Choose a house plan that suits the clients needs and ensure that it will help meet the Gold Coast City Councils requirements for land use. Construct a scaled diagram of the house plan using its external measurements (length and width) and determine the best position of the house on the block of land taking into account Gold Coast City Council requirements.

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Problem-solving and modelling task

Task
You are an employee for ‘Regency Lifestyle’. Your company is responsible for building and selling house and land packages. As part of the service, ‘Regency Lifestyle’ makes adjustments to the site and floor plans based on client’s needs. Adam and Belle Smith are your personal clients. The Smiths have purchased a rectangular block of land 32m by 22m and have requested that their floor plan is a Metricon Home design.

Client Information

  •  Both clients work full-time.
  •  They have two children, two boys aged seven and five. They are not planning on having any more children.
  •  They enjoy entertaining.

Based on the Client Information above, you will:
-Choose a house plan that suits the clients needs and ensure that it will help meet the Gold Coast City Councils requirements for land use.

-Construct a scaled diagram of the house plan using its external measurements (length and width) and determine the best position of the house on the block of land taking into
account Gold Coast City Council requirements.

-The clients have a budget of $7000 to both tile the outdoor room and paint the walls of the master bedroom. Determine the total cost of tiling the outdoor room and painting the
master bedroom and evaluate whether they will meet their budget.

The profit maximization problem is given by max 𝐾𝑡,𝐿𝑡 𝜋𝑡 = 𝑌𝑡 − 𝑅𝑡𝐾𝑡 − 𝑤𝑡𝐿𝑡 By solving the profit maximization problem, derive the definition of the value of 𝜎 mathematically. By solving the maximization problem, characterize the saving function depending on the value of 𝜃, i.e., there are three cases. By solving the maximization problem, derive the definition of the value of 𝜃 mathematically.

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Quiz 2 Topics in Macroeconomics

In submission, you need to type all equations, and submit your answers as a PDF file. Notations are the same as the lecture notes.

Problem 1
We consider a CES production function.
𝑌𝑡 = 𝐴 (𝛼𝐾𝑡
𝜎−1
𝜎 + (1 − 𝛼)𝐿𝑡
𝜎−1
𝜎 )
𝜎
𝜎−1
.
Q1: As 𝜎 → 1, prove the Cobb–Douglas production function 𝐴𝐾𝑡 𝛼𝐿𝑡 1−𝛼. (10 marks)

Q2: As 𝜎 → 0, prove the Leontief production function 𝑌𝑡 = 𝐴 min(𝐾𝑡, 𝐿𝑡). (10 marks)

Q3: The profit maximization problem is given by max
𝐾𝑡,𝐿𝑡
𝜋𝑡 = 𝑌𝑡 − 𝑅𝑡𝐾𝑡 − 𝑤𝑡𝐿𝑡
By solving the profit maximization problem, derive the definition of the value of 𝜎 mathematically. (10 marks)

Problem 2
The utility maximization problem is given by max
𝑐1𝑡,𝑐2𝑡,𝑠𝑡
𝑢𝑡 = (𝑎1
1
𝜃(𝑐1𝑡)𝜃−1
𝜃 + 𝑎2
1
𝜃(𝑐2𝑡)𝜃−1
𝜃 )
𝜃
𝜃−1
subject to 𝑐1𝑡 + 𝑠𝑡 = 𝑤𝑡 + 𝑒
𝑐2𝑡 = (1 + 𝑟𝑡+1)𝑠𝑡

Q4: By solving the maximization problem, characterize the saving function depending on the value of 𝜃, i.e., there are three cases. (30 marks)

Q5: By solving the maximization problem, derive the definition of the value of 𝜃 mathematically. (10 marks)

Problem 3
Consider a CES utility function.
𝑢𝑡 = (𝑎1
1
𝜃(𝑐1𝑡)𝜃−1
𝜃 + 𝑎2
1
𝜃(𝑐2𝑡)𝜃−1
𝜃 )
𝜃
𝜃−1

Q6: Derive 𝑢𝑡 as 𝜃 → 1. (10 marks)
Problem 4
Consider the following CES production function.
𝑌𝑡 = 𝐴 (𝛼 (𝐾𝑡
ℎ1
)
𝜎−1
𝜎
+ (1 − 𝛼) (𝐿𝑡
ℎ2
)
𝜎−1
𝜎
)
𝜎
𝜎−1

Q7: Derive factor prices 𝑅𝑡 and 𝑤𝑡. (10 marks)

Q8: Compute the values of 𝑅𝑡 and 𝑤𝑡, respectively, as 𝜎 → 0. (10 marks)

Write a paragraph to answer the following questions: What was the purpose of your study? What population did you sample from? Who/What made up your sample? When and where was the sample obtained? What Method Sampling did you use to select the sample? Give some detail about this.

