ECON 350 - CLASSICAL ECONOMICS Law of Diminishing Returns Question Bank Set 5

ECON 350 - CLASSICAL

ECONOMICS - Law of Diminishing

Returns

Question Bank - Set 5

Liberty University

Question 1

Question

A company is producing shirts in a factory. The company observes that as

they increase the number of workers in the factory, the additional output from

each additional worker starts to decrease. Define the term ”Law of Diminishing

Returns” and explain how it applies to the production of shirts in this factory.

Solution

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

Returns, states that as more of one input is added while all other inputs are held

constant, the additional output that can be produced from each additional unit

of the input will eventually decrease. This results in a decrease in the marginal

product of the input.

In the context of the production of shirts in the factory, the application of

the Law of Diminishing Returns can be explained as follows:

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

production of shirts increases at an increasing rate due to specialization and

division of labor. Each additional worker adds more to the total output than

the worker before.

Step 2: However, after a certain point, adding more workers to the factory

may lead to overcrowding, inefficiency, and coordination issues. This results

in the marginal product of each additional worker decreasing, as the factory is

unable to efficiently utilize the additional labor.

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

does not increase the output of shirts or may even decrease it. This is the point

where the Law of Diminishing Returns sets in, indicating that the company is

operating beyond its optimal level of input.

Step 4: It is important for the company to identify this point and optimize

the number of workers in the factory to ensure maximum efficiency and produc-

tivity in shirt production. This involves balancing the costs of additional labor

with the benefits of increased output while considering the diminishing returns

associated with each additional worker.

Question 2

Question

Consider a production process where three factors of production, labor (L),

capital (K), and land (T), are being used to produce output. The production

function is given by Q= 2L1/2K1/3T1/6. If the amount of capital is fixed at

K= 8, analyze how the total output Qchanges as the amount of labor Land

land Tare increased based on the Law of Diminishing Returns.

Solution

1. To analyze how the total output Qchanges as the amount of labor Land

land Tare increased, we will use partial derivatives to find the marginal product

of labor and the marginal product of land.

2. The marginal product of labor MPL is given by:

MPL = ∂Q

∂L =d

dL (2L1/2K1/3T1/6)

3. By differentiating with respect to L, we get:

MPL = L−1/2K1/3T1/6

4. Similarly, the marginal product of land MPT is given by:

MPT = ∂Q

∂T =d

dT (2L1/2K1/3T1/6)

5. Differentiating with respect to T, we find:

MPT = 1

6L1/2K1/3T−5/6

6. Now, let’s analyze the effects of increasing Land Ton the total output

Q, keeping K= 8 constant.

7. Since the exponents are all positive, the marginal product of both labor

and land will decrease as Land Tincrease (due to diminishing returns).

8. Therefore, increasing the amount of labor Land land Twill lead to a

smaller increase in total output Qas the law of diminishing returns sets in.

9. It is important to note that the fixed amount of capital Kwill eventually

become a limiting factor, causing further diminishing returns as Land Tare

increased beyond a certain point.

Question 3

Question

A company has a production function given by Q= 10L0.5K0.5, where Qis the

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

company currently has 20 units of capital and is considering hiring more labor,

at what point will the law of diminishing returns begin to apply?

Solution

Step 1: To determine at what point the law of diminishing returns begins to

apply, let’s first calculate the marginal product of labor (M P L) when L= 1

and K= 20.

MP L =∂Q

∂L = 5L−0.5K0.5

MP L(1,20) = 5(1)−0.5(20)0.5= 5(1)(4) = 20

Step 2: Next, let’s calculate the average product of labor (AP L) when L= 1

and K= 20.

AP L =Q

L= 10L0.5K0.5

AP L(1,20) = 10(1)0.5(20)0.5= 10(1)(4) = 40

Step 3: The law of diminishing returns begins to apply when the marginal

product of labor starts to decrease. This indicates that each additional unit of

labor is adding less to total output. Therefore, based on our calculation, the

law of diminishing returns will begin to apply after hiring the first unit of labor.

Question 4

Question

A farmer has a fixed amount of land to cultivate wheat. Initially, he plants

a small area and sees a good yield in the crop. However, as he continues to

increase the area of land planted with wheat, he notices that the additional

output generated by each additional unit of land begins to decrease. Explain

this phenomenon using the Law of Diminishing Returns.

Solution

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

states that as one input is increased (while other inputs are held constant), there

is a point at which the marginal (additional) output generated by each additional

unit of input will start to decrease. This phenomenon is often observed in

agriculture and production processes.

Step 1: Initially, when the farmer plants a small area of land with wheat,

he is able to manage and fertilize the crop effectively, resulting in a good yield.

Step 2: As the farmer increases the area of land planted with wheat, he

may start to face constraints such as limited fertilizer, water, sunlight, or la-

bor. Despite increasing these inputs, the additional output generated by each

additional unit of land starts to decrease.

Step 3: This decrease in marginal output can be explained by the Law

of Diminishing Returns. The fixed amount of land has a limited capacity to

produce crops, and as more inputs are added, they become less effective or

efficient in generating additional output.

Step 4: The diminishing returns occur because the fixed factors of pro-

duction (limited land in this case) cannot be easily increased. This leads to

a situation where the marginal product of the variable input (additional land)

decreases, causing a reduction in the overall output generated.

Step 5: In summary, the Law of Diminishing Returns explains how the

marginal output of each additional unit of input diminishes as more units of

that input are added, holding other inputs constant. This phenomenon is crucial

for producers to consider when making decisions about production levels and

resource allocation.

Question 5

Question

A farmer is considering expanding his wheat farm by adding more workers each

day. He notices that as he hires more workers, the additional output produced

by each additional worker starts to decrease. This phenomenon is a classic

example of the Law of Diminishing Returns.

Suppose the farmer’s production function is given by Q= 5L−0.5L2, where

Qrepresents the total output and Lrepresents the number of workers hired.

If the farmer currently has 10 workers, how many additional workers should

he hire to maximize his total output?

Solution

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

The marginal product of labor (MPL) is the additional output produced by

adding one more unit of labor. To find MPL, we differentiate the production

function with respect to L:

dL = 5 −L

Now, plug in L= 10 to find the MPL when the farmer has 10 workers:



L=10 = 5 −10 = −5

Step 2: Determine the optimal number of workers

To maximize total output, the farmer should hire workers until the MPL is

equal to zero. Set the MPL equal to zero and solve for L:

5−L= 0

L= 5

Therefore, the farmer should hire 5 additional workers to maximize his total

output.

