Figure 1 Angela’s production function shifts up.

3.6.1 MODELLING TECHNOLOGICAL CHANGEFor a farmer like Angela, technological change happens when shemanages to harvest more grain from the same amount of work.Mathematically, we can model technological change as a changein the parameters of the production function.

In the text, we illustrated technological progress with Figure 3.12, which wereproduce below as Figure 1. Angela’s production function shifts up,because she is able to produce more grain per hour of work.

The production function we have been working with so far (in particular inLeibniz 3.1.2) is:

In this example, stands for Angela’s daily hours of work on her farm, andher daily grain output. and are two parameters that describe the

specific shape and position of her production function; we have assumedthat and . The first assumption simply means that worktranslates into creating, not destroying grain. The second assumption

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Figure 2 Two ways of shifting the production function.

imposes diminishing marginal product of labour, so the production func-tion has a concave shape.

Technological progress is being able to produce more with the samequantity of inputs: the production function will now match a higherquantity of grain to the same number of hours worked. Mathematically,there are two ways to achieve this with Angela’s production function.

If increases, then output also increase for any given level of hoursworked. Thus an increase in can be interpreted as technological progress.

If increases, then output decreases if and increases if . Ifwe interpret the unit of measurement of literally to be an hour, thenwould appear to be the normal case, so an increase in may also beinterpreted as technological progress.

The two different ways to model technological progress are sketched inthe two panels of Figure 2. In the left-hand panel, an increase in increases

by the same multiple at every value of . For example, if andincreases from to , then changes from to , a 100% increase inoutput for any given amount of work. In the right-hand panel, an increasein , with held constant, increases for any : if, for example, ,

and increases from to , then Angela’s daily production of grainrises from to units, a little more than 40%.

The two panels of Figure 2 show that increasing and increasing havesimilar but not identical effects. The parameter determines the curvatureof the function. When , as we have assumed, it is concave. But ifit is a straight line, and when it is convex. If is initially less than 1,and then increases a little, the curve becomes less concave. The economicmeaning of this is that if Angela increases her daily hours of work,diminishing marginal returns kick in less rapidly than they would havedone with the smaller value of .

Indeed, if continued to rise, becoming equal to and then greater than ,diminishing returns would cease to apply. This suggests that modellingtechnological progress as an increase in is problematic for a reason rathermore profound than the one stated above about what happens if :human society has been experiencing technological progress for centuriesnow, and yet we still face diminishing returns to our labour. Assuming thattechnological change can free us from concavity of the production functionis simply not very plausible.

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Modelling technical progress as a change in avoids this problem. Inthis case increases at any given and the production function remainsconcave: the property of diminishing marginal product still holds. This iswhy economists normally use and not to model technological change.

We can model technological change in the same way for any productionfunction. Suppose the production function is:

where is any increasing concave function taking positive values for all. Then an increase in the parameter implies that output increases by

the same proportion for every level of .The property of diminishing marginal product is preserved under tech-

nological change in this general case too. The marginal product of labour(MPL) is . Thus if hours of work change from to , theproportional change in the MPL is:

which does not depend on . Thus if increases, there is no effect on theproportional fall in the MPL caused by an increase in hours.

Read more: Sections 4.3 and 7.3 of Malcolm Pemberton and Nicholas Rau.2015. Mathematics for economists: An introductory textbook, 4th ed.Manchester: Manchester University Press.

3.6.1 MODELLING TECHNOLOGICAL CHANGE

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