/in /by

Body weight and Calorie intake

Purpose Statement:
The aim of the study is to find out if there is an association between body weight and calorie intake.

Research Question:
Is the daily calorie intake associated with an increased risk of obesity (increase in body weight)?

Data must have 30+ samples pieces of data (included in the attachment, survey data with survey questions)

I. Introduction

Write a paragraph to answer the following questions: What was the purpose of your study? What population did you sample from? Who/What made up your sample? When and where was the sample obtained? What Method Sampling did you use to select the sample? Give some detail about this.

Do you think you obtained a random sample (did everyone/thing in the population have an equal opportunity to be surveyed)? Explain.

II. Looking at a yes/no question

For the yes/no questions and answer the following questions:

A. State the question

B. Create a pie graph of the yes/no responses.

C. State the sample proportion of “yes” responses, .

 

III. Looking at a numerical questions (Make Sure that you have TWO Quantitative Variables (calorie intake and body weight)

For the each numerical questions answer the following questions:

 

A. State the question.

B. Draw a histogram, box-and-whisker plot,

C. Calculate the summary statistics (for each data set) Which measure of Central Tendency best describes the data?

D. Discuss the shape, center, and spread of the distribution of responses to this question.

 

IV. Scatter Plot

A. State the two numerical questions. Indicate which variable is x, and which is the dependent variable y. Explain why you assigned the variables in this way.

B. Draw the scatter plot

C. Find the Correlation Coefficient. Do you think that is a Strong, Moderate or Weak Correlation.

D. Find the Line of Best Fit. Draw it on The Scatter Plot.

E. Use the Regression Line to make a Prediction.

What happens if conditions change, so that the supply and demand sets are altered slightly? If the equilibrium is disturbed for some reason, what is the result? How do the suppliers and consumers arrive at the equilibrium?

/in /by

1. Mathematical models in economics

1.1 Introduction
In this book we use the language of mathematics to describe situations which occur in economics. The motivation for doing this is that mathematical arguments are logical and exact, and they enable us to work out in precise detail the consequences of economic hypotheses. For this reason, mathematical modelling has become an indispensable tool in economics, finance, business and management. It is not always simple to use mathematics, but its language and its techniques enable us to frame and solve problems that cannot be attacked effectively in other ways. Furthermore, mathematics leads not only to numerical (or quantitative) results but, as we shall see, to qualitative results as well.

1.2 A model of the market
One of the simplest and most useful models is the description of supply and demand in the market for a single good. This model is concerned with the relationships between two things: the price per unit of the good (usually denoted by p), and the quantity of it on the market (usually denoted by
q). The ‘mathematical model’ of the situation is based on the simple idea of representing a pair of numbers as a point in a diagram, by means of coordinates with respect to a pair of axes. In economics it is customary to take the horizontal axis as the q-axis, and the vertical axis as the p-axis. Thus, for example, the point with coordinates (2000, 7) represents the situation when 2000 units are available at a price of $7 per unit.
How do we describe demand in such a diagram? The idea is to look at those pairs (q, p) which are related in the following way: if p were the selling price, q would be the demand, that is the quantity which would be sold to consumers at that price. If we fill in on a diagram all the pairs (q, p) related in this way, we get something like Figure 1.1.

