ASSIGNMENT #1

Business Simulations

2013-14, Sem. 2

Eric Eisenstat

Faculty of Business Administration

University of Bucharest

Due Date: 7 April 2014

Please complete the following exercises. In answering the questions, be sure to show allsteps and calculations involved. Also, please submit all working documents such as the Excelspreadsheets. Note that the last exercise is extra credit—you will not loose points if you donot do it, but you can gain up to 10 extra points if you do. These points will applied to thetotal points accumulated for the course, which means they may be used to offset points loston any assignment(s) and/or the final exam. You may submit this assignment by email to:eric.eisenstat@faa.ro

Exercise 1 (10 points) Implement Newton’s Method in Excel to find the roots of the fol-lowing equation

f(x) = x3 − 2x2 − x+ 2, (1)

given

f ′(x) = 3x2 − 4x− 1. (2)

Based on this, answer the following questions:

(a) How many roots did you find?

(b) What stopping criteria did you use?

(c) For each root, please list what starting values (x0) that resulted in those roots andhow many iterations were needed to find it.

Exercise 2 (5 points) Implement Newton’s Method in Excel to find the roots of the followingequation

f(x) = x3 − x− 3, (3)

given

f ′(x) = 3x2 − 1. (4)

Run the algorithm with the following starting values: -100, 0, 1. For each starting value,what root did the algorithm find and how many iterations were needed?

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Exercise 3 (10 points) Implement Newton’s Method in Excel to find the roots of the fol-lowing equation

f(x) = xe−x, (5)

given

f ′(x) = (1− x)e−x. (6)

Based on this, answer the following questions:

(a) Which roots does the algorithm find for starting values x0 < 1?

(b) Which roots does the algorithm find for starting values x0 ≥ 1?

(c) If someone was planning to incorporate Newton’s Algorithm to numerically solve (5)as part of a more complex simulation algorithm, what warnings or advice would youoffer?

Extra Credit (10 points) After reading carefully the example in Section 4.4, consider thefollowing. As before, preferences for two types of consumers are described by the utilityfunctions

u(xi, zi; θi) = xθii z1−θii , (7)

for i ∈ {1, 2}. Assume that for consumers of type I, the preference parameter is known toinitially be θ1 = 0.5, while for type II consumers, it is θ2 = 0.

Such a model might be applied to characterize a market where type II consumers representa segment that does not consume product x. Accordingly, suppose consumer behaviouris consistent with maximizing utility subject the budget constraint pxi + zi = yi. Thismaximization problem may be simplified as

maxxi

xθii (yi − pxi)1−θi , (8)

which result in the first order conditions (FOC)

(

θ11

x1− (1− θ1)

p

y1 − px1

)

u(x1, z1; θ1) = 0, (9)

(

θ21

x2− (1− θ2)

p

y2 − px2

)

u(x2, z2; θ2) = 0. (10)

In particular, solving the FOCs above for x1 and x2, respectively, provides the optimalconsumption levels for the two consumer types, given the price (p), incomes (y1, y2), andpreferences (θ1, θ2). The total demand for the product is then

x(p, y1, y2, θ1, θ2) = n1x1(p, y1, θ1) + n2x2(p, y2, θ2). (11)

Suppose now that you operate the only producer of this good and follow a price rule thatequates the price to marginal production costs, such that

p =ex/10

5−5

√x

. (12)

Notice that (9)-(12) form a system of equations that yields solutions for x1, x2, x, p giveny1, y2, θ1, θ2, n1, n2. If it is further known that incomes (in thousands of EUR) for the twosegments are given by y1 = 25 and y2 = 30, while the market sizes are n1 = 1.5× 105 and

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n2 = 105, then a little algebra reveals that total revenues are R = 1.875 × 106, and totaldemand is x = R/p.

Now suppose that you are considering investing in a marketing campaign that wouldattract type II consumers to your product. In particular, the marketing strategy wouldtarget influencing type II consumer preferences in such a way as to induce a new value ofθ2 > 0. Provided the above information, find the market price (p) and total quantity (x)

(a) at the moment with θ2 = 0

and those that would result if the marketing campaign succeeds to yield

(b) θ2 = .10

(c) θ2 = .25

In implementing the Newton’s algorithm to approximate the prices and quantities forvarious values θ2, keep in mind that R = px is constant and that total quantity producedis in the range of one to two million units. After obtaining the solution,

(d) discuss how you might approach evaluating whether the various level of investmentsin the marketing campaign are worthwhile (assume that achieving higher levels of θ2requires larger investments)

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