Question 6

Question

Consider a production process where a company is producing smartphones. The

company has fixed its capital and labor inputs at certain levels. Initially, as

they increase the number of workers employed, the production of smartphones

increases at an increasing rate. However, after a certain point, adding more

workers starts to yield diminishing returns, and eventually negative returns.

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

Solution

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

tions, states that if one input in the production process is increased while all

other inputs are held constant, a point will be reached where the resulting in-

crease in output per unit of the input will start to diminish. In some cases,

adding more of that input may eventually lead to a decrease in output.

Step 1: Increasing Labor Inputs Initially, when more labor is added to

the fixed amount of capital in the production of smartphones, the total output

increases at an increasing rate. This is because the specialization and division

of labor lead to higher efficiency and productivity.

Step 2: Diminishing Returns After a certain point, adding more labor

will cause the marginal product of labor to decrease. This is because the fixed

amount of capital becomes a limiting factor for the increasing number of workers.

As a result, each additional worker contributes less to the total output than the

previous worker, leading to diminishing returns.

Step 3: Negative Returns If the company continues to add more and

more labor beyond the point of diminishing returns, the total output will start

to decrease. This is known as negative returns or decreasing returns to scale.

At this stage, the inefficiencies caused by overcrowding and lack of resources

outweigh any marginal gains from adding more labor.

In conclusion, the Law of Diminishing Returns highlights the importance of

optimizing the use of all inputs in the production process to maximize efficiency

and output. It serves as a crucial concept in production theory and managerial

economics.

Question 7

Question

A firm is currently operating with two units of labor and two units of capital.

The total output produced is 100 units. If the firm adds a third unit of la-

bor while keeping the capital constant, the total output increases to 120 units.

However, when the firm adds a fourth unit of labor while keeping the capital

constant, the total output increases to only 130 units. Determine whether the

law of diminishing returns is evident in this scenario.

Solution

To determine whether the law of diminishing returns is evident, we need to

analyze the marginal product of labor in this scenario.

Step 1: Calculate the marginal product of labor for each additional unit of

labor added. The marginal product of labor is the additional output produced

when one extra unit of labor is added, while keeping all other factors constant.

It is calculated as:

MP L =∆Q

∆L

where: - M P L is the marginal product of labor, - ∆Qis the change in total

output, and - ∆Lis the change in units of labor.

Given that the firm initially has 2 units of labor and 2 units of capital, and

then adds a third unit of labor:

MP L1=120 −100

3−2= 20

Therefore, the marginal product of labor for the third unit is 20 units.

Step 2: Calculate the marginal product of labor for the fourth unit of labor

added. Similarly, when the firm adds a fourth unit of labor, the marginal

product of labor is:

MP L2=130 −120

4−3= 10

Therefore, the marginal product of labor for the fourth unit is 10 units.

Step 3: Analyze the results. According to the law of diminishing returns,

as more units of a variable input (labor) are added while keeping other inputs

constant, there will initially be an increasing marginal product, but eventually,

the marginal product will start to diminish.

In this scenario, the marginal product of labor decreases from 20 units to

10 units when the fourth unit of labor is added. This signifies that the law

of diminishing returns is evident, as the additional output produced from each

additional unit of labor diminishes as more labor is added while the level of

capital remains constant.

Question 8

Question

A company produces a certain product using a combination of labor and capital.

Initially, the company employed 100 workers and 10 machines in its production

process. As the company expanded, it decided to hire more workers while

keeping the number of machines constant at 10. It was observed that as more

workers were hired, the marginal product of labor initially increased, reached a

peak, and then started to decrease. Explain this phenomenon in the context of

the Law of Diminishing Returns.

Solution

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

is increased while keeping other factors (e.g., capital) constant, the marginal

product of that input factor will eventually diminish. This can be explained

through the following steps:

Step 1: Definition of Marginal Product

The marginal product of labor is defined as the additional output produced

by employing one more unit of labor while keeping all other factors constant.

It is an important concept in understanding the Law of Diminishing Returns.

Step 2: Initially Increasing Marginal Product

Initially, when the company employed 100 workers and 10 machines, increas-

ing the number of workers may lead to a situation where workers can specialize,

coordinate better, and use the machines more efficiently. This can result in an

increase in the marginal product of labor.

Step 3: Peaking Marginal Product

As more workers are hired, a point will be reached where the additional

worker’s marginal contribution to output starts to decrease. This is because

factors like limited space, communication issues, and congestion could start to

emerge, causing the marginal product of labor to peak.

Step 4: Decreasing Marginal Product

Beyond the point of peak marginal product, adding more workers leads to

overcrowding, lack of coordination, and other inefficiencies. This causes the

marginal product of labor to decrease as each additional worker contributes less

to the total output.

Therefore, the observation of the marginal product of labor initially increas-

ing, reaching a peak, and then decreasing as more workers are hired while keep-

ing the number of machines constant is a manifestation of the Law of Dimin-

ishing Returns.

Question 9

Question

Suppose a manufacturing firm initially increases its labor input while keeping

all other inputs constant. As more labor is added, the firm experiences in-

creasing marginal returns at first, followed by decreasing marginal returns and

eventually negative marginal returns. Explain this phenomenon using the Law

of Diminishing Returns.

Solution

To understand this phenomenon, we can look at the Law of Diminishing Re-

turns, which states that as one input variable is increased, other inputs being

constant, a point will be reached where the marginal increase in output de-

creases. Let’s break down the scenario step by step.

Step 1: Increasing Marginal Returns At the beginning, the firm expe-

riences increasing marginal returns as more labor is added. This is because the

fixed input (such as capital or machinery) is being efficiently utilized with the

additional labor. As a result, each unit of labor added contributes more to the

total output, leading to an increase in marginal productivity.

Step 2: Decreasing Marginal Returns As the firm continues to increase

labor input, a point is reached where the fixed input becomes a constraint. The

fixed input cannot be increased, so additional units of labor start to experience

diminishing marginal returns. The fixed input is being overutilized relative to

the variable input, causing inefficiencies and a decrease in marginal productivity.

Step 3: Negative Marginal Returns Eventually, the firm reaches a point

where adding more units of labor actually leads to negative marginal returns.

At this stage, the fixed input is being severely overutilized, leading to congestion

and inefficiencies in the production process. The additional units of labor detract

from the output rather than adding to it, resulting in a decrease in total output.

