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    Linear Algebra: Interest Rates and Cramer's Rule

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    For each output level Y, the IS curve defines the interest rate r at which the goods market clears:


    where b is the marginal propensity to consume, G is the government spending, I^0 is the maximum investment level, and a is the responsiveness of investment to interest rates. The LM curve defines the interest rate at which the money market clears:

    mY + M^0 - hr = M^8,

    Where m is the responsiveness of the transactions demand for money to output, M^0 is the maximum liquidity demand, h is the responsiveness of liquidity demand to interest rates, and M^8 is the money supply.

    a) Write down this system of equations in matrix form. Under what condition on the exogenous parameters can this system of two equations be solved for Y and r?
    b) Using Cramer's rule, solve the system for Y and r when the condition in (a) is met.
    c) What happens to the equilibrium interest rate r if government spending increases by ?G?

    Please see the attached file for full equations.

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    The Railway Ticket System has been established in order to design the fully automated software for issuing the railway ticket. With the help of this software, acceptance of user information and issuing of Passenger Name Record can be done. It has been analyzed that the online ticketing system is helpful in reducing the time of issuing ticket. Nowadays, passengers can get all information related to the train through the internet. A railway ticketing system is efficient and provides easy and quick services to the passengers. Here, inconsistencies, ambiguities and omissions in the given plan of ticket issuing system have been analyzed. Along with this, recommendations in order to improve the railway ticketing system have been discussed in this paper.
    There are several inconsistencies, ambiguities and omissions in the ticketing system. Ambiguities and omission include different points such as: customer can buy several tickets for the same destination together or they have to buy one at a time. In this type of situation, customers face problem and are not able to get full information related to ticket issuing. Moreover, other questions such as: Can customers cancel the request? If mistake has been done and how system will respond, if wrong PIN number has been inserted?" (Graham & Veenendaal, 2008). All these are the types of omissions that have been majorly found in ticketing system in railway.
    These omissions create problems in the services of railway ticketing system. Moreover, these problems also disturb the overall system of railway ticketing system. It has been analyzed that customers also make delay in the purchasing the tickets due to the confusion and ambiguities. Passengers face inconsistencies and ambiguities in different areas such as ticket prices, rooms in trains, seat assignment and train departure and arrival time. Railway ticketing system has to work on all these areas in order to make more efficient and passengers will also get efficient services. Various problems have been found such as if user wanted to buy the multiple tickets then in such a case user do not have option. Moreover, ambiguity has been found in how potential destination can be activated (Faulkner, Finlay & Détienne, 2002). These are the different kind of inconsistencies, ambiguities and omissions that has been majorly found in railway ticketing system.
    Strategies to Correct the Problems
    An automated ticket issuing system sells rail tickets and provides quick services to the users. User can select the destination; it has been analyzed that while selecting the destination, options for the numbers of tickets will be displayed. With the help of this, a user can be able to buy several tickets for the same destination. Moreover, it will also reduce the problem of ambiguity. After selecting the destination and number of tickets, user will insert the credit card and personal identification number (Jawadekar, 2004). It has been analyzed that if the user insert the card before selecting destination, then system will not accept the card and it will automatically come out.
    The user has to follow the proper process of buying the railway tickets. After following the proper process, a rail ticket will be issued to the user and credit card will charged with the overall cost. In order to buy the ticket for another destination, user has to press the start button and menu will be displayed providing information related to potential destinations, along with the message to select the destination (Ni, 2012). After selecting the destination, users have to put the valid credit card. It is identified that system will not accept the invalid card and display the information regarding invalidity of the card. After validating the credit transaction, tickets will be issued to the user.
    In the issued ticket, user will get the information regarding the ticket prices, room on train, seat assignments, train departure and train arrival times etc. All these information on the ticket will be helpful in reducing omission, inconsistencies and ambiguities problems in the ticketing system. It has been analyzed that railway ticketing system also has to formulate the customer feedback program in order to identify the problem areas. With the help of this, corrective actions can be taken, so that improvement in the ticketing system can be made in order to provide efficient services to the users (Zhou & Li-Min, 2011).
    Faulkner, X., Finlay, J. & Détienne, F. (2002). People and Computers XVI: Memorable Yet Invisible. USA: Springer.

    Graham, D. & Veenendaal, E.V. (2008). Foundations of Software Testing: ISTQB Certification. USA: Cengage Learning EMEA.

    Jawadekar, W.S. (2004). Software Engineering: Principles and Practice. USA: McGraw-Hill.

    Ni, Y.Q. (2012). Proceedings of the 1st International Workshop on High-Speed and Intercity Railways. USA: Springer.

    Zhou, W. & Li-Min, J. (2011). The Theory and Method of Design and Optimization for Railway Intelligent Transportation Systems. USA: Bentham Science Publishers.

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