Production
 Robinson Crusoe gets shipwrecked on an island.
He has no food, but luckily the island is full of
coconut trees. He can harvest the coconuts using
his labor as the only input according to the
following production function:
Total Physical Product = f(x) = .4x + .09x2 - .003x3
Production
 Robinson would like to do some analysis to make sure
he is spending his labor wisely; he wants have enough
coconuts to survive while still having time to relax on
the beach. He’s terrible at math though, so our AAE
215 class has agreed to help him out.
How Many Coconuts?
Graph the Total Physical Product if Robinson uses 5, 10, 15, 20,
and 25 units of his labor. Do the same for Average Physical
Product. Remember TPP=f(x)=.4x+.09x2-.003x3
2. Using Calculus (if you can), find the equation for Marginal
Physical Product (Hint: Take the derivative of the production
function with respect to x). Graph the MPP curve at 5, 10, 15,
20, and 25 units of labor on the same graph as the Average
Physical Product Curve. If you do not remember calculus, do
the above using MPP=  O u tp u t
 In p u t
3. What are the maximum values of TPP, APP, and MPP? At what
levels of labor use are the maximum values reached?
4. Look at the graph of APP and MPP. Which regions on this
graph represent Stages 1, 2, and 3 of production? Why?
1.
1. TPP/APP
Coconuts (Output)
TPP
APP
0
0
5
3.875
0.775
10
10
1
15
16.125
1.075
20
20
1
25
19.375
0.775
25
20
15
10
5
0
Labor Use (Input)
Average Physical Product
(APP)
1.2
Coconuts (Output)
Input
Total Physical Product
(TPP)
1
0.8
0.6
0.4
0.2
0
Labor Use (Input)
2. MPP
Using Calculus,
MPP=.4+.18x-.009x2
APP and Discrete MPP
Average Physical Product (APP)
TPP
APP
MPP
Discrete Marginal Physical Product (MPP)
1.4
0
0
5
3.875
0.775
1.075
0.775
10
10
1
1.3
1.225
15
16.125
1.075
1.075
1.225
20
20
1
0.4
0.775
25
19.375
0.775
-0.725
-0.125
1.2
1
Coconuts (Output)
Input
Discrete
MPP
0.8
0.6
0.4
0.2
0
5
-0.2
10
15
Labor Use (Input)
20
25
3. Max TPP, APP, MPP
APP and Discrete MPP
Total Physical Product (TPP)
25
Average Physical Product (APP)
Max
TPP=20
at x=20
Discrete Marginal Physical Product (MPP)
1.4
20
Max APP=1.075
at x=15
1
15
Coconuts (Output)
Coconuts (Output)
1.2
Max
MPP=1.2
at x=10,
15
10
0.8
0.6
0.4
0.2
5
0
5
-0.2
0
Labor Use (Input)
10
15
Labor Use (Input)
20
25
4. Stages of Production
 Stage 1: Average productivity
is increasing; if it is worth
producing at all, it is always
worth producing MORE. Do
not produce here.
 Stage 2: A decision has to be
made. Must balance
additional input use against
the additional output
produced. We would like to
produce here!
 Stage 3: BAD! If we are
producing here we are
spending inputs to LOSE
output. Do not produce here.
Average/Marginal Physical
Product
APP
MPP
1.4
1.2
1
0.8
0.6
0.4
0.2
0
5
-0.2
10
15
20
25