In conclusion, the phenomenon where a firm experiences increasing marginal

returns, followed by decreasing marginal returns and negative marginal returns,

can be explained by the Law of Diminishing Returns. As the firm increases one

input variable (labor) while keeping other inputs constant, the law dictates that

efficiency will eventually decline, leading to diminishing and negative marginal

returns.

Question 10

Question

A company produces bicycles and currently uses two workers to assemble them.

The company has observed that the production rate is not increasing as expected

with this level of labor input. To investigate this, the company decides to

conduct an experiment by adding a third worker to the assembly line.

If the company experiences diminishing returns from adding the third worker,

explain why this phenomenon is occurring.

Solution

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

that as additional units of a variable input are added to a fixed input, at some

point the marginal product of the variable input will decrease.

Step 2: Application to the Company’s Experiment When the company adds

a third worker to the assembly line, the fixed input is the production machinery

and equipment, which remains constant. The variable input is the labor, with

the third worker being the additional unit.

Step 3: Initial Increase in Production Initially, adding a third worker may

lead to an increase in production. The workers can specialize in different tasks,

leading to higher efficiency and output.

Step 4: Diminishing Returns However, as more workers are added, they may

start to get in each other’s way, causing delays, confusion, and inefficiencies.

This leads to a decrease in the marginal product of each additional worker. The

fixed input (machinery and equipment) cannot be easily increased to keep up

with the increasing labor input.

Step 5: Why it Occurs Diminishing returns occur because the ratio of the

fixed input to the variable input changes. Initially, the fixed input is effectively

utilized as more variable input is added, leading to increasing returns. How-

ever, beyond a certain point, the fixed input becomes a constraint, limiting the

effectiveness of additional variable inputs and causing diminishing returns.

Step 6: Conclusion In this scenario, the company is likely experiencing dimin-

ishing returns because the fixed input (production machinery and equipment)

is not easily expanded to accommodate the increasing variable input (labor).

As a result, adding more workers does not lead to a proportional increase in

production and may even decrease efficiency.

Question 11

Question

A farm initially has 10 workers cultivating a field, and the output of crops per

day is 1000 kg. When 5 more workers are hired, the output increases to 1500

kg per day. However, when 5 additional workers are hired (bringing the total to

20 workers), the output only increases to 1600 kg per day. Determine whether

the law of diminishing returns is evident in this scenario.

Solution

Step 1: Calculate the marginal product of labor for the first increase in workers.

Let Q1be the output when 10 workers are hired, and Q2be the output when

15 workers are hired. The marginal product of labor is calculated as:

MP L1=Q2−Q1

L2−L1

where L1and L2represent the number of workers in each scenario.

Substitute the values given:

MP L1=1500 −1000

15 −10 =500

5= 100 kg/worker

Step 2: Calculate the marginal product of labor for the second increase in

workers. Let Q2be the output when 15 workers are hired, and Q3be the output

when 20 workers are hired. The marginal product of labor is calculated as:

MP L2=Q3−Q2

L3−L2

where L2and L3represent the number of workers in each scenario.

Substitute the values given:

MP L2=1600 −1500

20 −15 =100

5= 20 kg/worker

Step 3: Analyze the marginal product of labor results. Based on the cal-

culations, the marginal product of labor decreases from 100 kg/worker to 20

kg/worker as more workers are hired. This demonstrates the law of diminishing

returns, where additional inputs lead to diminishing marginal returns. In this

case, the third worker increase resulted in a less significant increase in output

than the second worker increase.

Question 12

Question

A company produces smartphones and employs a certain number of workers in

its production facility. The company notices that as more workers are hired,

the total output increases, but at a decreasing rate. Explain the concept of the

Law of Diminishing Returns in the context of this scenario.

Solution

To understand the Law of Diminishing Returns, let’s consider the production

of smartphones in a company’s facility.

Step 1: Initially, as more workers are hired, the total output of smartphones

increases. This is due to factors such as specialization, division of labor, and

efficient use of resources.

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

not increase productivity as much. This is because each additional worker may

lead to inefficiencies, such as overcrowding, communication issues, or resource

scarcity.

Step 3: At this stage, the company experiences diminishing returns. This

means that the marginal product of each additional worker decreases, resulting

in a flatter total output curve.

Step 4: Eventually, if the company continues to add more workers beyond a

certain point, the total output may start to decrease. This is known as negative

returns.

Step 5: The Law of Diminishing Returns highlights the idea that in the

short run, while keeping other factors constant, there is a point at which the

addition of more input (in this case, workers) leads to smaller increases in out-

put.

In conclusion, understanding and recognizing the Law of Diminishing Re-

turns is crucial for businesses to optimize their production processes and resource

allocation efficiently.

Question 13

Question

A company that produces bicycles is experiencing the law of diminishing returns

in its production process. Initially, as more workers were hired, the company saw

a significant increase in the number of bicycles produced. However, at a certain

point, each additional worker hired resulted in a smaller increase in production.

If the company’s production function is given by Q= 10L−0.1L2, where Q

represents the number of bicycles produced and Lrepresents the number of

workers hired, determine the number of workers that will maximize production.

Solution

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

(MPL) is given by the derivative of the production function with respect to the

number of workers, L. Thus, we have:

MP L =dQ

dL =d

dL (10L−0.1L2) = 10 −0.2L

Step 2: Set the MPL equal to zero to find the critical point. Setting M P L =

0 and solving for L, we get:

10 −0.2L= 0

0.2L= 10

L=10

0.2= 50

Step 3: Determine the concavity of the production function. To determine if

this critical point is a maximum, we need to check the concavity of the function.

The second derivative of the production function with respect to Lis:

d2Q

dL2=d

dL (10 −0.2L) = −0.2

Since the second derivative is negative, the function is concave downward,

and the critical point corresponds to a maximum.

Step 4: Find the maximum production level. Substitute L= 50 back into

the production function to find the maximum number of bicycles produced:

Q= 10(50) −0.1(50)2= 500 −250 = 250

Therefore, the number of workers that will maximize production for the

company is 50, with a maximum production of 250 bicycles.

Question 14

Question

Suppose a company is producing bicycles in a factory with a fixed amount of

capital and labor. The company notices that as they hire more workers, the

output of bicycles initially increases at an increasing rate, but eventually starts

to increase at a decreasing rate. Explain the concept of the Law of Diminishing

Returns in this scenario.

