an animated instructional module

An Animated Instructional ModuleAn Animated Instructional Module

for Teaching Production Economicsfor Teaching Production Economics

with the Aid ofwith the Aid of

Three-Dimensional GraphicsThree-Dimensional Graphics

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David L. DebertinUniversity of Kentucky

This instructional module is based on theThis instructional module is based on the

polynomial production functionpolynomial production function

y = x + x - 0.05 x + x + xy = x + x - 0.05 x + x + x

- 0.05 x + 0.4 x x .- 0.05 x + 0.4 x x .

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This production function was chosenThis production function was chosen

for a number of reasons.for a number of reasons.

1. It possesses a region of increasing1. It possesses a region of increasingmarginal returns,marginal returns,

a region of diminishing marginal returnsa region of diminishing marginal returns

and a region of negative marginal returnsand a region of negative marginal returns

to the variable inputsto the variable inputs

x and x .x and x .11 22

dy = 1 + 2 x - 0.15 x + 0.4 x .dy = 1 + 2 x - 0.15 x + 0.4 x .

dy = 1 + 2 x - 0.15 x + 0.4 x .dy = 1 + 2 x - 0.15 x + 0.4 x .

dxdx

dxdx

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2. Since the function has a finite maximum,2. Since the function has a finite maximum,

there are "ring" isoquants, centered at thethere are "ring" isoquants, centered at the

maximum output level.maximum output level.

At the maximum output level,At the maximum output level,

dy and dy are both zero.dy and dy are both zero.

dxdx dxdx11 22

Maximum output is 237.21 unitsMaximum output is 237.21 units

corresponding tocorresponding to

x = x = 16.41 unitsx = x = 16.41 units11 22

3. There is a global point of profit3. There is a global point of profitmaximization.maximization.

This point occurs at an output levelThis point occurs at an output levelless than the global point ofless than the global point of

profit maximization.profit maximization.

For example, if the price of the product (y)For example, if the price of the product (y)is $1.00 per unit, the price of x is $3.00is $1.00 per unit, the price of x is $3.00per unit, and the price of x is $1.50 per unit,per unit, and the price of x is $1.50 per unit,

then profit is maximum at x =15.21 and x = 15.71then profit is maximum at x =15.21 and x = 15.71

units. Total Revenue is $234.85;units. Total Revenue is $234.85;Total Cost for x and x is $69.20;Total Cost for x and x is $69.20;

Profit is Total Revenue - Total Cost = $165.20Profit is Total Revenue - Total Cost = $165.20

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In the following sequence, the production surfaceIn the following sequence, the production surface

for the polynomial is sliced horizontally atfor the polynomial is sliced horizontally at

various levels.various levels.

The isoquant at each output level appears in red.The isoquant at each output level appears in red.

Note that isoquants at low output levelsNote that isoquants at low output levels

are concave to the origin, but as the outputare concave to the origin, but as the output

level increases, the isoquants becomelevel increases, the isoquants becomeconvex to the origin.convex to the origin.

You are looking at the 3-D surface from theYou are looking at the 3-D surface from the

origin. x is at your right; x at your left.origin. x is at your right; x at your left.

Output (y) is measured on theOutput (y) is measured on the vertical axis.vertical axis.11 22

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