Don’t forget to turn in your homework! Today’s lecture notes are available on the website, in PowerPoint format (WS10 last part & WS11) Good Morning! Announcements: Next Week: WS Tues: Activity 3 Fri: Hwk 4

Finishing off WS 10 We discussed 3 graphical methods to compute max profit: 1)Given the graphs of TR and TC, max profit occurs when we see the greatest max distance between the graphs of TR and TC, with TR on top. Summary of Part I: Looks like max profit occurs at about 650 paperweights.

Method 2: Given the graphs of TR and TC, we look for matching slopes (parallel tangent lines) at the same q, since we want MR=MC. (Of course, we still want TR>TC, otherwise we find the max loss instead!) In our example, since TR is already a straight line, its slope (=MR=p) is always 2.5. So we can align the ruler with TR and move it parallel until it becomes tangent to TC: Once again, looks like max profit occurs at about 650 paperweights.

Method 3: Given the graphs of MR and MC, we look for their intersection point. If MR greater than MC before, and smaller after, that q gives max profit. Otherwise it gives max loss. In our example, we can compute the values of MC from the slopes of tangents to the graph of TC and fill in the table below, then plot the graph of MC. MR is always 2.5 so how do we plot MR versus q? MR(q)=2.5 MR So max profit is at q=640 (notice: more accurate answer!) MR=MC

Two more uses for the graph of MC: BEP= 1.8 Recall that on the graph of TC vs q: 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 by looking where MC equals (crosses) AC.

2 nd 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 the graph of 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.

NEW MINDSET: LINEAR ANALYSIS We’ll be using functional notation and algebra to answer questions Most 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=3.5 TR(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=3.5q TC=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 m and very quickly compute the quantity at which we’ll break even without having to redo all the work each time! Algebra Rocks!