Solution

To understand the concept of the Law of Diminishing Returns in this scenario,

let’s break it down into steps:

Step 1: Increasing Returns Initially, as the company hires more workers,

the output of bicycles increases at an increasing rate. This is because each

additional worker can specialize and contribute efficiently to the production

process, leading to higher productivity.

Step 2: Diminishing Returns However, as more and more workers are

hired, the factory may become overcrowded and the capital may not be sufficient

to support the increasing number of workers. This leads to inefficiencies, such

as workers getting in each other’s way, leading to a situation where the output

of bicycles starts to increase at a decreasing rate. This is the stage where the

Law of Diminishing Returns comes into play.

Step 3: Negative Returns If the company continues to hire more workers

beyond a certain point, the output of bicycles may even start to decrease. This

is known as the stage of negative returns, where each additional worker leads

to a decrease in overall production due to severe overcrowding and inefficiencies

in the production process.

In conclusion, the Law of Diminishing Returns states that as one input factor

(in this case, labor) is increased while other factors (such as capital) are held

constant, the marginal product of that input will eventually decrease, leading

to diminishing overall returns.

Question 15

Question

A firm is currently operating in the short run and is experiencing diminishing

returns to labor. Explain how the Law of Diminishing Returns manifests in the

short run production process, and illustrate this concept with a hypothetical

production scenario.

Solution

Step 1: The Law of Diminishing Returns states that as one input variable is

increased, while other inputs are held constant, a point will be reached at which

the overall output will increase at a diminishing rate.

Step 2: In the short run production process, a firm experiences diminishing

returns to labor when the quantities of all inputs fixed except for labor. Initially,

as more units of labor are added to a fixed amount of capital, output increases

at an increasing rate due to the specialization of labor and the division of tasks.

Step 3: However, at some point, adding more units of labor becomes less

productive as the fixed capital input becomes a constraint. This leads to dimin-

ishing returns to labor, where each additional unit of labor contributes less to

the total output than the previous unit.

Step 4: To illustrate this concept, let’s consider a hypothetical scenario of

a bakery with a fixed amount of ovens and other machinery. Initially, with one

baker, the bakery produces 50 loaves of bread per day. When a second baker is

hired, the output increases to 120 loaves per day due to specialization and more

efficient use of equipment.

Step 5: Adding a third baker results in a total output of 160 loaves per day.

However, when a fourth baker is hired, the total output only increases to 175

loaves per day, indicating diminishing returns to labor.

Step 6: This decrease in the additional output per unit of labor is a clear

manifestation of the Law of Diminishing Returns in the short run production

process, where the fixed capital input acts as a limiting factor to overall pro-

ductivity.

Question 16

Question

A company produces bicycles in a factory. The company recently hired addi-

tional workers and noticed that the output of bicycles initially increased as more

workers were added. However, after a certain point, the company observed that

the additional output from each new worker started to decrease. Explain why

this phenomenon occurs and discuss the implications of the Law of Diminishing

Returns in the context of bicycle production.

Solution

Step 1: The Law of Diminishing Returns 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

capital), beyond a certain point, the marginal product of the variable input

will start to diminish. This occurs because the fixed input (capital) becomes a

limiting factor in the production process.

Step 2: Application to Bicycle Production Initially, when the company hired

additional workers, there were more individuals available to work on producing

bicycles. This led to an increase in the output of bicycles as each new worker

contributed positively to the production process. However, as more workers

were hired and the factory became crowded, there were constraints such as

limited space, machinery, and materials. These fixed inputs started to limit the

efficiency of each additional worker, leading to diminishing returns.

Step 3: Implications The implications of the Law of Diminishing Returns

for bicycle production are significant. Beyond a certain point, hiring more

workers may not lead to a proportional increase in output. In fact, the company

may experience negative returns if it continues to add more workers without

addressing the constraints of fixed inputs. This can result in decreased efficiency,

higher production costs, and potentially lower profits.

In conclusion, understanding and managing the Law of Diminishing Returns

is crucial for businesses to optimize their production processes and resources

effectively. By recognizing the point of diminishing returns, companies can make

informed decisions about resource allocation, avoid inefficiencies, and ensure

sustainable growth and profitability.

Question 17

Question

A manufacturing company is producing electronic devices in a factory. The

company has fixed the number of workers at 100. Initially, as more workers

were hired, the production output increased significantly. However, after reach-

ing a certain point, the company noticed that the marginal product of each

additional worker started to diminish. Using the concept of the Law of Dimin-

ishing Returns, explain why this phenomenon occurs and how it impacts the

company’s production efficiency.

Solution

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

while keeping the other factors constant, the output will eventually begin to

increase at a decreasing rate. In the case of the manufacturing company with

a fixed number of workers, there are several factors contributing to this phe-

nomenon.

Step 1: Increasing Marginal Product Initially, when the company hired

more workers, the marginal product of each additional worker was positive and

significant. This is because with more workers, there was better division of

labor, specialization, and increased efficiency.

Step 2: Diminishing Marginal Product As the number of workers

reached a certain point (in this case, 100 workers), the marginal product of

each additional worker started to diminish. This occurs due to factors such as

limited space, equipment, and managerial oversight. Each additional worker

may not be as productive as the previous hires, leading to a decrease in the

marginal product.

Step 3: Impact on Production Efficiency The diminishing marginal

returns have a direct impact on the company’s production efficiency. As the

marginal product decreases, the company experiences diminishing returns on

each additional unit of the input (workers). This can lead to inefficiencies,

increased costs, and lower overall productivity.

Step 4: Optimal Workforce To maximize production efficiency and min-

imize costs, the company needs to find the optimal number of workers. This

is the point where the marginal cost of hiring an additional worker equals the

marginal product of that worker. Beyond this point, hiring more workers will

lead to diminishing returns and decreased efficiency.

In conclusion, the Law of Diminishing Returns explains the phenomenon

where the marginal product of each additional worker decreases as the work-

force size increases. This understanding is crucial for companies to optimize

production processes and maintain efficiency.

Question 18

Question

A company produces bicycles in a small factory. The company finds that as they

hire more workers, the marginal increase in the number of bicycles produced

starts to diminish. Explain how the Law of Diminishing Returns applies to this

situation.

Solution

The Law of Diminishing Returns states that as one input is increased while

keeping all other inputs constant, there will be a point at which the marginal

increase in output will start to decrease.

Step 1: Initially, when the company hires more workers, the production of

bicycles increases. This is because these workers can specialize in different tasks

and be more efficient in producing bicycles.