2 Mathematical models in economics
pDq

Figure 1.1: The demand set
We shall refer to this as the demand set D for the particular good. In economics you will learn reasons why it ought to look rather like it does in our diagram, a smooth, downward sloping curve.
Suppose the demand set D contains the point (30,5). This means that when the price p = 5 is given, then the corresponding demand will be for q = 30 units. In general, provided D has the ‘right’ shape, as in Figure 1.1, then for each value of p there will be a uniquely determined value of q. In this situation we say that D determines a demand function, qD. The value written qD(p) is the quantity which would be sold if the price were p, so that qD(5) = 30, for example.
Example Suppose the demand set D consists of the points (q,p) on the straight line 6q + 8p = 125. Then for a given value of p we can determine the corresponding q; we simply rearrange the equation of the line in the form q = (125 – 8p)/6. So here the demand function is D( ) _ 125 – 8p
q p – 6 .
For any given value of p we find the corresponding q by substituting in this formula. For example, if p = 4 we get
q = qD(4) = (125 – 8 x 4)/6 = 93/6.
o

A model of the market 3
There is another way of looking at the relationship between q and p. If we suppose that the quantity q is given, then the value of p for which (q, p) is
in the demand set D is the price that consumers would be prepared to pay if q is the quantity available. From this viewpoint we are expressing p in
terms of q, instead of the other way round. We write pD(q) for the value of p corresponding to a given q, and we call pD the inverse demand function. Example (continued) Taking the same set D as before, we can now rearrange the equation of the line in the form p = (125 – 6q)/8. So the inverse demand function is
D( ) _ 125 – 6q
p q – 8 .

Next we turn to the supply side. We assume that there is a supply set S consisting of those pairs (q,p) for which q would be the amount supplied to
the market if the price were p. There are good economic reasons for supposing that S has the general form shown in Figure 1.2.
p
q
Figure 1.2: The supply set
If we know the supply set S we can construct the supply function qS and the inverse supply function pS in the same way as we did for the demand function and its inverse. For example, if S is the set of points on the line 2q – 5p = -12, then solving the equation for q and for p we get
S( ) _ 5p – 12 q p – 2 ‘ S( ) _ 2q + 12 p q – 5 .

4 Mathematical models in economics

1.3 Market equilibrium
The usefulness of a mathematical model lies in the fact that we can use mathematical techniques to obtain information about it. In the case of supply and demand, the most important problem is the following. Suppose we know all about the factors affecting supply and demand in the market for a particular good; in other words, the sets Sand D are given. What values of q and p will actually be achieved in the market? The diagram (Figure 1.3) makes it clear that the solution is to find the intersection of D and S, because that is where the quantity supplied is exactly balanced by the quantity required.
p
D s
Figure 1.3: The equilibrium set E = S n D
q
The mathematical symbol for the intersection of the sets Sand D is S n D, and economists refer to E = S n D as the equilibrium set for the given market. Fortunately, there is a simple mathematical technique for finding the equilibrium set; it is the method for solving ‘simultaneous equations’.
Example Suppose the sets D and S are, respectively, the sets of pairs (q, p) such that’
q + 5p = 40 and 2q -15p = -20.
Then a point (q*, p*) which is in the equilibrium set E = S n D must, by definition, be in both Sand D. Thus (q*, p*) satisfies the two equations
q* + 5p* = 40, 2q* – 15p* = -20.
The standard technique for solving these equations is to multiply the first one by 2 and subtract it from the second one. Working through the algebra,

Excise tax 5
we get q* == 20 and p* == 4. In other words the equilibrium set E is the single point (20,4). D It is worth remarking that in this example we get a single point of equilibrium, because we took the sets D and S to be straight lines. It is possible to imagine more complex situations, such as that we shall describe in Example 2.5, where the equilibrium set contains several points, or no points at all.

1.4 Excise tax
Using only the simple techniques developed so far we can obtain some interesting insights into problems in economics. In this section we study the
problem of excise tax. Suppose that a government wishes to discourage its citizens from drinking too much whisky. One way to do this is to impose a fixed tax on each bottle of whisky sold. For example, the government may decide that for each bottle of whisky the suppliers sell, they must pay the government $1. Note that the tax on each unit of the taxed good is a fixed amount, not a percentage of the selling price. Some very simple mathematics tells us how the selling price changes when an excise tax is imposed. Example In the previous example the demand and supply functions are given by
S 15 q (p) == 2 P – 10 ,
and the equilibrium price is p* == 4. Suppose that the government imposes an excise tax of T per unit. How does this affect the equilibrium price?
The answer is found by noting that, if the new selling price is p, then, from the supplier’s viewpoint, it is as if the price were p – T, because the supplier’s revenue per unit is not p, but p – T. In other words the supply function has changed: when the tax is T per unit, the new supply function qST is given by Of course the demand function remains the same. The new equilibrium values qT and pT satisfy the equations

