allocation, better land utilization, and

economies of scale.

Step 2: However, as the farmer decides to plant a second crop per square

meter, he may still see an increase in total yield, but the rate of increase will

start to diminish. This is because the land may become more crowded, leading

to limited access to sunlight, nutrients, and water for each crop.

Step 3: In the scenario of planting a third crop per square meter, the

farmer is likely to experience diminishing marginal returns. The additional crop

may lead to overcrowding, competition for resources, increased susceptibility to

diseases, and reduced individual crop productivity.

Step 4: Eventually, there may come a point where adding more crops per

square meter could result in a negative impact on total yield. This is known

as the point of diminishing returns, where the costs (such as additional seeds,

labor, and resources) outweigh the benefits (in terms of increased yield).

Therefore, the farmer might experience diminishing marginal returns when

planting multiple crops per square meter due to the limited availability of re-

sources per crop, leading to decreased efficiency and productivity per additional

unit of input.

Question 8

Question

A company produces electronic gadgets in a factory where the production pro-

cess is subject to the Law of Diminishing Returns. Initially, increasing the

number of workers in the factory led to a significant increase in output. How-

ever, at a certain point, adding more workers started to have a diminishing effect

on the overall production efficiency.

Assume that the production with 10 workers resulted in a total output of

500 gadgets per day, while the production with 15 workers resulted in a total

output of 650 gadgets per day.

Using this information, calculate the marginal product of labor (MPL) when

moving from 10 workers to 15 workers in the factory.

Solution

Step 1: Calculate the initial production per worker with 10 workers. Given that

the total output with 10 workers is 500 gadgets per day, the production per

worker can be calculated as:

Production per worker with 10 workers = 500 gadgets

10 workers = 50 gadgets/worker

Step 2: Calculate the initial production with 10 workers. The total produc-

tion with 10 workers is given as 500 gadgets per day.

Step 3: Calculate the total production with 15 workers. Given that the total

output with 15 workers is 650 gadgets per day, the production per worker can

be calculated as:

Production per worker with 15 workers = 650 gadgets

15 workers =130

3gadgets/worker

Step 4: Calculate the marginal product of labor (MPL) moving from 10

workers to 15 workers. The MPL is the additional production gained by adding

one more worker. It can be calculated as the difference between the production

with 15 workers and the production with 10 workers, divided by the additional

workers added:

MPL = Change in Output

Change in Labor =650 −500

15 −10

MPL = 150

5= 30 gadgets/worker

Therefore, the marginal product of labor when moving from 10 workers to

15 workers in the factory is 30 gadgets per worker.

Question 9

Question

Suppose a company produces electronic devices in a factory. The company finds

that when they increase the number of workers in the factory, the production

initially increases at a faster rate, but eventually starts to slow down. Explain

how the Law of Diminishing Returns applies to this scenario and discuss its

implications for the company’s production process.

Solution

The Law of Diminishing Returns states that as one input variable (e.g., labor) is

increased while keeping other input variables (e.g., capital, technology) constant,

the marginal output or return on that input will eventually decrease.

Step 1: Initially, as the company increases the number of workers in the

factory, the production increases at an increasing rate. This is because each

new worker can specialize in a specific task, leading to improved efficiency and

productivity.

Step 2: However, there comes a point where adding more workers becomes

counterproductive. The factory may become overcrowded, leading to inefficien-

cies such as bottlenecks, communication issues, and congestion in workspace.

As a result, the marginal output of each additional worker starts to decrease.

Step 3: This phenomenon can have several implications for the company’s

production process. Firstly, the company may experience diminishing returns

where adding more workers does not proportionally increase output. This can

lead to increased production costs without a significant increase in production

levels.

Step 4: Secondly, the company may need to reevaluate its production pro-

cess and consider other factors besides labor to improve efficiency. This could

involve investing in new technology, optimizing the workflow, or reorganizing

the factory layout to eliminate bottlenecks and improve productivity.

Step 5: In conclusion, understanding and applying the Law of Diminishing

Returns is crucial for companies to optimize their production processes and

maximize output efficiency. By recognizing when adding more of a certain input

variable no longer brings proportional benefits, companies can make informed

decisions to improve their overall performance and sustainability.

Question 10

Question

A company produces bicycles in a factory. The company discovers that after

hiring more workers, the output of bicycles initially increases but eventually

starts to diminish. Explain the concept of the Law of Diminishing Returns in

the context of this scenario.

Solution

The Law of Diminishing Returns, also known as the Law of Variable Propor-

tions, states that if one factor of production is increased while the others remain

constant, the overall returns will eventually decrease.

Step 1: Initially, as the company hires more workers in the factory, the

output of bicycles increases because there are more workers available to work

on the production line. This may lead to better division of labor, increased

efficiency, and higher productivity.

Step 2: However, there comes a point where adding more workers becomes

counterproductive. This is because the factory has a limited amount of machin-

ery, work space, and resources. Adding more workers beyond this point can

lead to overcrowding, inefficiencies, and conflicts.

Step 3: Eventually, the Law of Diminishing Returns sets in, and the ad-

ditional workers start to get in each other’s way, causing bottlenecks in the

production process. This leads to a decrease in the marginal product of each

additional worker, resulting in a diminishing overall output of bicycles.

Step 4: In the context of this scenario, the company needs to find the opti-

mal number of workers that maximizes production efficiency without reaching

the point of diminishing returns. This may involve balancing the number of

workers with available resources and machinery to achieve the highest level of

productivity.

Therefore, the Law of Diminishing Returns highlights the importance of

careful resource allocation and planning to maximize production efficiency and

output in a factory setting.