Step 2: However, as the factory becomes more crowded with workers, there

will be less space and resources available for each worker. This can lead to

inefficiencies, overlaps in work, and a decrease in productivity.

Step 3: Eventually, the company will reach a point where adding more

workers will not result in a proportional increase in the number of bicycles

produced. In fact, there may even be a decrease in production due to the

inefficiencies caused by overcrowding and lack of resources.

Step 4: This situation illustrates the Law of Diminishing Returns, as the

marginal increase in output (number of bicycles produced) diminishes as more of

the input (workers) is added. The company will need to find the optimal number

of workers to maximize production efficiency without experiencing diminishing

returns.

Question 19

Question

A company produces widgets in a factory. The company notices that after hiring

additional workers, the marginal product of each new worker decreases.

Given the total product of labor (TPL) function of the factory as TPL =

10L−0.5L2, where Lis the number of workers, determine: a) The average

product of labor (APL) b) The marginal product of labor (MPL) c) At what

value of Lwill the marginal product of labor start to decrease according to the

law of diminishing returns?

Solution

a) The average product of labor (APL) can be calculated as the total product

of labor (TPL) divided by the number of workers (L). Thus,

AP L =TPL

L=10L−0.5L2

L= 10 −0.5L

b) The marginal product of labor (MPL) is the derivative of the total product

of labor (TPL) with respect to the number of workers (L). So,

MP L =d(TPL)

dL =d(10L−0.5L2)

dL = 10 −L

c) According to the law of diminishing returns, the marginal product of labor

starts to decrease when MP L < 0.

Setting MP L < 0:

10 −L < 0

L > 10

Thus, the marginal product of labor will start to decrease when the number

of workers exceeds 10.

Question 20

Question

A company produces widgets and currently operates with two machines. The

company notices that as they add more machines, the production rate per ma-

chine starts to decrease. The production rate for each machine is given by the

function P(x) = 100x−5x2, where xis the number of machines. Determine the

optimal number of machines the company should use to maximize production.

Solution

Step 1: Find the total production function by multiplying the production rate

per machine by the number of machines.

Total production, T(x) = P(x)·x= (100x−5x2)·x= 100x2−5x3

Step 2: Calculate the marginal product of the total production function.

Marginal product, T′(x) = d

dx (100x2−5x3) = 200x−15x2

Step 3: Set the marginal product equal to zero to find the critical points.

200x−15x2= 0 ⇒15x(200 −15x)=0⇒x= 0, x =200

15 = 13.3

Step 4: Calculate the second derivative to determine the nature of the critical

point at x= 13.3.

T′′ (x) = d

dx (200x−15x2) = 200 −30x

Step 5: Substitute x= 13.3 into the second derivative to determine the

concavity.

T′′ (13.3) = 200 −30(13.3) <0

Since the second derivative is negative at x= 13.3, the critical point is a

maximum.

Step 6: Therefore, the optimal number of machines the company should use

to maximize production is 13 machines.

Question 21

Question

Consider a firm that produces agricultural products. The firm decides to in-

crease its labor input while keeping all other factors constant. After a certain

point, the firm observes that the marginal product of labor starts to decrease.

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

Solution

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

that as a firm increases one input while keeping all other inputs constant, a

point will be reached where the marginal product of that input will start to

decrease.

2. Initially, when the firm increases its labor input while keeping land,

capital, and other factors constant, the marginal product of labor increases.

This means that each additional unit of labor contributes more to the total

output of the firm.

3. However, as the firm continues to increase the labor input, a point will

be reached where the marginal product of labor starts to decrease. This occurs

because the fixed factors (such as land and capital) become a constraint on how

efficiently additional units of labor can contribute to the total output.

4. The diminishing marginal returns imply that the additional unit of labor

adds less to total output than the previous unit of labor. This can lead to

inefficiencies and increase in production costs per unit.

5. To summarize, the Law of Diminishing Returns highlights the importance

of balancing different inputs in production to achieve optimal output levels. It

suggests that there is an optimal combination of inputs that maximizes out-

put efficiency, beyond which adding more of a particular input may lead to

decreasing returns.

Question 22

Question

A company is producing smartphones using a fixed amount of capital. The ini-

tial number of workers is 10 and the company is observing the law of diminishing

returns. The total output of smartphones is increasing at a decreasing rate as

more workers are hired. If the total output in a given time period is modeled by

the function Q(w) = 100w−5w2, where Qis the total output of smartphones

and wis the number of workers, determine the maximum total output of smart-

phones that the company can achieve and the number of workers required to

achieve this maximum output.

Solution

Step 1: To find the maximum total output of smartphones and the number of

workers required to achieve this maximum output, we need to find the maximum

point of the function Q(w) = 100w−5w2.

Step 2: To find the maximum point of the function, we first need to find

the critical points by taking the derivative of Qwith respect to wand setting

it equal to zero.

Q′(w) = d

dw (100w−5w2) = 100 −10w

Step 3: Setting Q′(w) equal to zero and solving for w:

100 −10w= 0

10w= 100

w= 10

Step 4: To determine whether this critical point is a maximum, we need to

perform the second derivative test. By taking the second derivative of Q:

Q′′ (w) = d2

dw2(100w−5w2) = −10

Step 5: Since Q′′ (10) = −10 <0, the critical point at w= 10 corresponds

to a maximum point.

Step 6: Therefore, the maximum total output of smartphones the company

can achieve is:

Q(10) = 100(10) −5(10)2= 1000 −500 = 500 smartphones

Step 7: The number of workers required to achieve this maximum output is

10 workers.

Question 23

Question

Suppose a farm is cultivating a field, and the application of fertilizer is resulting

in increased crop yield. However, after a certain point, adding more fertilizer

leads to diminishing returns, where the additional crop yield decreases with

each additional unit of fertilizer. Explain the concept of the Law of Diminishing

Returns in this agricultural context.

Solution

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

states that as more of one input is added while keeping all other inputs con-

stant, the marginal product of that input will eventually decrease. This concept

is commonly observed in agricultural production, as in the case of applying

fertilizer to a field to increase crop yield.

Step 1: Initially, when fertilizing the field, the increase in crop yield per

additional unit of fertilizer is significant. This is because the first units of

fertilizer help to improve soil fertility and provide essential nutrients for plant

growth. As a result, the marginal product of fertilizer is high.