6 Mathematical models in economics Eliminating qT we get
T 15 T
40 – 5p = 2(P – T) – 10.
Rearranging this equation, we obtain and so we have a new equilibrium price of The corresponding new equilibrium quantity is
qT =40-5pT =20-3T.
For example, if T = 1, the equilibrium price rises from 4 to 4.6 and the equilibrium quantity falls from 20 to 17. Unsurprisingly, the selling price has
risen and the quantity sold has fallen. But note that, although the tax is T per unit, the selling price has risen not by the full amount T, but by the fraction 3/5 of T. In other words, not all of the tax is passed on to the consumer. D

1.5 Comments
1. Economics tells us why the supply and demand sets ought to have certain properties. Mathematics tells us what we can deduce from those properties and how to do the calculations.

2. Mathematics also enables us to develop additional features of the model. In the case of supply and demand, we might ask questions such as the following:

  •  What happens if conditions change, so that the supply and demand sets are altered slightly?
  •  If the equilibrium is disturbed for some reason, what is the result?
  •  How do the suppliers and consumers arrive at the equilibrium? A typical instance of the first question is the excise tax discussed above. In this book we shall develop the mathematical techniques needed to deal withmany other instances of these questions.

Analyze your data for the mean, median, and mode of each questions. Create a visual from this chapter: bar graph, box and whisker plot, histogram, stem and leaf plot. etc. Compile the information into a slide presentation, of at least 5 slides, to present at the next town hall. The presentation should present: the mean, median, and mode of each question, the visual of the data, and conclusions based on the statistics you found in the survey.

/in /by

College Mathematics: Gather and analyze data

You have been charged with creating a survey for your community! The community is interested in having you create a survey and present the results at the next town hall meeting.

To complete this project you will:

Think of a problem within your community or workplace. The problem needs to be something others will have an interest in solving or will want to share reactions to.

1. Create a 10 question survey with quantitative variables (number) on a topic you are interested in. Think of questions where 0 is dislike there is a scale to 4- like. Another way to do this is using 0-never, 1 sometimes, 2 frequently, and 3 always.

2. Administer the survey to a minimum of 10 people.

3. Analyze your data for the mean, median, and mode of each questions.

4. Create a visual from this chapter: bar graph, box and whisker plot, histogram, stem and leaf plot. etc.

5. Compile the information into a slide presentation, of at least 5 slides, to present at the next town hall. The presentation should present: the mean, median, and mode of each question, the visual of the data, and conclusions based on the statistics you found in the survey.

What happens if conditions change, so that the supply and demand sets are altered slightly?  If the equilibrium is disturbed for some reason, what is the result?  How do the suppliers and consumers arrive at the equilibrium?

/in /by

1. Mathematical models in economics

1.1 Introduction
In this book we use the language of mathematics to describe situations which occur in economics. The motivation for doing this is that mathematical arguments are logical and exact, and they enable us to work out in precise detail the consequences of economic hypotheses. For this reason, mathematical modelling has become an indispensable tool in economics, finance, business and management. It is not always simple to use mathematics, but its language and its techniques enable us to frame and solve problems that cannot be attacked effectively in other ways. Furthermore, mathematics leads not only to numerical (or quantitative) results but, as we shall see, to qualitative results as well.

1.2 A model of the market
One of the simplest and most useful models is the description of supply and demand in the market for a single good. This model is concerned with the relationships between two things: the price per unit of the good (usually denoted by p), and the quantity of it on the market (usually denoted by
q). The ‘mathematical model’ of the situation is based on the simple idea of representing a pair of numbers as a point in a diagram, by means of coordinates with respect to a pair of axes. In economics it is customary to take the horizontal axis as the q-axis, and the vertical axis as the p-axis. Thus, for example, the point with coordinates (2000, 7) represents the situation when 2000 units are available at a price of $7 per unit.
How do we describe demand in such a diagram? The idea is to look at those pairs (q, p) which are related in the following way: if p were the selling price, q would be the demand, that is the quantity which would be sold to consumers at that price. If we fill in on a diagram all the pairs (q, p) related in this way, we get something like Figure 1.1.