Question 11

Question

A company that manufactures smartphones is currently producing 1000 units

per week using 10 workers. The company is considering hiring more workers to

increase production. However, based on their analysis, they expect to experience

diminishing returns after a certain point.

If the company hires 5 additional workers, the weekly production is expected

to increase by 400 units. On the other hand, if they hire 10 additional workers,

the weekly production is expected to increase by 600 units.

Determine the point at which the law of diminishing returns sets in for this

company, and explain the implications of this point on their production strategy.

Solution

Step 1: Let xbe the number of workers currently employed by the company. Let

f(x) be the weekly production in units when the company employs xworkers.

Step 2: From the problem, we have the following information: - f(x+ 5) −

f(x) = 400 - f(x+ 10) −f(x) = 600

Step 3: We can express f(x+5)−f(x) and f(x+ 10) −f(x) in terms of f(x)

as follows: - f(x+ 5) −f(x)=(f(x+ 5) −f(x+ 10)) + (f(x+ 10) −f(x)) =

200 + 600 = 800 - f(x+ 10) −f(x) = 2(400) = 800

Step 4: Now, we can solve the equations: - 800 = 400 - 800 = 600

Step 5: Since 800 = 400, there seems to be an inconsistency in the infor-

mation provided. However, in the context of the law of diminishing returns,

this could imply that adding more workers beyond a certain point may not lead

to proportional increases in production. This point is where the diminishing

returns set in.

Step 6: The implications of this point on the production strategy of the

company are that they should carefully consider the marginal benefits of hiring

additional workers. Once they reach the point of diminishing returns, the in-

crease in production per additional worker becomes less significant, and it may

not be cost-effective to continue hiring more workers beyond that point.

Question 12

Question

A company is producing bicycles using a fixed amount of factory space and a

variable number of workers. Initially, the company had one worker and was

able to produce 50 bicycles per day. As more workers were hired, the company

observed the following increase in daily production until reaching a maximum

of 200 bicycles per day with 5 workers. However, when the company hired a

6th worker, the daily production increased only to 210 bicycles.

Given this scenario, analyze the company’s production output in terms of

the Law of Diminishing Returns.

Solution

Step 1: Calculate the marginal product of labor (MPL) for each additional

worker. Let the production output be denoted as Qand the number of workers

as L. The MPL can be calculated using the formula:

MP L =∆Q

∆L

For the initial increase from 1 to 2 workers:

MP L1−2=100 −50

2−1= 50

For the increase from 2 to 3 workers:

MP L2−3=150 −100

3−2= 50

For the increase from 3 to 4 workers:

MP L3−4=175 −150

4−3= 25

For the increase from 4 to 5 workers:

MP L4−5=200 −175

5−4= 25

For the increase from 5 to 6 workers:

MP L5−6=210 −200

6−5= 10

Step 2: Analyze the trend in MPL. Initially, the MPL remains constant at

50 as more workers are added. This suggests increasing returns to scale up to

the 2nd worker. However, after this point, the MPL begins to decline. There is

a noticeable decrease in production efficiency when going from 4 to 5 workers

and from 5 to 6 workers.

Step 3: Interpretation based on the Law of Diminishing Returns. The Law

of Diminishing Returns states that as more units of a variable input are added to

fixed inputs, after a certain point, the marginal product of the variable input will

decrease. This is precisely what we observe in this scenario. The diminishing

returns set in after the 2nd worker is hired, leading to a decline in MPL as

more workers are added. This explains why the company’s production output

increased at a decreasing rate as more workers were employed.

Question 13

Question

A farm has been cultivating a certain crop on a plot of land. By applying more

and more units of a certain input (such as fertilizer), the farm initially saw an

increase in crop yield. However, after a certain point, the increase in crop yield

began to diminish. Explain this phenomenon using the concept of the Law of

Diminishing Returns.

Solution

Step 1: The Law of Diminishing Returns states that as additional units of a

variable input (e.g., fertilizer) are applied to a fixed quantity of a fixed input

(e.g., land), the marginal product of the variable input will eventually decrease.

Step 2: Initially, when the farm applies more units of fertilizer, the crop yield

increases because the additional fertilizer helps to improve the growth conditions

for the crop, leading to a higher yield.

Step 3: However, as more and more fertilizer is added to the fixed plot of

land, there will come a point where the soil becomes saturated with nutrients.

At this stage, adding more fertilizer does not significantly improve the crop yield

as the soil can no longer absorb or utilize the excess fertilizer efficiently.

Step 4: Consequently, the marginal product of each additional unit of fer-

tilizer begins to decrease, leading to diminishing returns. This means that the

increase in crop yield per unit of additional fertilizer applied becomes smaller

and smaller.

Step 5: Ultimately, there may even be a point where adding more fertilizer

can have a negative impact on crop yield, as the soil becomes over-fertilized,

causing harm to the crop rather than benefiting it.

Step 6: Therefore, the phenomenon observed in the scenario, where the

increase in crop yield begins to diminish after a certain point despite applying

more fertilizer, can be explained by the Law of Diminishing Returns.

Question 14

Question

A company manufacturing smartphones is analyzing the production of a new

model in their factory. After a certain point, they observe that adding more

workers to the assembly line leads to a decrease in the marginal product of each

additional worker. Explain the concept of the Law of Diminishing Returns in

this context and how it impacts the company’s production efficiency.

Solution

The Law of Diminishing Returns is a fundamental concept in economics that

states that as more units of a variable input (such as labor) are added to a

fixed input (such as capital), the marginal product of the variable input will

eventually decrease.

Step 1: Production Function In the context of the smartphone manu-

facturing company, the production function relates the inputs (labor, capital,

etc.) to the output of smartphones. Let’s denote the production function as

Q=f(L, K), where Qis the quantity of smartphones produced, Lis the quan-

tity of labor employed, and Kis the quantity of capital.