Step 2: As more fertilizer is added beyond a certain point, the field may

reach a saturation level where the soil is already rich in nutrients. In this

situation, adding extra fertilizer does not lead to a proportional increase in crop

yield. The marginal product of fertilizer starts to diminish.

Step 3: Continuing to increase the amount of fertilizer beyond the satu-

ration point can even lead to negative returns, where the additional fertilizer

reduces the crop yield instead of increasing it. This occurs due to factors such

as nutrient imbalances, soil compaction, or toxicity.

In summary, the Law of Diminishing Returns explains how in agricultural

production, the marginal product of an input (such as fertilizer) will decrease

as more of that input is added, ultimately leading to diminishing returns and

potentially negative impacts on crop yield.

Question 24

Question

Suppose a company is producing bicycles. The company has a fixed amount

of land and labor. The company notices that as they increase the number of

workers beyond a certain point, the additional output from each extra worker

starts to diminish. Describe how the Law of Diminishing Returns applies to this

scenario.

Solution

The Law of Diminishing Returns is an economic principle stating that if one

input in the production of a commodity is increased while all other inputs are

held fixed, a point will be reached at which the marginal product of the variable

input starts to decrease.

Step 1: Define the Law of Diminishing Returns The Law of Dimin-

ishing Returns implies that adding more of a variable input (such as labor) to

fixed inputs (such as land) will eventually yield smaller per-unit returns. This

occurs because the fixed inputs become overused or inefficient in conjunction

with the variable input.

Step 2: Apply the Law to the bicycle production scenario In the

context of bicycle production, the company has a fixed amount of land and

labor. Initially, as the company hires more workers, the output of bicycles will

increase since each worker can specialize in a specific task in the production

process, leading to increased efficiency.

Step 3: Identify the turning point However, there comes a point where

hiring additional workers becomes counterproductive. This is the turning point

where the Law of Diminishing Returns sets in. At this stage, the fixed inputs

(land and capital) are being overused in conjunction with the variable input

(labor), leading to a decrease in marginal product of labor.

Step 4: Implications The company may experience inefficiencies, such

as worker congestion, communication breakdowns, or production bottlenecks,

which ultimately reduce the overall output of bicycles. This inefficiency occurs

because the additional workers are not contributing as much to the production

process as the initial workers.

Step 5: Optimal production level To maximize efficiency and output,

the company must identify the optimal level of labor input where the marginal

product of labor is highest before the Law of Diminishing Returns sets in. This

will ensure that the company operates at its most cost-effective and productive

level.

Question 25

Question

Suppose a company is producing smartphones and the production function is

given by Q= 5L0.5K0.3, where Qis the quantity of smartphones produced, Lis

the quantity of labor, and Kis the quantity of capital. If the company currently

has 6 units of capital, determine the maximum output of smartphones before

the law of diminishing returns sets in for labor.

Solution

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

MP L =∂Q

∂L = 2.5L−0.5K0.3

Step 2: Calculate the average product of labor (APL):

AP L =Q

L= 5L−0.5K0.3

Step 3: Determine the point at which the law of diminishing returns sets in,

which is when the MPL equals APL:

MP L =AP L

2.5L−0.5K0.3= 5L−0.5K0.3

Step 4: Simplify the equation and solve for L:

2.5=5

L=?

Therefore, the maximum output of smartphones before the law of diminish-

ing returns sets in for labor is unknown as the current level of labor was not

provided in the question.

Question 26

Question

A manufacturing company produces computer chips in a factory. The company

has noticed that when they increase the number of workers in the factory, the

production output initially increases. However, after a certain point, adding

more workers starts to have a diminishing return on the production output.

Suppose the production output (measured in computer chips per day) is

modeled by the function P(x) = 300x−5x2, where xis the number of work-

ers added to the factory. Find the number of workers that will maximize the

production output according to this model.

Solution

Given the production output function P(x) = 300x−5x2, to find the number of

workers that will maximize the production output, we need to find the critical

point of the function. This can be done by taking the derivative of P(x) and

setting it equal to zero.

Step 1: Find the derivative of the production output function.

dx =d

dx (300x−5x2) = 300 −10x

Step 2: Set the derivative equal to zero and solve for x.

300 −10x= 0

10x= 300

x= 30

Step 3: Test the critical point to determine if it is a maximum. To

determine whether x= 30 corresponds to a maximum or minimum, we can use

the second derivative test. We will find the second derivative of P(x).

d2P

dx2=d

dx (300 −10x) = −10

Since the second derivative is negative, the critical point x= 30 corresponds

to a maximum.

Step 4: Conclusion The maximum production output will be achieved by

having 30 workers in the factory.

Question 27

Question

An agricultural firm is cultivating a field with fixed size. The firm is observing

the law of diminishing returns as it adds more fertilizer to the field. The total

output of wheat (in tons) is given by the function Q(f) = 100f−2f2, where

fis the amount of fertilizer in kilograms. Determine the optimal amount of

fertilizer that the firm should use to maximize the total output of wheat.

Solution

Step 1: Find the derivative of the total output function Q(f) with respect to f.

df = 100 −4f

Step 2: Set the derivative equal to zero to find the critical points.

100 −4f= 0

4f= 100

f= 25

Step 3: To determine whether this critical point is a maximum or a minimum,

we will use the second derivative test. Find the second derivative of the total

output function.

d2Q

df2=−4

Step 4: Evaluate the second derivative at the critical point f= 25.

d2Q

df2



f=25

=−4

Since the second derivative is negative at f= 25, the critical point is a local

maximum.

Step 5: Therefore, the optimal amount of fertilizer that the firm should use

to maximize the total output of wheat is 25 kilograms.

Question 28

Question

A company operates a factory where they produce widgets. The factory has a

fixed size and a fixed number of workers. Initially, adding more workers led to

a significant increase in widget production. However, as they continue to hire

more workers, the increase in widget production begins to diminish.

Suppose the production function of the factory can be modeled by Q=

10L−0.5L2, where Qis the total quantity of widgets produced and Lis the

number of workers hired.

If the company currently has 15 workers, how many more workers should

they hire to maximize widget production?

Solution

Step 1: To find the number of workers the company should hire to maximize

widget production, we need to find the point where the rate of change of pro-

duction with respect to workers is zero. This point corresponds to the maximum

value of the production function.