2 Mathematical models in economics
pDq

Figure 1.1: The demand set
We shall refer to this as the demand set D for the particular good. In economics you will learn reasons why it ought to look rather like it does in our diagram, a smooth, downward sloping curve.
Suppose the demand set D contains the point (30,5). This means that when the price p = 5 is given, then the corresponding demand will be for q = 30 units. In general, provided D has the ‘right’ shape, as in Figure 1.1, then for each value of p there will be a uniquely determined value of q. In this situation we say that D determines a demand function, qD. The value written qD(p) is the quantity which would be sold if the price were p, so that qD(5) = 30, for example.
Example Suppose the demand set D consists of the points (q,p) on the straight line 6q + 8p = 125. Then for a given value of p we can determine the corresponding q; we simply rearrange the equation of the line in the form q = (125 – 8p)/6. So here the demand function is D( ) _ 125 – 8p
q p – 6 .
For any given value of p we find the corresponding q by substituting in this formula. For example, if p = 4 we get
q = qD(4) = (125 – 8 x 4)/6 = 93/6.
o

A model of the market 3
There is another way of looking at the relationship between q and p. If we suppose that the quantity q is given, then the value of p for which (q, p) is
in the demand set D is the price that consumers would be prepared to pay if q is the quantity available. From this viewpoint we are expressing p in
terms of q, instead of the other way round. We write pD(q) for the value of p corresponding to a given q, and we call pD the inverse demand function. Example (continued) Taking the same set D as before, we can now rearrange the equation of the line in the form p = (125 – 6q)/8. So the inverse demand function is
D( ) _ 125 – 6q
p q – 8 .

Next we turn to the supply side. We assume that there is a supply set S consisting of those pairs (q,p) for which q would be the amount supplied to
the market if the price were p. There are good economic reasons for supposing that S has the general form shown in Figure 1.2.
p
q
Figure 1.2: The supply set
If we know the supply set S we can construct the supply function qS and the inverse supply function pS in the same way as we did for the demand function and its inverse. For example, if S is the set of points on the line 2q – 5p = -12, then solving the equation for q and for p we get
S( ) _ 5p – 12 q p – 2 ‘ S( ) _ 2q + 12 p q – 5 .

4 Mathematical models in economics

1.3 Market equilibrium
The usefulness of a mathematical model lies in the fact that we can use mathematical techniques to obtain information about it. In the case of supply and demand, the most important problem is the following. Suppose we know all about the factors affecting supply and demand in the market for a particular good; in other words, the sets Sand D are given. What values of q and p will actually be achieved in the market? The diagram (Figure 1.3) makes it clear that the solution is to find the intersection of D and S, because that is where the quantity supplied is exactly balanced by the quantity required.
p
D s
Figure 1.3: The equilibrium set E = S n D
q
The mathematical symbol for the intersection of the sets Sand D is S n D, and economists refer to E = S n D as the equilibrium set for the given market. Fortunately, there is a simple mathematical technique for finding the equilibrium set; it is the method for solving ‘simultaneous equations’.
Example Suppose the sets D and S are, respectively, the sets of pairs (q, p) such that’
q + 5p = 40 and 2q -15p = -20.
Then a point (q*, p*) which is in the equilibrium set E = S n D must, by definition, be in both Sand D. Thus (q*, p*) satisfies the two equations
q* + 5p* = 40, 2q* – 15p* = -20.
The standard technique for solving these equations is to multiply the first one by 2 and subtract it from the second one. Working through the algebra,

Excise tax 5
we get q* == 20 and p* == 4. In other words the equilibrium set E is the single point (20,4). D It is worth remarking that in this example we get a single point of equilibrium, because we took the sets D and S to be straight lines. It is possible to imagine more complex situations, such as that we shall describe in Example 2.5, where the equilibrium set contains several points, or no points at all.