Step 2: Marginal Product of Labor The marginal product of labor

(MPL) is the additional output gained by employing one more unit of labor

while holding all other factors constant. Mathematically, it is expressed as:

MP L =∆Q

∆L

Step 3: Diminishing Returns Initially, as workers are added to the as-

sembly line, the MPL may increase due to specialization and division of labor.

However, at some point, adding more workers may lead to overcrowding, ineffi-

ciencies, and bottlenecks, causing the MPL to eventually decrease.

Step 4: Impact on Production Efficiency As the company experiences

diminishing returns, the cost of producing each additional smartphone will in-

crease, as more labor is required to produce the same amount of output. This

can lead to inefficiencies, higher costs, and a decrease in overall production

efficiency.

By understanding and managing the Law of Diminishing Returns, the com-

pany can optimize its production process by determining the optimal number of

workers to employ on the assembly line to maximize productivity and minimize

costs.

Question 15

Question

Suppose a firm is producing bicycles and initially hires more workers to increase

production. At some point, the firm experiences diminishing returns to labor.

Explain the concept of the Law of Diminishing Returns in this context and

discuss how it affects the firm’s production efficiency.

Solution

1. Law of Diminishing Returns: The Law of Diminishing Returns states

that as a firm increases one input (e.g., labor) while keeping all other inputs

constant, there will reach a point where the marginal product of that input will

decrease. This means that each additional unit of input will yield diminishing

additional output.

2. When a firm increases the number of workers in the production of bicycles,

initially the output will increase at an increasing rate due to specialization and

division of labor. However, at some point, the firm will reach a stage where

adding more workers would lead to a decline in the marginal product of each

worker.

3. As the firm hires more and more workers beyond the point of diminishing

returns, inefficiencies start to arise. Workers may get in each other’s way, com-

munication breakdowns may occur, and the firm may face logistical challenges

in managing a large workforce effectively.

4. Furthermore, the firm may experience increased production costs as it

hires more workers beyond the optimal level. This could be due to the need for

more supervision, higher wages, or additional training costs.

5. Overall, the Law of Diminishing Returns highlights the importance of

optimizing the use of resources in production. Firms need to find the right

balance in utilizing inputs to maximize output efficiently without falling into

the trap of diminishing returns.

Question 16

Question

A company is producing a certain product using a fixed amount of capital

and variable amounts of labor. The company has observed that initially, as

more units of labor are hired, the marginal product of labor (MPL) increases.

Eventually, however, the MPL starts to decrease.

Given this information, explain the concept of the Law of Diminishing Re-

turns and its implications for production in the short run.

Solution

Step 1: Explanation of Law of Diminishing Returns The Law of Diminish-

ing Returns states that as more units of a variable input (in this case, labor) are

added to a fixed amount of another input (in this case, capital) in the produc-

tion process, the marginal product of the variable input will eventually decrease.

This occurs because the fixed input becomes a limiting factor as more of the

variable input is added.

Step 2: Implications for production in the short run In the short

run, where at least one input is fixed, the Law of Diminishing Returns has

several implications: - Initially, as more units of the variable input (labor)

are added, the total product and marginal product of labor increase due to

specialization and division of labor. - However, as more units of labor are

added beyond a certain point, the marginal product of labor starts to decrease.

This signifies inefficiency and diminishing returns to labor. - The total product

of the company will continue to increase as long as the MPL is positive, but

at a decreasing rate due to diminishing returns. - To maximize profitability,

the company should continue to produce up to the point where the marginal

cost equals the marginal revenue, which will be at a level of labor utilization

before the MPL starts decreasing significantly. - If the company continues to

increase the variable input (labor) beyond the point of diminishing returns, it

may experience negative impacts such as increased costs, reduced profit margins,

and overall inefficiency in production.

Therefore, understanding and applying the Law of Diminishing Returns is

crucial for firms to optimize their production processes and resource allocation

in the short run.

Question 17

Question

A manufacturing plant currently employs 50 workers to produce a certain prod-

uct. The plant manager is considering hiring more workers to increase pro-

duction. However, the plant is already experiencing the effects of the Law

of Diminishing Returns. If each additional worker hired beyond 50 leads to

a smaller increase in production output, how can the manager determine the

optimal number of workers to hire to maximize productivity?

Solution

To determine the optimal number of workers to hire and maximize productivity

while considering the Law of Diminishing Returns, the plant manager can follow

these steps:

Step 1: Calculate the marginal product of labor (MPL). The MPL is the

change in output resulting from adding one more unit of labor.

Step 2: Determine the average product of labor (APL). The APL is the

total output divided by the number of workers.

Step 3: Compare the MPL and APL values. Initially, the MPL should

be greater than the APL, indicating increasing returns to labor. As we hire

more workers, the MPL will eventually start to decrease, showing diminishing

returns.

Step 4: Identify the point where the MPL equals the APL. At this point,

the production output per worker is maximized, indicating the optimal number

of workers to hire for maximum productivity.

Step 5: Beyond this point, where the MPL is less than the APL, further

hiring will result in diminishing returns and reduced productivity. Hence, the

optimal number of workers is the point where the MPL equals the APL.

Question 18

Question

A company produces smartphones and has a production function given by Q=

100L−2L2, where Qis the quantity of smartphones produced and Lis the

amount of labor input. Use the concept of the Law of Diminishing Returns to

explain why there is a point at which adding more labor will lead to a decrease

in the production of smartphones.

Solution

Step 1: To find the point at which adding more labor leads to a decrease in

production, we need to calculate the marginal product of labor (M PL) and

identify where it becomes negative. The marginal product of labor is defined as

the change in output per unit change in labor input.