Step 2: Given the production function Q= 10L−0.5L2, we need to find the

derivative of Qwith respect to Lto determine the rate of change of production:

dL = 10 −L

Step 3: Set the derivative equal to zero to find the critical point:

10 −L= 0

Step 4: Solve for Lto find the number of workers that maximizes production:

L= 10

Step 5: Since the company currently has 15 workers, the company should

hire 10 - 15 = -5 more workers to maximize widget production.

Note: In practice, it is not possible to hire negative workers, so the com-

pany should simply maintain the current number of workers (15) to maximize

production based on the given production function.

Question 29

Question

A company is producing smartphones and experiencing diminishing returns to

labor. The total output of smartphones (Q) produced per day is given by the

function Q= 100L−5L2, where Lis the number of workers hired. Calculate

the marginal product of labor when the company employs 20 workers.

Solution

Step 1: To find the marginal product of labor, we first need to find the total

product of labor. Step 2: The total product of labor is given by Q= 100L−5L2.

Step 3: Substituting L= 20 into the total product function, we get:

Q= 100(20) −5(20)2= 2000 −5(400) = 2000 −2000 = 0.

Step 4: Now, we find the marginal product of labor, which is the derivative of

the total product function with respect to labor. Step 5: Taking the derivative

of Qwith respect to L, we have:

MPL=dQ

dL = 100 −10L.

Step 6: Substituting L= 20 into the marginal product of labor function, we

get:

MPL= 100 −10(20) = 100 −200 = −100.

Step 7: Therefore, the marginal product of labor when the company employs

20 workers is -100 smartphones per additional worker.

Question 30

Question

Suppose a firm is producing shirts in a factory. The firm observes that as it adds

more workers to the production process while keeping all other inputs constant,

the total output of shirts initially increases at an increasing rate, reaches a peak,

and then starts to increase at a decreasing rate. Define the Law of Diminishing

Returns in the context of this situation and discuss how it applies to the firm.

Solution

The Law of Diminishing Returns states that as one input variable is increased

while all other variables are held constant, the marginal output of that variable

will eventually decrease. In the context of the firm producing shirts in a factory,

this law can be observed in the following way:

Step 1: Increasing Total Output Initially, as more workers are added,

the firm experiences increasing total output of shirts. This is because the workers

are specialized in their tasks, leading to improved efficiency and productivity.

Step 2: Peak Output After a certain point, the firm reaches a peak level

of total output. This is the point where the Law of Diminishing Returns comes

into play. At this stage, the additional workers enhance the production process,

but the marginal increase in output begins to decrease.

Step 3: Decreasing Marginal Returns As more workers are added be-

yond the point of peak output, the firm experiences decreasing marginal returns.

This means that each additional worker contributes less to the total output of

shirts than the workers added before them. This decrease can be attributed to

factors such as overcrowding, inefficiency, and limited resources.

Step 4: Optimal Input Level To maximize efficiency and productivity,

the firm must identify the optimal number of workers to employ in the produc-

tion process. This optimal input level ensures that the firm operates at peak

efficiency and avoids the diminishing returns associated with overstaffing.

In conclusion, the Law of Diminishing Returns highlights the importance of

resource allocation and input optimization in production processes. By under-

standing this principle, firms can make informed decisions regarding input levels

to achieve optimal output and efficiency.

Question 31

Question

A company is currently producing 100 units of a product per day using a certain

amount of labor. The company decides to increase the amount of labor used by

Solution

The Law of Diminishing Returns states that as one input variable is increased,

while all other input variables are held constant, a point will be reached at which

the marginal increase in output decreases. In this case, the input variable being

increased is the amount of labor.

Step 1: Initially, the company is producing 100 units of the product per

day with a certain level of labor.

Step 2: When the company increases the amount of labor used by 20

Step 3: This situation can be explained by the Law of Diminishing Returns.

As more labor is added to the production process, eventually the additional

units of output produced by each additional unit of labor will diminish. This

is because other factors, such as machinery, space, or management, may not be

increasing at the same rate as the labor input.

Step 4: Therefore, the company is experiencing diminishing returns on

labor: adding more labor does not result in proportionally more output. This

concept is important for businesses to understand in order to optimize their

production processes and resource allocation.

Question 32

Question

Suppose a company is analyzing the production of a certain product. Initially,

as more units of labor are added, the company experiences increasing returns

to scale. However, after a certain point, adding more units of labor results in

diminishing returns. If the total output initially increases at a rate of 10 units

per additional worker and then decreases at a rate of 5 units per additional

worker, how many units of labor should the company employ to maximize total

output?

Solution

Let the number of units of labor be Land the total output be Q. From the

given information:



Initially, the company experiences increasing returns to scale, so the rate

of increase in total output is 10 units per additional worker. This can be

represented as:

Q′= 10



After a certain point, the company experiences diminishing returns to

scale, so the rate of decrease in total output is 5 units per additional

worker. This can be represented as:

Q′=−5

We want to find the number of units of labor that maximizes total output. This

can be found by setting the rate of increase in total output equal to the rate of

decrease and solving for L:

10 = −5

L= The number of units of labor that maximizes total output

Since the equation 10 = −5 has no solution, this means that there is no number

of units of labor that will maximize total output. The company should employ

just enough labor to avoid diminishing returns.

Question 33

Question

Suppose a company is producing smartphones and experiencing the law of di-

minishing returns in its production process. Initially, the company hired 10

workers and was able to produce 100 smartphones per day. When the company

hired 5 more workers, the daily production increased to 150 smartphones. If

each worker is paid

100 per day, at what point does the company’s marginal

cost start to increase?

Solution

Let xbe the number of additional workers hired beyond the initial 10 workers.

The total production of smartphones can be represented by the function Q(x) =

100 + 10x+ 5x2, where Q(x) is the number of smartphones produced per day.

The company’s total cost (TC) can be calculated as:

T C(x) = 1000 + 100x

The marginal cost (MC) is the derivative of the total cost with respect to x:

MC(x) = d

dx T C(x) = d

dx (1000 + 100x) = 100

Since the marginal cost remains constant at

100 per additional worker, the

point where the company’s marginal cost starts to increase is not reached within

the range considered in this scenario.

Question 34

Question

A company is producing smartphones and has a fixed capital for its production.

Initially, as more workers are hired, the output of smartphones increases at an

increasing rate due to specialization. However, after a certain point, the com-

pany starts to experience diminishing returns. Explain the concept of the Law

of Diminishing Returns and discuss why it occurs in the context of smartphone

production.

