MBA 201A

Section 1

Overview

Introduction

Section Agenda

- Math Review

- Class Concepts (briefly)

- Problem Set #1

- Answer Additional Questions

Introduction

Objective: teach the how to and to learn by doing

Office Hours – By email appointment

sharat_raghavan@haas.berkeley.edu

My Job:

Get you through 201A with as little stress as possible

Go through problem sets; review for exams

Problem Set Questions Exam Questions

Your Job:

Ask Questions and suggest improvements

Work though problem sets

Section is open ended, so don’t be afraid to speak up

Math Review

Can you solve this problem?

Do you understand the basic ideas?

What is a function and how do I solve one for an unknown “x” (Algebra)

How / why is the slope (derivative) of a function associated with its maximum? (Simple Calculus)

a) What value of x maximizes profits ( in the following function?b) What is the value of the profit function at its maximum?

(x) = (10 – x)(x – 4)

What is a function?

Formal definition

A function is a mathematical expression of the relationship between two variables. Changes to the independent variable (right hand side) cause changes in the dependent variable (left hand side).

Examples:DEPENDENT VAR. (F(x)) INDEPENDENT VAR (X)Profits for a firm Quantity producedDemand for mortgages Interest rateBatting average Hits in a season

Algebra

Step 1: Simply the left side

10X – X – 2 + 3X – 3 = 19

12X – 5 = 19

Add 5 to both sides

12X – 5 + 5 = 19 + 5

12X = 24

Divide 12 by both sides

12X/12 = 24/12

X = 2

10X – (X + 2) + 3(X – 1) = 19What is X?

Lines and Slope

Slope

“Steepness of a line”

“rise / run”

What are signs of the slopes of these lines?

Derivatives

Derivative

“Slope” for a curve at a point (ie the slope of the tangent line)

X(x = 3)

dy /dx = y’(x)Y

(note that: y’(3) > 0)

y(x)

Derivative Formulas

If f(x) = 5x, then df/dx = 5 If f(x) = 5x2, then df/dx = 10x For this class, if f(x) = axb, then df/dx =

(ba)x(b-1)

*Note: typically, we will only use equations where b=1 or 2 in this class

If f(x) = 5x2 + 5x, then df/dx = 10x + 5 Product Rule: d[f(x)g(x)]/dx = f(x)[dg/dx]+

[df/dx]g(x)

Derivatives and Maximizing – Some Intuition

Remember that the derivative of f(x) w.r.t. X evaluated at any X is the slope of the curve at X.

Notice that at the top of the curve the curve is flat i.e. the slope is zero.

This is why to find the value of X that maximizes f(x) we set the derivative equal to zero and solve for X.

In other words, we find the value of X where the slope of the curve is zero.

Derivative Example

f(X) = 16X + 2 – 2X2

What is the value of X that maximizes f(X)?

Step 1: Take the derivative of f(x) w.r.t. X

df(x)/dX = 16 – 4X Step 2: Set the derivative of f(x) equal to

zero

df(x)/dX = 16 – 4X = 0 Step 3: Solve for X

X = 4

Derivative Example

Now let’s solve that problem:

Step 1: Use Algebra to simplify

(x) = (10 – x)(x – 4)(x) = 10x – x2 – 40 + 4x

(x) = – x2 + 14x - 40

a) What value of x maximizes profits ( in the following function?b) What is the value of the profit function at its maximum?

(x) = (10 – x)(x – 4)

Derivative Example (cont’d)

Step 2: Use Calculus to find maximum

(x) = – x2 + 14x – 40ddx= -2x + 14

At max, -2x+14 = 0-2x = -14

x* = 7 Step 3: Evaluate function at Maximum

(use original function)at x=7 *(x) = -49 + 98 – 40 = 9

a) What value of x maximizes profits ( in the following function?b) What is the value of the profit function at its maximum?

Derivative Example (cont’d)

Step 4: Check for Consistency (Optional)

(x) = (10-x)(x-4)

(7) = 3*3 = 9(8) = 2*4 = 8(6) = 4*2 = 8

Note: You can also use the product rule of derivates to answer the question, this would save you the effort of simplifying

√

√

Class Concepts – Setting up Decision Trees Node Types

: Chance Node

: Decision Node

:Terminal Node

Setting up a Decision Tree

- Timeline

- What has happened already? Sunk Costs: ignore but record?

Expected value vs. sunk decision- but equivalence with sunk cost

- Is a given node a decision, chance, or terminal node?

- How many branches from the node? Are you sure? Continue

until every branch has come to a dead end (i.e., a terminal node)

Class Concepts – Solving Decision Trees

Backward Induction

Start from each terminal node and work towards the beginning

Decision Nodes

- Choose the best of the options (and mark it!)

Chance Nodes

- Find the expected value of the outcomes

Finally, calculate the EV of the entire tree: each POSSIBLE terminal node X probability of each POSSIBLE terminal node (note all probabilities added together better equal 100% and generally NOT all terminal nodes are possible)

Launch

Don’t Launch

50%

50%

2.5

-3

2.5

EV = 5*0.4 + -10*0.6

EV = 2 – 6

EV = -4

Customer Makes a Purchase

No Purchase

Invest in Acquiring a Customer

-10

-4

540%

60%

Problem 3 from Problem Set 1

Should AK steel undertake a multi-step R&D Project?

3 sequential steps, each with a .8 probability of success (.2 probability of failure)

Each step costs $500k

If all three are successful, then save $4,000k

Need all three steps to have any savings.

Can decide whether to do the next step after seeing the result of the previous step.

Problem 3 from Problem Set 1: Decision Tree part a

Problem 3 from Problem Set 1: Understanding the Tree

b) Probability of Success? 0.8^3 = 0.512

c) Expected gross return: .512*4mm= $ 2,048,000

d) Should they do it?Expected net return: 0.8*0.8*0.8 * (4,000,000 – 500,000 – 500,000 – 500,000) + 0.8*0.8*0.2 * (-500,000 – 500,000 – 500,000) + 0.8*0.2 * (-500,000 – 500,000) + 0.2 * (-500,000) = $ 828,000

e)Should it quit if there are no failures? Continue even if there is a failure? No and No (value is 0 if have one step that fails)

f) What if there is an alternative that AK steel leans about before the third step? With it, AK steel can spend $150k on a technology that saves $1,000k with certainty. The technologies cannot be used together.

Problem 3 from Problem Set 1: Decision Tree part f

2,750,000

Problem 3 from Problem Set 1: Decision Tree

g) What is the chance the alternative is valuable? .2 chance valuable (i.e., if step 3 of original fails)

h) How much is it worth? CONDITIONIAL on having the first 2 step’s succeed, the option value is: EV[|option value] = 0.2 * 1,000,000 = 200k

i) What should it do?Firm should pursue both simultaneously (highest EV- follow the arrow)

j) What to do if it knew about the alternative technology to begin with? (careful here)Only the 3-step project: EV= $828,000Only the alternative project: EV=(1,000,000-150,000)=$850,000Do Both? EV[marginal value of 3 step] = 3,000,000 * 0.8^3 = 1,536,000EV[costs] = 0.2 * 500,000 + 0.8* 0.2 * 1,000,000 + 0.8 * 0.8 * 1,500,000 = 1,220,000 THUS, do both still…

What if risk averse?

Questions on anything else?