MPL=dQ

dL = 100 −4L

Step 2: To determine when adding more labor leads to a decrease in pro-

duction, we set MPLto 0 and solve for L.

100 −4L= 0

4L= 100

L=100

4= 25

Step 3: So, at L= 25, the marginal product of labor (M PL) becomes 0.

This means that adding more labor beyond 25 units will lead to a decrease in

production. This is in line with the Law of Diminishing Returns, which states

that as you add more of a variable input (in this case, labor), while keeping other

inputs constant, the marginal product of that input will eventually decrease.

Question 19

Question

Consider a farm that produces wheat. The farm has a total of 100 acres of land.

Initially, the farmer hires 5 workers to work on the farm and applies a fixed

amount of fertilizer per acre. The total output of wheat is 700 bushels. When

the farmer hires an additional worker, the total output of wheat increases to

900 bushels. However, when the farmer hires another worker, the total output

only increases to 950 bushels. Calculate the marginal product of labor for each

additional worker and discuss the concept of the Law of Diminishing Returns

in this scenario.

Solution

Let’s denote the number of workers as Land the total output of wheat as Qin

bushels.

Step 1: Calculate the Marginal Product of Labor



When L= 5, Q= 700



When L= 6, Q= 900



When L= 7, Q= 950

To find the marginal product of labor, we will calculate the additional output

produced when an additional worker is hired:



For the first additional worker (L= 6):

MPL1=Q(L= 6) −Q(L= 5) = 900 −700 = 200 bushels



For the second additional worker (L= 7):

MPL2=Q(L= 7) −Q(L= 6) = 950 −900 = 50 bushels

Therefore, the marginal product of labor for the first additional worker is

200 bushels and for the second additional worker is 50 bushels.

Step 2: Discussion of Law of Diminishing Returns The Law of Di-

minishing Returns states that as more of a variable input (such as labor) is

added to a fixed input (such as land), while holding other inputs constant, the

marginal product of that input will eventually decrease. This can be seen in the

scenario of the wheat farm: as more workers are hired, the additional output

gained from hiring each additional worker decreases. In this case, the marginal

product of labor decreases from 200 bushels to 50 bushels as more workers are

hired, indicating the operation is experiencing diminishing returns.

Question 20

Question

A company is producing widgets and currently employs 10 workers in its pro-

duction process. The company finds that as it hires more workers, the output

of widgets initially increases at an increasing rate, then at a decreasing rate.

The total product of labor is given by the function T P = 100L−2L2, where L

is the number of workers.

1. Determine the marginal product of labor.

2. Calculate the average product of labor.

3. Identify the point at which the law of diminishing returns sets in.

Solution

1. To find the marginal product of labor, we need to calculate the derivative of

the total product with respect to labor, dT P

dL .

dT P

dL =d

dL (100L−2L2) = 100 −4L

2. The average product of labor is given by the total product divided by the

number of workers:

AP =T P

L=100L−2L2

L= 100 −2L

3. The point at which the law of diminishing returns sets in is when the

marginal product of labor starts decreasing. This occurs when dT P

dL = 0.

100 −4L= 0

4L= 100

L= 25

Therefore, the law of diminishing returns sets in when the company employs

25 workers.

Question 21

Question

A company produces cars in a factory where there are three inputs: labor, capi-

tal, and raw materials. The production function is given by Q= 5L0.4K0.6M0.3,

where Lis the units of labor, Kis the units of capital, Mis the units of raw

materials, and Qis the total output of cars.

If the company currently has 50 units of labor, 25 units of capital, and 10

units of raw materials, determine the marginal product of labor.

Solution

Step 1: Calculate the total product (T PL) by varying labor while holding capital

and raw materials constant.

T PL= 5(50)0.4(25)0.6(10)0.3

Step 2: Calculate the total product obtained by adding one more unit of

labor, denoted as T PL+1.

T PL+1 = 5(51)0.4(25)0.6(10)0.3

Step 3: Calculate the marginal product of labor (MPL) as the difference

between T PL+1 and T PL.

MP L =T PL+1 −T PL

Step 4: Substitute the calculated values to find the marginal product of

labor.

Question 22

Question

A company operates a production plant which produces electronic devices. The

company hires additional workers to increase the output of devices. Initially, as

more workers are hired, the output increases at an increasing rate. However,

after a certain point, the output starts to increase at a decreasing rate. Explain

this phenomenon in the context of the Law of Diminishing Returns.

Solution

To understand the phenomenon described in the question, we will discuss the

Law of Diminishing Returns and its implications on the production process.

Step 1: Law of Diminishing Returns The Law of Diminishing Returns

states that as one input is increased while keeping other inputs constant, the

marginal output will eventually decrease. This law assumes that at least one

input is fixed while increasing the quantity of another input.

Step 2: Production Stages When a company hires additional workers to

increase production, it goes through three stages related to the Law of Dimin-

ishing Returns:



Stage 1: Increasing Returns (Marginal Productivity ¿ 0) Initially,

adding more workers increases the total output of devices at an increas-

ing rate. This is because the fixed input (plant and machinery) is being

complemented by more labor, leading to increased efficiency and special-

ization.



Stage 2: Diminishing Returns (Marginal Productivity ¡ 0) After a

certain point, adding more workers contributes less to the total output of

devices. This is because the fixed input becomes a limiting factor, causing

overcrowding and inefficiency among workers. As a result, the marginal

productivity of each additional worker decreases.



Stage 3: Negative Returns (Marginal Productivity ¡ 0) If the

company continues to hire more workers beyond a certain threshold, it

may experience negative returns. This means that the total output of

devices will decrease with each additional worker, leading to inefficiencies

in the production process.

