Total Physical Product = f(x) = .4x + .09x x3

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Total Physical Product = f(x) = .4x + .09x2 - .003x3 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 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= What are the maximum values of TPP, APP, and MPP? At what levels of labor use are the maximum values reached? Look at the graph of APP and MPP. Which regions on this graph represent Stages 1, 2, and 3 of production? Why?

1. TPP/APP Input TPP APP 5 3.875 0.775 10 1 15 16.125 1.075 20 25 19.375

2. MPP Using Calculus, MPP=.4+.18x-.009x2 Input TPP APP MPP Discrete MPP 5 3.875 0.775 1.075 10 1 1.3 1.225 15 16.125 20 0.4 25 19.375 -0.725 -0.125

3. Max TPP, APP, MPP

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.