1.4 Excise tax
Using only the simple techniques developed so far we can obtain some interesting insights into problems in economics. In this section we study the
problem of excise tax. Suppose that a government wishes to discourage its citizens from drinking too much whisky. One way to do this is to impose a fixed tax on each bottle of whisky sold. For example, the government may decide that for each bottle of whisky the suppliers sell, they must pay the government $1. Note that the tax on each unit of the taxed good is a fixed amount, not a percentage of the selling price. Some very simple mathematics tells us how the selling price changes when an excise tax is imposed. Example In the previous example the demand and supply functions are given by
S 15 q (p) == 2 P – 10 ,
and the equilibrium price is p* == 4. Suppose that the government imposes an excise tax of T per unit. How does this affect the equilibrium price?
The answer is found by noting that, if the new selling price is p, then, from the supplier’s viewpoint, it is as if the price were p – T, because the supplier’s revenue per unit is not p, but p – T. In other words the supply function has changed: when the tax is T per unit, the new supply function qST is given by Of course the demand function remains the same. The new equilibrium values qT and pT satisfy the equations

6 Mathematical models in economics Eliminating qT we get
T 15 T
40 – 5p = 2(P – T) – 10.
Rearranging this equation, we obtain and so we have a new equilibrium price of The corresponding new equilibrium quantity is
qT =40-5pT =20-3T.
For example, if T = 1, the equilibrium price rises from 4 to 4.6 and the equilibrium quantity falls from 20 to 17. Unsurprisingly, the selling price has
risen and the quantity sold has fallen. But note that, although the tax is T per unit, the selling price has risen not by the full amount T, but by the fraction 3/5 of T. In other words, not all of the tax is passed on to the consumer. D

1.5 Comments
1. Economics tells us why the supply and demand sets ought to have certain properties. Mathematics tells us what we can deduce from those properties and how to do the calculations.

2. Mathematics also enables us to develop additional features of the model. In the case of supply and demand, we might ask questions such as the following:

  •  What happens if conditions change, so that the supply and demand sets are altered slightly?
  •  If the equilibrium is disturbed for some reason, what is the result?
  •  How do the suppliers and consumers arrive at the equilibrium? A typical instance of the first question is the excise tax discussed above. In this book we shall develop the mathematical techniques needed to deal withmany other instances of these questions.

Write a paragraph to answer the following questions: What was the purpose of your study? What population did you sample from? Who/What made up your sample? When and where was the sample obtained? What Method Sampling did you use to select the sample? Give some detail about this.

/in /by

Body weight and Calorie intake

Purpose Statement:
The aim of the study is to find out if there is an association between body weight and calorie intake.

Research Question:
Is the daily calorie intake associated with an increased risk of obesity (increase in body weight)?

Data must have 30+ samples pieces of data (included in the attachment, survey data with survey questions)

I. Introduction

Write a paragraph to answer the following questions: What was the purpose of your study? What population did you sample from? Who/What made up your sample? When and where was the sample obtained? What Method Sampling did you use to select the sample? Give some detail about this.

Do you think you obtained a random sample (did everyone/thing in the population have an equal opportunity to be surveyed)? Explain.

II. Looking at a yes/no question

For the yes/no questions and answer the following questions:

A. State the question

B. Create a pie graph of the yes/no responses.

C. State the sample proportion of “yes” responses, .

 

III. Looking at a numerical questions (Make Sure that you have TWO Quantitative Variables (calorie intake and body weight)

For the each numerical questions answer the following questions:

 

A. State the question.

B. Draw a histogram, box-and-whisker plot,

C. Calculate the summary statistics (for each data set) Which measure of Central Tendency best describes the data?

D. Discuss the shape, center, and spread of the distribution of responses to this question.

 

IV. Scatter Plot

A. State the two numerical questions. Indicate which variable is x, and which is the dependent variable y. Explain why you assigned the variables in this way.

B. Draw the scatter plot

C. Find the Correlation Coefficient. Do you think that is a Strong, Moderate or Weak Correlation.

D. Find the Line of Best Fit. Draw it on The Scatter Plot.

E. Use the Regression Line to make a Prediction.

Develop a model for estimating the average or expected average cost of benefits based on the number of employees a company has. If you develop a parametric model, provide the model.