In summary, the Law of Diminishing Returns explains the diminishing marginal

productivity of an input (in this case, labor) as more of it is used in the produc-

tion process. This is why the output of electronic devices initially increases at

an increasing rate but eventually starts to increase at a decreasing rate as more

workers are hired.

Question 23

Question

A company produces smartphones and currently operates with two production

workers. The company experiences diminishing returns when it hires a third

worker. The total output with 2 workers is 400 smartphones per day, and with

3 workers, the total output increases to 420 smartphones per day. If the marginal

product of the third worker is 10 smartphones per day, calculate the marginal

product of the second worker.

Solution

Step 1: Define the terms. Let Q2be the total output with two workers, Q3be

the total output with three workers, MP3be the marginal product of the third

worker, and MP2be the marginal product of the second worker.

Step 2: Apply the Law of Diminishing Returns. According to the Law

of Diminishing Returns, when all factors of production are increased, after a

certain point, the marginal product of an additional worker will decrease.

Step 3: Write the given information as equations. From the question: Q2=

400 smartphones, Q3= 420 smartphones, MP3= 10 smartphones/day.

Step 4: Use the definition of marginal product. The marginal product of

a worker is the additional output gained by adding one more worker. Hence,

MP3=Q3−Q2.

Step 5: Calculate the marginal product of the second worker. We know

MP3= 10 smartphones/day and Q3= 420 smartphones. Therefore, Q2=Q3−

MP3= 420−10 = 410 smartphones. Thus, MP2=Q2−400 = 410−400 = 10

smartphones/day.

Question 24

Question

Suppose a company is producing widgets and initially hires 3 workers to work

on the production line. The company notices that as they hire more workers,

the marginal product of each additional worker starts to decrease.

The total output of widgets with respect to the number of workers is given by

the function Q= 100L−2L2, where Qis the total quantity of widgets produced

and Lis the number of workers.

Determine the point at which the company experiences diminishing returns

to labor.

Solution

Step 1: Find the marginal product of labor (MPL). The marginal product of

labor is given by the derivative of the total output function with respect to the

number of workers (L).

MP L =dQ

dL = 100 −4L

Step 2: Set MPL equal to zero and solve for Lto find the critical point.

100 −4L= 0

4L= 100

L= 25

Step 3: Analyze the critical point. When L= 25, the marginal product of

labor is equal to zero. This indicates that the company is experiencing dimin-

ishing returns to labor at this point.

Therefore, the point at which the company experiences diminishing returns

to labor is when they hire 25 workers.

Question 25

Question

Suppose a production process is initially operating under the Law of Diminishing

Returns. As more units of a variable input are added to the fixed input, the total

output increases at a decreasing rate. At some point, the total output starts

to decrease. Explain this concept in the context of a hypothetical production

scenario, illustrating the diminishing returns using a graph.

Solution

Step 1: The Law of Diminishing Returns states that as more units of a variable

input are added to a fixed input in the production process, the extra output

produced per additional unit of the variable input will eventually diminish.

Step 2: In the context of a hypothetical production scenario, consider a farm

with a fixed amount of land being the fixed input and labor being the variable

input. Initially, as more workers are hired, the total output (total crop yield)

will increase. This is because the labor inputs are effectively utilized on the

available land.

Step 3: However, at some point, adding more labor becomes less effective as

the fixed land resource cannot support an unlimited number of workers. This

results in diminishing returns, where each additional worker contributes less to

the total output.

Step 4: Eventually, the total output may even start to decrease as too many

workers overcrowd the limited land, leading to inefficient use of labor and a

lower overall crop yield. This is an example of the law of diminishing returns in

action.

Step 5: To illustrate this concept graphically, we can plot a graph with the

quantity of labor on the x-axis and the total output (crop yield) on the y-axis.

Initially, the curve will rise steeply, indicating increasing returns. However,

it will eventually level off and start to slope downwards, showing diminishing

returns and eventually negative returns.

Step 6: Thus, the Law of Diminishing Returns highlights the importance

of optimizing the use of inputs in the production process to achieve maximum

efficiency and output.

Question 26

Question

Suppose a farmer is cultivating a field and employing labor to plant corn. Ini-

tially, with the first few workers hired, the yield of corn per worker increases

steadily. However, at a certain point, adding more workers starts to result in

a diminishing marginal product of labor. Explain this phenomenon using the

Law of Diminishing Returns.

Solution

The Law of Diminishing Returns states that as more of a variable input (in this

case, labor) is added to a fixed input (such as land and capital), the marginal

product of the variable input will eventually decrease. This phenomenon can

be explained as follows:

Step 1: Initially, when the farmer assigns a few workers to plant corn,

the fixed inputs (land, capital) are being effectively utilized. Each additional

worker can easily communicate with others, have sufficient space to work, and

have enough tools to perform efficiently. As a result, the marginal product of

labor increases.

Step 2: However, as more workers are hired beyond a certain point, the fixed

inputs become overutilized. Workers start getting in each other’s way, leading

to inefficiencies like congestion, miscommunication, or conflicts over resources.

As a result, the marginal product of labor starts to diminish.

Step 3: Eventually, adding even more workers reaches a point where the

drawbacks of overutilization outweigh the benefits of additional labor. At this

stage, the marginal product of labor becomes negative, meaning each additional

worker actually reduces the total output of corn.

In conclusion, the Law of Diminishing Returns explains how the marginal

product of a variable input will decrease as more of that input is added to a

fixed input. This concept is important for businesses and farmers to optimize

their production processes and resource allocation.

