Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

Question

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Answer 1

Question

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

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Answer 2

Question

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

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Answer 3

Question

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

Answer 6

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

Answer 7

Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

Answer 8

a

Expert Solution

Answer to Problem 15.2P

C will accept the contract because EU=75 is greater than the utility from the next best alternative, U=0

Answer 9

a

Expert Solution

Answer 10

a

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

U=100 .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

EU=0.5[500−100]+0.5[200−100] =0.5[400]+0.5[100] =200+50 =250

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since EU=250 , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

EU=0.5[250−100]+0.5[100−100] =0.5[150]+0.5[0] =75+0 =75

In this scenario also, contract will be accepted because EU=75 is greater than the utility from the next best alternative, U=0

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

0.5(1,0000s)+0.5(400s)−100≥0500s+200s−100≥0700s−100≥0700s≥100s≥100700s≥17

Hence, the share which needs to be given in order to make C accept the contract will be

s≥17

Answer 13

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

Answer 14

b)

To determine

Maximum amount paid to shareholders.

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

Answer 15

b)

To determine

Maximum amount paid to shareholders.

Answer 16

b)

Expert Solution

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Answer 17

b)

Expert Solution

Answer 18

b)

Expert Solution

Answer 19

Expert Solution

Answer 20

Explanation of Solution

Given:

C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

The highest amount which C will be ready to pay to buy out the store will be the expected value (EV) of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

EV=0.5(1,000)+0.5(400) =500+200 =700

Now to see whether EV=700 maximizes the expected utility or not, should calculate the expected utility by assuming EV=700=w

EU=700−100 =600

Thus, the expected utility of EV=700 is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Answer 21

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

Answer 22

c)

To determine

Fixed salary to be paid to C.

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100

Hence, C needs to be paid a fixed salary of $100 in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility

Answer 23

c)

To determine

Fixed salary to be paid to C.

Answer 24

c)

Expert Solution

Answer to Problem 15.2P

C needs to be paid a fixed salary of $100 in addition to $100 bonus in order to accept the contract.

Answer 25

c)

Expert Solution

Answer 26

c)

Expert Solution

Answer 27

Expert Solution

Answer 28

Explanation of Solution

Given C’s utility function: U=w−100 and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is U=100 .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of sx which needs to be paid by C so as to induce to accept the contract

0.5(b+SX)−100≥00.5(100+X)−100≥050+0.5X−100≥0−50+0.5X≥0X≥500.5X≥100