/in /by

Download the dataset Health Care Cost per Employee.csv . In this dataset you will find data on small to mid sized local business and they’re health care costs (in thousands, and actually it’s a bunch of other benefit costs as well, but just pretend its health care costs and ignore why the numbers seem so high). There are two variables, the first is the number of employees that a company has. This number ranges from a single employee up to about 100 employees. The second variable represents the average cost in benefits associated with employees.
You’ll notice if you scatter plot, benefits for a small number of employees is quite high (image paying single payer health insurance for a few people and their families in addition to insuring them at work Ect. Ect.).

You are tasked with the following:
1. Develop a model for estimating the average or expected average cost of benefits based on the number of employees a company has. If you develop a parametric model, provide the model. If you develop a non parametric model, graphically represent your model overlaid on top of a scatterplot of the data. In either case please document how you arrived at your final model.

2. Create a 95% confidence interval for E(avg. cost|55 employees). That is, compute a 95% confidence interval for the average cost of benefits per employee for all companies that have 55 employees.

3. Create a 95% prediction interval for E(avg.cost|55 employees).

4. Add your results from part 2 and 3 to your scatterplot

Sort and re-index the bidders so that bitvi > b2w2 > > brawn. Pick winners in this order until one doesn’t fit, and then halt. Return either the solution from the previous step or the highest bidder, whichever has larger social welfare. Call this allocation rule “Method X”. Does a payment rule exist to make Method X DSIC?

/in /by

Imagine you are the captain of a cargo ship and different companies want you to transport their cargo. Each piece of cargo has a certain weight and valuation below. Suppose the ship can carry 17 tons of cargo. Item’s valuations are kept private by the companies but item weights are known publically.

Item/Bidder Valuation Weight

A                 10                10

B                  9                   8

C                  6                    5

D                 4.5                 5

 

(a) (3 pts) Run the Greedy Algorithm and compare that to the Social Welfare Maximizing Allocation.

(b) (5 pts) Determine the price bidder C will pay under the Greedy Algorithm.

(c) (5 pts) Sort and re-index the bidders so that bitvi > b2w2 > > brawn. Pick winners in this order until one doesn’t fit, and then halt. Return either the solution from the previous step or the highest bidder, whichever has larger social welfare. Call this allocation rule “Method X”. Does a payment rule exist to make Method X DSIC?

(d) (5 pts) For the Greedy Knapsack Heuristic, we are guaranteed to obtain at least 1/2 of the maximum social welfare. Discover the analogous guaranteed social welfare of the allocation rule of Method X or prove that no such guarantee exists. You may assume truthful bidding.

Is this a single-parameter environment? Explain fully. Does the greedy allocation rule maximize social welfare? Prove the claim or construct an explicit counterexample.

/in /by

Exercise adapted from Problem 4.3:

Consider a set M of distinct items. There are n bidders, and each bidder i has a publicly known subset Ti ⊆ M of items that it wants, and a private valuation vi for getting them. If bidder i is awarded a set Si of items at a total price of p, then her utility is vixi − p, where xi is 1 if Si ⊇ Ti and 0 otherwise. Since each item can only be awarded to one bidder, a subset W of bidders can all receive their desired subsets simultaneously if and only if Ti ∩ Tj = ∅ for each distinct i, j ∈ W .

  • (a) Is this a single-parameter environment? Explain fully.
  • (b) The allocation rule that maximizes social welfare is well known to be NP hard (as the Knapsack auction was) and so we make a greedy allocation rule. Given a reported truthful bid bi from each player i, here is a greedy allocation rule:
  1. (i) Initialize the set of winners W = ∅, and the set of remaining items X = M.
  2. (ii) Sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn.
  3. (iii) For i = 1, 2, 3, . . . , n :

If Ti ⊆ X, then:
– Delete Ti from X.
– Add i to W .

       4. (iv) Return W (and give the bidders in W their desired items).

Is this allocation rule monotone (bidder smaller leads to a smaller cost)? If so, find a DSIC auction based on this allocation rule. Otherwise, provide an explicit counterexample.

  • (C) Does the greedy allocation rule maximize social welfare? Prove the claim or construct an explicit counterexample.