Question 27

Question

A company is analyzing the production of wheat on a farm. Initially, they

found that by adding more labor to the farm, the output of wheat increased at

an increasing rate. However, after a certain point, adding more labor started to

yield diminishing returns. Define the Law of Diminishing Returns and explain

why it occurs in the context of wheat production on the farm.

Solution

Law of Diminishing Returns: The Law of Diminishing Returns states that

as more of a variable input (such as labor or capital) is added to a fixed quantity

of another input (such as land), beyond a certain point, the marginal product

of that variable input will decrease.

Explanation:



Step 1: Initially, when the farm has a fixed amount of land, adding more

labor allows for specialization and division of labor. This leads to an

increase in efficiency and productivity, resulting in an increasing marginal

product of labor.



Step 2: However, as more labor is added beyond a certain point, factors

such as overcrowding, lack of resources, and inefficiencies in coordination

start to hinder the productivity of each additional unit of labor. This

leads to diminishing marginal returns.



Step 3: In the context of wheat production on the farm, the initial in-

crease in labor may result in tasks being completed faster, with more

attention to detail and better utilization of resources. This can lead to a

higher yield of wheat per unit of labor input.



Step 4: When the farm reaches a point where adding more labor com-

promises the quality of work, reduces the available resources per worker

(such as land or tools), or causes logistical issues, the marginal product

of each additional unit of labor decreases. This is the Law of Diminishing

Returns in action.

Therefore, in the context of wheat production on the farm, the Law of Dimin-

ishing Returns occurs due to the trade-off between the benefits of specialization

and efficiency gained initially by adding more labor, and the negative effects

of overcrowding and resource limitations experienced as more labor is added

beyond a certain threshold.

Question 28

Question

A company producing smartphones has a production function given by Q(K, L) =

4K1/3L2/3, where Qis the output quantity, Kis the capital input, and Lis the

labor input. Given that the company has a fixed amount of capital K= 64,

analyze the impact of increasing the amount of labor on the marginal product

of labor and average product of labor.

Solution

1. To find the marginal product of labor, we first need to calculate the total

product of labor (TPL) and then differentiate with respect to L.

TPL(L) = Q(64, L)

= 4(64)1/3L2/3

= 4 ×4L2/3

= 16L2/3

2. The marginal product of labor (MP L) is the rate of change of total

product with respect to labor, given by dT P L

dL .

MP L(L) = dT P L

=d(16L2/3)

=32

3L−1/3

3. The average product of labor (AP L) is the total product divided by the

amount of labor used, given by TPL

AP L(L) = TPL(L)

=16L2/3

= 16L−1/3

4. Analyzing the impact of increasing the amount of labor on the marginal

product of labor and average product of labor: - As more labor is employed: -

The marginal product of labor initially decreases, indicating the law of dimin-

ishing returns. - The average product of labor also initially decreases, reflecting

diminishing returns to labor.

Hence, both the marginal and average product of labor exhibit diminishing

returns as more labor is added to the fixed amount of capital.

Question 29

Question

Suppose a company has a production function given by Q= 5L1

2K3

4, where

Qis the total output, Lis the amount of labor input, and Kis the amount

of capital input. Explain how the Law of Diminishing Returns applies to this

production function.

Solution

The Law of Diminishing Returns states that as additional units of one input

variable are added to production, while holding all other inputs constant, the

marginal return from that input will eventually decrease. In other words, there

is a point where adding more of a variable input will yield diminishing additional

output.

Step 1: Calculate the marginal product of labor The marginal product

of labor (MPL) is the additional output produced when one more unit of labor

is added, while holding the amount of capital constant. It is calculated as the

derivative of the production function with respect to labor:

MPL = ∂Q

∂L =5

2L−1

2K3

Step 2: Calculate the marginal product of capital The marginal

product of capital (MPK) is the additional output produced when one more

unit of capital is added, while holding the amount of labor constant. It is

calculated as the derivative of the production function with respect to capital:

MPK = ∂Q

∂K =15

4L1

2K−1

Step 3: Determine the applicability of the Law of Diminishing

Returns The Law of Diminishing Returns applies when MPL and MPK ex-

hibit diminishing marginal returns. This occurs when MPL and MPK decrease

as more of one input is added, while holding the quantity of the other input

constant.

In this case, we can see that MPL and MPK are dependent on the quantities

of labor and capital in the production function. Depending on the values of L

and K, it is possible for the production function to exhibit diminishing marginal

returns of labor and/or capital, thus illustrating the Law of Diminishing Returns

in action.

Question 30

Question

Consider a factory that produces furniture. The factory currently has 10 work-

ers, each working 8 hours a day. It has been observed that when the number of

workers is increased by 2, the total output of furniture decreases. Explain this

phenomenon in the context of the Law of Diminishing Returns.

Solution

The Law of Diminishing Returns states that as more units of a variable input

(such as labor) are added to a fixed input (such as capital), the marginal output

of each additional unit of the variable input will eventually decrease.

Step 1: Initially, with 10 workers, the factory is able to produce a certain

amount of furniture. Let’s assume this output is 100 units of furniture per day.

Step 2: When the number of workers is increased by 2 to 12 workers, the total

output of furniture decreases. This decrease in total output occurs because the

fixed input (capital, machinery, etc.) is unable to efficiently utilize the additional

labor input. As a result, the marginal productivity of each additional worker

diminishes.

Step 3: In this scenario, the output might decrease to 95 units of furniture

per day when 12 workers are employed. This decrease in total output despite an

increase in labor input exemplifies the Law of Diminishing Returns. As more

units of labor are added, the additional output generated by each extra unit

begins to diminish.

Step 4: Eventually, if more workers are continuously added beyond the point

of diminishing returns, the total output of furniture may start to decrease,

leading to inefficiencies and wastage of resources.

