Today’s lecture notes are available on the website, in PowerPoint format Announcements: Next Week: WS 11-13 Tues: Activity 3 Fri: Hwk 3 (WS 8->12) We aim.

Today’s lecture notes are availableon the website, in PowerPoint format

Announcements:

Next Week: WS 11-13Tues: Activity 3Fri: Hwk 3 (WS 8->12)

We aim to return your midterms on Tuesday.

Today

• Finish off Worksheet 9– Recap of MC/AC/AVC– How to read BEP & SDP from MC/AC/AVC

• Worksheet 10: Breaking Even– Linear TR and TC

q 100 200 300 400 500 600 700 800 900 1000

MC 1.88 0.68 0.08 0.08 0.68 1.88 3.68 6.08 9.08 12.68

AC 7.73 4.48 2.90 2.33 1.93 1.81 1.94 2.30 2.88 3.68

AVC 2.73 1.98 1.43 1.08 0.93 0.98 1.14 1.68 2.33 3.18

200

q 100 200 300 400 500 600 700 800 900 1000

MC 1.88 0.68 0.08 0.08 0.68 1.88 3.68 6.08 9.08 12.68

AC 7.73 4.48 2.90 2.33 1.93 1.81 1.94 2.30 2.88 3.68

AVC 2.73 1.98 1.43 1.08 0.93 0.98 1.14 1.68 2.33 3.18

Amounts ($)

TangentSlopes($/item)

($ per item)

q 100 200 300 400 500 600 700 800 900 1000

MC 1.88 0.68 0.08 0.08 0.68 1.88 3.68 6.08 9.08 12.68

AC 7.73 4.48 2.90 2.33 1.93 1.81 1.94 2.30 2.88 3.68

AVC 2.73 1.98 1.43 1.08 0.93 0.98 1.14 1.68 2.33 3.18

Amounts ($)

DiagonalSlopes

($/item)

($ per item)

Reading BEP from MC/AC:

BEP=1.8

On TC vs q graph: the Breakeven Price corresponds to the slope of the lowest diagonal line which is tangent to the graph of TC. At that point, BEP corresponds to the AC (slope of diag) AND to MC (slope of tangent).

So on the graph of MC vs q we can locate BEP at the point where MC equals (crosses) AC.(y-coordinate)

(Recall that BEP is also smallest value of AC)

2nd use of MC:

Similarly: on the graph of VC vs q: the SDP corresponds to the slope of the lowest diagonal line which is tangent to VC.

SDP=0.9

At that point, SDP corresponds to the AVC (slope of diag) AND to MC (slope of tangent).

So on the graph of MC vs q we can locate SDP by looking where MC equals (crosses) AVC.

(Recall that SDP is also smallest value of AVC)

NEW MINDSET: LINEAR ANALYSIS

We’ll be using functional notation and algebra to answer questionsMost expressions will be linear: f(x)=ax+b

• f(x) means f is a function of (depends on) x.•x is the independent variable, f is the dependent

• The graph (f versus x) of such a linear expression is a line• of slope = a• and y-intercept = b

Example: TC(q)=mq+c

The Total Cost TC is a function of the quantity you produce, q.Its graph will be a line of slope m and y-intercept c

What if we write this:TC(q)=z+qm?

Same thing:The functional notation clearly shows that

the Total Cost TC is a function of the quantity you produce, q.So its graph is a line of slope m and y-intercept z.

TC(q)=z+qm=mq+z

WS 11: Breaking Even

Story I: You make and sell toothbrushes. • The market price (aka selling price) is $3.50 per toothbrush.• You have fixed costs of $200, • and the marginal cost is always $1

Let’s start with something familiar: Describe the graphs of TR and TC.

Since the selling price p=$3.5 is constant for all q, MR(q)=p=3.5TR(q)=pq=3.5q

So the graph of TR is a diagonal line of slope 3.5.

Since the marginal cost is always $1, VC(q)=1xq=q, so TC(q)=VC(q)+FC=q+200

So the graph of TC is a line of slope 1 and y-intercept 200.

TR=3.5qTC=q+200

GOAL: To determine the quantity q for which we break even

Meaning: profit=loss=0

??

We’ll start to solve things using algebraic expressions and getting more accurate answers!

Since TR(q)=3.5q & TC(q)=q+200

& we break even when profit=TR-TC=0:

we can set TR=TC and solve for q

Do it! What do you get? Answer: q=80

Story II: Same setup, but • your marginal cost is doubled: it now takes $2 per toothbrush

TR(q)=pq=3.5q

TC(q)=VC(q)+FC=2q+200

We break even when TR=TC, so 3.5q=2q+200. Solve for q.

Answer: q=133.33

Story, generalized: Same setup, but everything is in letters now

TC(q)=VC(q)+FC=mq+c

We break even when TR=TC, so pq=mq+c. Solve for q.

Answer: q=c/(p-m)

•The market price (aka selling price) is $p per toothbrush.• You have fixed costs of $c• and the marginal cost is always $m.

TR(q)=pq

pq=mq+c

First, separate all terms involving the variable q:pq-mq=c

Now, factor out the variable q on the left:(p-m)q=c

Yay, we obtained our first formula! : q=c/(p-m)

Why the excitement?Since we made a general formula to fit our story problem we can now plug in any numbers for the fixed cost c, market price p & the marginal cost mand very quickly compute the quantity at which we’ll break evenwithout having to redo all the work each time!

Algebra Rocks!