answer the following questions...

In equilibrium, the aggregate private investment corresponds to the aggregate private saving:

t= t

Let t t/t denote the output per capita, t t/t stands for the private investment per capita, t t/t stands for the private physical capital per capita, t t/t represents for the public physical capital per capita and t t/t stands for the consumption per capita.

a. Using the aggregate production function, show that the output/income per capita at time t is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

b. From your answer to a, show that the consumption per capita at time t and the equilibrium private investment per capita at time t are functions of the private physical capital per capita at time t and the public physical capital per capita at time t. (10 points)

c. Show that the change in the public physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

d. In equilibrium, show that the change in the private physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

e. Derive the steady-state formulas for the private physical capital per capita and for the public physical capital in per capita. (20 points)

f. Derive the steady-state formulas for the output per capita and the consumption per capita. (10 points)

g. Using your answer to f., derive the tax rate that maximizes the steady-state consumption per capita. (10 points)

Just about every event these days is expected to have a professionally catered menu to accompany it. Weddings, conferences, corporate events, anniversaries, graduations and parties are just some of the different types of events where people want to have that special food and drink experience.

Weddings, Parties, Everything (WPE) is a rapidly-growing online business that links together customers who need a caterer for their event with businesses that provide catering. Customers input their location and requirements, and are presented with a shortlist of potential caterers who may be able to provide what they need.

WPE has been partly manual for the first few months of their operation, but now needs a new system to automate their business of putting together the right customer with the right caterer. They intend to roll the system out nationally, from their original base in WA. Edison Guacamole, the owner/manager of WPE, has contacted you for help.

The main two sides to the system are, of course, the caterers and the customers. Caterers register on the WPE system by providing their business name, description, types of services they provide (selected from a list), their location (city and postcode), link to their own website, and contact email. 'Verified' caterers are those that have provided their ABN, business name, logo, and proof of current Public Liability or Professional Indemnity insurance to the system. Using a verified caterer provides additional confidence to customers that they are using a legitimate business.

If they choose, the caterer can also advertise themselves as having an 'On-Time Guarantee'. This means that they guarantee to provide the service on time, and if they are more than half an hour after the scheduled time will reimburse the customer $100. Caterers can also use reviews from previous satisfied customers in their profile, and their average rating by customers. Caterers pay to use the WPE system: this is done through purchasing a number of 'credits' when they register. Payment for credits is done through PayPal outside of the WPE system, although WPE also keeps a record of the transactions. Credits are used for preparing quotes for customers, with every quote costing the same number of credits. Credits expire after three months, and the system will notify the caterer that they need to purchase more credits in order to be able to quote. The system also notifies the caterer if they have run out of credit.

Caterers can also buy extra credits in order for all or any of their quotes and operation procedures to be assessed by third party private Covid-19 Safety Assessors (CSA). In that case, the quotes and operation procedure are sent to a CSA and then CSA provides their approval in 24 hours. The seals of those approvals are then visible in the CSA approved quotes' pages. CSAs are paid for the assessment. In case a CSA fails to provide assessment in 24 hours, the request for assessment goes to another CSA.

Customers do not need to pay in order to use the WPE system. When a customer requires a quote for a catering job, they submit their requirements through an online form. They supply their postcode location and a list of requirements, including type of event (buffet, dinner party, food truck, cocktail party, etc); broad type of cuisine (Australian, French, Chinese, Mexican, Indian, etc); number of people to be catered for; budget per person; service required (delivery only, wait staff, bar staff, etc), date of event, and any further instructions. They also supply an email address they can be contacted through. This email address also provides them with access to their account where they can see the status of all their job requests. The account is created when they submit their first job request.

Once the customer request for a catering job has been submitted, the WPE system matches the request with the catering services that are registered with it, and determines the three caterers that could best meet the customer's requirements. These caterers are then notified via email that there is an open request for quotation.

The caterers then respond to the customer with a detailed quote. The quote includes the cost estimate, a flag indicating whether further information is required for a complete quote, and any comments. This information is provided in an online form that the caterer completes and the system sends to the customer.

The customer receives an email including the same information that the caterer has provided in the quote form. The caterer and the customer may then discuss the job directly (outside the WPE system) in order to arrive at a final decision. To help make up their mind, the customer may also read more about the caterer in the profile they have provided to WPE, the caterer's own website, and in the reviews that have been posted to WPE.

Once the customer has decided on a caterer, they return to their account on the WPE site and choose the 'hired' option against the relevant caterer's quote for the job request. When this has been done, the other caterers who quoted are automatically notified that their quote was unsuccessful. If none of the quotes suits the customer, they may choose to enter 'unsuccessful' directly against the quotes.

If no quotes are received within 48 hours of posting the job request the customer is notified and the request is closed. A customer may also decide that they wish to cancel the job request any time before making a booking, and record this against the job request in their account.

Customers are encouraged to provide feedback on the service that their caterer has provided. After the job is completed, the customer can log in to their account and leave a 5-star rating and associated comment against the job. To prevent spurious reviews, it is only possible to post a review for a service that is recorded as being booked by that customer.

When the review is posted, the feedback recorded against the caterer is updated for other potential customers to browse through on the WPE site.

Obviously, the success of WPE depends on its ability to provide the best caterers for a customer so that they can be confident in using it. Various summary reports are provided each month so that Edison can monitor the health of the business. These include the percentage of quotes that were successful in obtaining a booking for a catering job; and the average satisfaction rating of completed catering jobs. The revenue to WPE in terms of payments for credit is also reported, broken down for each capital city.

Read the complete WPE case study and answer the following questions.

What is the trigger for the new system? [2]

Draw a context level data flow diagram. [3]

Prepare a system vision document. [2]

A Solow Growth Model Augmented with Public Physical Capital (65 points)

Let us consider a production economy populated at the time t with t workers. The aggregate output/income ! is produced according to the following production function:

t= tt*t1- -.

where >0 denotes a productivity parameter , (0,1) stand for output elasticity parameters, ! represents the private physical capital and ! denotes the public physical capital. The aggregate population grows at a constant rate (1, +):

t+1 t= t

The law of motion for the private physical capital from time t to time t+1 can be written as:

t+1= t+ (1 )t

where ! denotes the aggregate private investment at time t and (0,1) stands for the depreciation rate. The law of motion for the public physical capital from time t to time t+1 is:

t+1= t+ (1 )t

where t denotes the aggregate tax revenue at time t:

t= t

where (0,1) stands for a time-invariant income tax rate.

The aggregate private saving at time t is a constant fraction (0,1) of the private disposable income dt:

t= dt

where td t t and represent their marginal propensity to save parameter.

The aggregate consumption at time t is a constant fraction 1 of the private disposable income dt

t= (1 )Ydt

In equilibrium, the aggregate private investment corresponds to the aggregate private saving:

t= t

Let t t/t denote the output per capita, t t/t stands for the private investment per capita, t t/t stands for the private physical capital per capita, t t/t represents for the public physical capital per capita and t t/t stands for the consumption per capita.

a. Using the aggregate production function, show that the output/income per capita at time t is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

b. From your answer to a, show that the consumption per capita at time t and the equilibrium private investment per capita at time t are functions of the private physical capital per capita at time t and the public physical capital per capita at time t. (10 points)

c. Show that the change in the public physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

d. In equilibrium, show that the change in the private physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

e. Derive the steady-state formulas for the private physical capital per capita and for the public physical capital in per capita. (20 points)

f. Derive the steady-state formulas for the output per capita and the consumption per capita. (10 points)

g. Using your answer to f., derive the tax rate that maximizes the steady-state consumption per capita. (10 points)