Therefore, the observed phenomenon of a decrease in total furniture output

when the number of workers is increased by 2 can be explained by the Law of

Diminishing Returns.

Question 31

Question

Agricultural economists are studying a farm where labor is the variable input

and land is the fixed input. The farm is currently cultivating wheat, and the

total output is given by the function Q= 10L−0.5L2, where Qis the total

output in bushels and Lis the number of laborers employed. Determine the

point at which the law of diminishing returns sets in.

Solution

1. To find the point at which the law of diminishing returns sets in, we need to

find the point where the marginal product of labor begins to decrease.

2. The marginal product of labor (MP L) is the derivative of the total output

(Q) with respect to labor input (L):

MP L =dQ

dL = 10 −L

3. The law of diminishing returns sets in when MP L starts to decrease.

Setting MP L = 0 and solving for Lgives us:

10 −L= 0

L= 10

4. Therefore, the law of diminishing returns sets in when the farm employs

10 laborers.

Question 32

Question

A company is producing smartphones in a factory. Initially, they had 5 workers

and the production was steadily increasing with each new worker hired. How-

ever, after hiring the 8th worker, the company noticed a decline in the rate of

production increase. Explain the concept of the Law of Diminishing Returns in

this scenario.

Solution

Step 1: The Law of Diminishing Returns states that as additional units of a

variable input are added to fixed inputs, the marginal product of the variable

input will eventually decline.

Step 2: Initially, when the company hired the first few workers, the fixed

input (factory space, machinery, etc.) was enough to support each new worker,

leading to an increase in production.

Step 3: However, as more workers were hired, the fixed inputs became a

constraint, leading to a less efficient use of each worker and a decrease in the

marginal product of labor.

Step 4: In this scenario, hiring the first few workers led to an increase in

production due to specialization, coordination, and efficient use of fixed inputs.

Step 5: When the 8th worker was hired, the fixed inputs were not able to

support the increasing number of workers efficiently, causing a decline in the

marginal product of labor.

Step 6: This decline in the rate of production increase after a certain point

is a clear demonstration of the Law of Diminishing Returns in action.

Question 33

Question

A company is analyzing the production of a new product in its factory. The

company currently has 20 workers and is producing 100 units of the product per

day. After hiring 10 more workers, the company is able to produce 150 units of

the product per day. If the company hires 10 more workers, but the production

only increases to 160 units per day, calculate the point at which the Law of

Diminishing Returns sets in.

Solution

Step 1: Calculate the production per worker before hiring more workers. Let P

be the production per worker before hiring more workers. We have:

P=100

20 = 5 units per worker

Step 2: Calculate the new production per worker after hiring 10 more work-

ers. Let Nbe the total number of workers after hiring 10 more workers. We

have:

N= 20 + 10 = 30 workers

P′=150

30 = 5 units per worker

Step 3: Calculate the production per worker after hiring 20 more workers.

Let N′be the total number of workers after hiring 20 more workers. We have:

N′= 30 + 10 = 40 workers

P′′ =160

40 = 4 units per worker

Step 4: Identify the point at which the Law of Diminishing Returns sets

in. The Law of Diminishing Returns sets in when the additional workers fail to

increase the production proportionally. In this case, after hiring 20 more workers

(N′), the production per worker decreases from 5 units to 4 units, indicating

the beginning of diminishing returns.

Therefore, the point at which the Law of Diminishing Returns sets in is when

the company hires 20 more workers.

Question 34

Question

Consider a firm that produces smartphones. The firm initially hires additional

workers and observes an increase in the number of smartphones produced. How-

ever, after a certain point, the firm notices that the marginal product of each

additional worker starts to decrease. Explain this phenomenon using the Law

of Diminishing Returns.

Solution

The Law of Diminishing Returns states that as more units of a variable input

(in this case, workers) are added to the production process while holding all

other inputs constant, the marginal product of the variable input will eventually

decrease. This phenomenon can be explained through the following steps:

Step 1: Initially, when the firm hires additional workers, the total product

of smartphones increases at an increasing rate. This is because the division of

labor and specialization lead to higher efficiency and productivity.

Step 2: As the firm continues to hire more workers beyond a certain point,

the total product of smartphones still increases, but at a decreasing rate. This is

because there may be a limited amount of machinery or workspace available that

the additional workers have to share, leading to diminishing marginal returns.

Step 3: Eventually, a point is reached where the marginal product of each

additional worker becomes negative. This means that each new worker is ac-

tually reducing the total output of smartphones. This occurs when the fixed

inputs (such as machinery, workspace, or managerial oversight) are being spread

too thin among a large number of variable inputs (workers).

Step 4: At this stage, the firm is experiencing diminishing returns to labor.

To maximize output efficiency, the firm needs to carefully assess the optimal

number of workers to hire to ensure that each worker contributes positively to

the total production process.

In conclusion, the Law of Diminishing Returns explains why there is a point

at which the addition of more workers leads to diminishing marginal product in

the production process.

Question 35

Question

A manufacturing company produces widgets in a factory. The company notices

that when they increase the number of workers in the factory, the total number

of widgets produced increases initially but eventually starts to decrease. Explain

the concept behind this observation using the Law of Diminishing Returns.

Solution

The Law of Diminishing Returns states that as more units of a variable input

(such as labor) are applied to a fixed amount of other inputs (such as capital),

the marginal product of the variable input will eventually decrease.

Step 1: Initially, adding more workers increases the total output. This

is because each additional worker can specialize and increase productivity by

focusing on a specific task. As a result, the marginal product of each new worker

is positive, leading to an increase in total output.