Solution

The Law of Diminishing Returns is an economic principle that states that as

more of one factor of production is added to a fixed quantity of other factors of

production, after a certain point, the marginal increase in output will decrease.

Step 1: Understanding the Law of Diminishing Returns Initially,

as more workers are hired in the smartphone production process, the company

benefits from specialization, efficiency gains, and increased division of labor,

leading to a higher output of smartphones. However, after a certain point,

the additional workers may lead to overcrowding, inefficiencies, and diminishing

returns.

Step 2: Why it occurs in smartphone production In the context of

smartphone production, the company may have a fixed set of machinery and

equipment. Initially, as more workers are added, the utilization of these fixed

assets improves, leading to higher productivity. However, after a certain point,

adding more workers may lead to overcrowding and competition for limited

resources such as machinery, resulting in inefficiencies and diminishing returns.

Step 3: Example For example, suppose a smartphone production company

has a fixed set of machinery that can efficiently handle a certain number of work-

ers. Initially, adding more workers may lead to a higher output of smartphones

due to specialization. However, beyond a certain point, adding more workers

may lead to congestion in the production process, resulting in inefficiencies, in-

creased waiting times, and ultimately diminishing returns where each additional

worker contributes less to the overall output.

Therefore, understanding the Law of Diminishing Returns is crucial for com-

panies to optimize their production processes and resource allocation effectively.

Question 35

Question

A company produces smartphones in a factory. The company has a fixed factory

size and hires additional workers to produce more smartphones. Initially, when

the company hires more workers, the overall productivity increases as the spe-

cialization of labor improves. However, at a certain point, the company starts

experiencing diminishing returns to labor as overcrowding and inefficiency set

in.

Suppose the company experiences diminishing returns after hiring 20 work-

ers. The total number of smartphones produced by the company is given by the

function P(w) = 80w−2w2, where wrepresents the number of workers hired.

Determine: 1. The maximum number of smartphones the company can

produce. 2. The number of workers at which the company reaches the maximum

productivity. 3. The rate of change in total smartphone production when the

company hires the 25th worker.

Solution

1. To find the maximum number of smartphones the company can produce,

we first need to determine the number of workers that maximizes the total

production function P(w) = 80w−2w2.

Step 1: Find the derivative of the production function with respect to the

number of workers, w.

dw =d

dw (80w−2w2) = 80 −4w

Step 2: Set the derivative equal to zero and solve for wto find the critical

point(s):

80 −4w= 0

4w= 80

w= 20

Step 3: To confirm that this is a maximum rather than a minimum, we will

use the second derivative test.

d2P

dw2=d

dw (80 −4w) = −4

Since the second derivative is negative, the critical point at w= 20 corre-

sponds to a maximum.

Step 4: Substitute w= 20 back into the production function to find the

maximum number of smartphones produced.

P(20) = 80(20) −2(20)2= 1600 −800 = 800

So, the maximum number of smartphones the company can produce is 800.

2. The number of workers at which the company reaches the maximum

productivity is 20 workers.

3. To find the rate of change in total smartphone production when the

company hires the 25th worker, we will find the derivative at w= 25.

Step 1: Find the rate of change by evaluating the derivative at w= 25.

dw = 80 −4w

where the Law of Diminishing Returns sets in, indicating that the company is

operating beyond its optimal level of input.

Step 4: It is important for the company to identify this point and optimize

the number of workers in the factory to ensure maximum efficiency and produc-

tivity in shirt production. This involves balancing the costs of additional labor

with the benefits of increased output while considering the diminishing returns

associated with each additional worker.

Question 2

Question

Consider a production process where three factors of production, labor (L),

capital (K), and land (T), are being used to produce output. The production

function is given by Q= 2L1/2K1/3T1/6. If the amount of capital is fixed at

K= 8, analyze how the total output Qchanges as the amount of labor Land

land Tare increased based on the Law of Diminishing Returns.

Solution

1. To analyze how the total output Qchanges as the amount of labor Land

land Tare increased, we will use partial derivatives to find the marginal product

of labor and the marginal product of land.

2. The marginal product of labor MPL is given by:

MPL = ∂Q

∂L =d

dL (2L1/2K1/3T1/6)

3. By differentiating with respect to L, we get:

MPL = L−1/2K1/3T1/6

4. Similarly, the marginal product of land MPT is given by:

MPT = ∂Q

∂T =d

dT (2L1/2K1/3T1/6)

5. Differentiating with respect to T, we find:

MPT = 1

6L1/2K1/3T−5/6

6. Now, let’s analyze the effects of increasing Land Ton the total output

Q, keeping K= 8 constant.

7. Since the exponents are all positive, the marginal product of both labor

and land will decrease as Land Tincrease (due to diminishing returns).

8. Therefore, increasing the amount of labor Land land Twill lead to a

smaller increase in total output Qas the law of diminishing returns sets in.

9. It is important to note that the fixed amount of capital Kwill eventually

become a limiting factor, causing further diminishing returns as Land Tare

increased beyond a certain point.

Question 3

Question

A company has a production function given by Q= 10L0.5K0.5, where Qis the

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

company currently has 20 units of capital and is considering hiring more labor,

at what point will the law of diminishing returns begin to apply?

Solution

Step 1: To determine at what point the law of diminishing returns begins to

apply, let’s first calculate the marginal product of labor (M P L) when L= 1

and K= 20.

MP L =∂Q

∂L = 5L−0.5K0.5

MP L(1,20) = 5(1)−0.5(20)0.5= 5(1)(4) = 20

Step 2: Next, let’s calculate the average product of labor (AP L) when L= 1

and K= 20.

AP L =Q

L= 10L0.5K0.5

AP L(1,20) = 10(1)0.5(20)0.5= 10(1)(4) = 40

Step 3: The law of diminishing returns begins to apply when the marginal

product of labor starts to decrease. This indicates that each additional unit of

labor is adding less to total output. Therefore, based on our calculation, the

law of diminishing returns will begin to apply after hiring the first unit of labor.

Question 4

Question

A farmer has a fixed amount of land to cultivate wheat. Initially, he plants

a small area and sees a good yield in the crop. However, as he continues to

increase the area of land planted with wheat, he notices that the additional

output generated by each additional unit of land begins to decrease. Explain

this phenomenon using the Law of Diminishing Returns.

Solution

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

states that as one input is increased (while other inputs are held constant), there

is a point at which the marginal (additional) output generated by each additional

unit of input will start to decrease. This phenomenon is often observed in

agriculture and production processes.