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Math 3660 - Spring 2017
Mathematical Models in Economics\
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Save a copy of this notebook, complete the exercises, save and submit on \
Blackboard your final version by start of class Tues. Jan. 17.\
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Edit the line above that reads Enter your name here, inserting instead your \
name. You may also want to save to a file name that includes your name. By \
adding your name into the file and file name, you reduce the chance I will \
mix-up your submission with someone else\[CloseCurlyQuote]s.\
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1. What is a 15% tip on a $40 meal? How many such tips an hour would a \
server need to make to earn as much as a $300 an hour lawyer?\
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This is a really simple (almost silly) mathematical model, but it points up \
some of the common elements of mathematical models. We compare a real \
situation, how much people leave in tips in relation to their bills, \
translated to a numerical relationship, that of a proportion between the \
amount of the tip and the amount of the bill, specified in this case as 15%. \
We can make a mathematical (numerical) prediction based on assumptions, \
namely the assumption that the proportion is a constant, the constant a \
parameter of the model whose value is assumed, with the predicted tip \
determined as a function of the bill, taken to be a particular value in this \
example. But the prediction is only an approximation to the real experience. \
Some tips are larger, some smaller. This model is so well known and \
understood that most people use it as a guide to tipping. But is it really \
an accurate prediction? If a server has to deal with too many tables, the \
average tip rate may decline due to a decline in the quality of service. Are \
relatively small individual bills likely to get a larger or smaller rate of \
tipping? Is there any reason for tips to be in proportion to bill, rather \
than say number of visits to a table? Similarly, is there any reason lawyers \
should be paid at an hourly rate rather than by the job? Is it reasonable \
that lawyers will sometimes work for a share of any award they win? What \
about doctors or economic consultants or ...? Compensation schemes is a fair \
topic for economists. The mathematical model is an analogy between some \
aspect of a real world situation and mathematical objects and relationships. \
It translates a real world question into a mathematical question where \
mathematical techniques can be applied. The model will likely only be an \
approximation to the truth, but it can illuminate the real world situation \
because it reveals the underlying logic relating the assumptions to the \
conclusions. \
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2. A small gourmet restaurant might serve 30 people at a time, for three \
sittings a night, five nights a week, 52 weeks a year, averaging say $40 a \
meal. Variable costs, that is all the costs that would decrease if the \
number of customers were reduced, presumably in proportion to a change in the \
number of customers, might be $15 per meal served, for food and cooking \
supplies, dish washing and the like. Fixed costs would include personnel, \
cooks, servers, busboys etc. (if an adjustment to personnel is not a likely \
response to a change in the number of customers), rent for the space for a 30 \
person dining area plus kitchen and perhaps office space, utilities, \
marketing and miscellaneous administrative expenses. Imagine 10 employees \
averaging $35000 a year salary (more for the head chef, less for a busboy). \
Suppose a retail space of 2500 sq. ft. renting for $5 per sq. ft. per month. \
For this analysis ignore other costs and taxes. What is the total annual \
revenue for the restaurant? What is the total annual expenses? And what \
profit does the restaurant make for the owner each year? \
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Give an income and expenses statement for the restaurant. Of course, I\
\[CloseCurlyQuote]ve simplified things considerably, and made up a lot of \
numbers. These numbers may be way off. The question is one of basic \
accounting, adding up total income, and subtracting total costs, to see what \
the net profit of a business should be. If the difference doesn\
\[CloseCurlyQuote]t show up in the owner\[CloseCurlyQuote]s bank account at \
the end of the year, the accountant should be fired or arrested for \
embezzlement. (Wait, we forgot to pay for an accountant, oh well. What if \
we could move to a larger location, or open seven days a week, but then we \
would need more employees, but would we have more customers, or could we \
lower prices, or...)\
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3. This small business probably doesn\[CloseCurlyQuote]t hold many assets, \
unless it owns a location instead of renting perhaps. But a reputation and \
supply of regular customers that yields an income is of value. A way of \
accounting for this value is to imagine how much money you would have to \
invest, at some rate of return, to get the same income. Alternatively, the \
value of a certain income stream is the principal of a loan whose interest \
payments would be just met by that income stream, the maximum amount you \
could pay if you were to borrow the money to buy the business and pay back \
the loan out of the income. This present value of future income depends \
critically on the discount rate applied to the future income, the interest \
rate in the calculation. Suppose you could count on the income in the above \
calculation year in and year out, and you use a 10% per year discount rate. \
Imagining simple interest on the value of the business each year is equal to \
the profit the business makes, what is the value of the business?\
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If a business made a certain $100,000 a year and you could borrow money for \
10% per year, wouldn\[CloseCurlyQuote]t you borrow $500,000 to buy the \
business pay $50,000 interest and pocket the $50,000 difference between the \
income and loan payment? Wouldn\[CloseCurlyQuote]t someone else be willing \
to outbid you, borrow and pay $700,000 for the business, pay $70,000 in \
interest, and pocket instead just $30,000? And then the bidding would \
continue up to the limiting value of $1,000,000 when the interest payed \
matches the income. This would work except that in all practical cases, the \
income is uncertain, potential difficulty in reselling the business at a \
future time or else a relatively short time horizon over which one would want \
to have payed back also the principal of a loan. All of these adjustments \
can be folded into an appropriate discount rate to apply; compare this to the \
P/E ratio, or its reciprocal, earnings yield, for stocks.\
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4. What happens to the income statement if during a recession the number of \
customers decreases by 10%? What is the value of the business then, assuming \
the 10% decrease is permanent?\
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"5. (Challenge) A better model of present value of an income stream would \
account for the uncertainty in future income and potentially include variable \
and uncertain rates of discounting over time. We will consider models with \
random elements soon enough. For now, consider a continuous periodically \
varying income stream, say from a predictable business cycle or other \
periodic event (Olympics, presidential campaigns, etc.). Say that the rate \
of income at a time ",
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" varying continuously over time. Over a seven year period, income averages \
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business cycle?"
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