Step 2: However, as more workers are added to the fixed amount of capital,

there is a point where the additional workers become less productive. This

could be due to resource constraints, limited space, communication issues, or

other factors. The fixed inputs cannot support an unlimited number of variable

inputs forever.

Step 3: Eventually, the marginal product of each new worker will start to

decrease. This means that each additional worker contributes less to the total

output than the worker before them. As a result, the total output will increase

at a decreasing rate until it reaches a maximum level.

Step 4: Beyond this point, adding more workers will actually cause the

total output to decrease. This is because the negative impact of overcrowding

or inefficiencies outweighs any benefits of additional workers. The company

will experience diminishing returns, where adding more workers leads to lower

productivity and increased costs.

In conclusion, the Law of Diminishing Returns explains why increasing the

number of workers in a factory initially leads to increased production, but even-

tually results in diminishing marginal returns and a decrease in total output.

much to the overall production. Eventually, adding more workers may even

lead to negative returns, where the additional workers actually decrease the

total output due to inefficiency and crowding.

In conclusion, the Law of Diminishing Returns explains why, after adding

the tenth worker, the company experienced a diminishing increase in production

on their stuffed animal production line.

Question 2

Question

An agricultural farm initially employs 20 workers to cultivate a wheat field. As

more workers are hired, the total output of wheat increases. However, after

a certain point, the additional output from each additional worker diminishes.

Define the Law of Diminishing Returns in this context and explain how it man-

ifests in the production process.

Solution

The Law of Diminishing Returns, also known as the Law of Diminishing Marginal

Returns, states that as additional units of a variable input (such as labor) are

added to a fixed input (such as land), the marginal product of the variable input

will eventually decrease. This occurs because the fixed input becomes a limiting

factor in the production process.

Step 1: Initially, with 20 workers, the farm sees a significant increase in

wheat output as each worker specializes in a particular task (e.g., planting,

weeding, harvesting). This is known as the stage of increasing returns, where

the marginal product of labor is rising.

Step 2: As more workers are hired beyond the optimal level, the farm

enters the stage of diminishing returns. Each additional worker adds less to the

total wheat output compared to the workers hired before. This is because the

fixed input (land, machinery) becomes a constraint, leading to inefficiencies and

wastage of resources.

Step 3: Eventually, if the farm continues to add more workers, it reaches

the stage of negative returns. In this phase, the total wheat output begins to

decrease, indicating that the additional workers are now causing more harm

than good to the production process. Overcrowding, lack of resources, and

coordination issues may lead to this decline in productivity.

Therefore, the Law of Diminishing Returns highlights the importance of

optimizing input levels in the production process to achieve maximum efficiency

and output levels.

Question 3

Question

A company has a production function described by Q= 2L0.5K0.5, where Qis

the total output, Lis the amount of labor, and Kis the amount of capital. If

the company currently has 2 units of capital, how many units of labor should

they hire in order to maximize output, according to the Law of Diminishing

Returns?

Solution

1. The marginal product of labor (MPL) is given by:

MP L =∂Q

∂L = 0.5×2×L−0.5K0.5=K0.5

2. The marginal product of capital (MPK) is given by:

MP K =∂Q

∂K = 0.5×2×L0.5K−0.5=L0.5

3. The company’s production is subject to the Law of Diminishing Returns

when MPL and MPK decrease as more labor or capital is added, respectively.

At the point of diminishing returns, MPL = MPK for maximizing output. 4.

Setting MPL = MPK:

K0.5=L0.5

5. Given that the company has 2 units of capital, we substitute K= 2:

20.5=L0.5

L= 22= 4

6. Therefore, the company should hire 4 units of labor to maximize output

according to the Law of Diminishing Returns.

Question 4

Question

A company operates a factory with fixed capital and a variable number of work-

ers. Initially, hiring more workers increased the output proportionally due to

specialization and division of labor. However, after reaching a certain point,

adding more workers leads to diminishing returns. Explain the concept of the

Law of Diminishing Returns using a production function and provide a graphical

representation of this concept.

Solution

The Law of Diminishing Returns states that if one input factor is increased

while keeping all other factors constant, the marginal output of that input will

eventually decrease. This law arises due to fixed factors of production (such as

capital) and variable factors of production (such as labor).

Let’s consider a production function Q=f(L), where Qis the total output,

Lis the number of workers, and fis the production function.

Step 1: Increasing Marginal Returns Initially, as more workers are

hired, there is an increase in output due to specialization and division of labor.

This leads to increasing marginal returns. Mathematically, this is represented

by a positive slope of the total product curve.

Step 2: Diminishing Marginal Returns After a certain point, adding

more workers causes the fixed factor (capital) to become a constraint, leading

to diminishing marginal returns. Each additional worker contributes less to the

total output, ultimately resulting in a decrease in the marginal product of labor.

Mathematically, this is represented by a decreasing slope of the total product

curve.

Step 3: Graphical Representation We can represent the concept of di-

minishing returns graphically using the total product curve. Initially, the curve

rises at an increasing rate, indicating increasing marginal returns. However, it

eventually flattens out and starts to slope downward, representing diminishing

marginal returns.

Overall, the Law of Diminishing Returns highlights the importance of effi-

ciency in production processes and the optimal allocation of resources to maxi-

mize output.

Question 5

Question

A firm is currently operating in the short run with one variable input and one

fixed input. The total product of the variable input is given by Q= 10L−0.5L2,

where Qis the total output and Lis the quantity of the variable input. Find

the marginal product of labor and determine at what level of labor does the law

of diminishing returns set in.

Solution

Step 1: To find the marginal product of labor, we differentiate the total product

function Q= 10L−0.5L2with respect to L.

dL = 10 −L

Step 2: The marginal product of labor is given by dQ

dL , so the marginal

product of labor is 10 −L.