This is written from the point of view of a mathematician. I make very little claim to an understanding of economics, having not followed well the course at AS way back, before decimalisation in Britain. Subsequent reading showed that the number of mathematicians also competent in economics was low but significant. The teaching of FM allowed me to encourage students into exploring that overlap of fields.
I have a particular favourite book on Calculus, unsurprisingly titled Calculus. I have only ever seen two copies; mine and another owned by Chris Compton at PMC. Mine went missing on leaving Xi’an and no amount of hunting, moaning and searching has brought it to light. I recommend it wholeheartedly Ron Larson? RA Adams? I looked at all 101 pages of Amazon’s listings and didn’t find it...
Growth theory — an economy grows at a constant rate: the classical model (c/o Roy Harrod and Evsey Dumar) assumes that the national saving rate (meaning that fraction of income saved) must equal the capital output ratio multiplied by the rate of growth of the labour force (meaning the effective labour force, not the population). Corollaries include that the plant/equipment stock would be in balance with labour supply, so steady growth would be unaffected by slippage in the balance between labour surplus or shortage (called unemployment) or by over or under supply of equipment (resources for industry).
Y = Output = Income; K = capital stock, S= total Savings, s the applicable rate, I = investment.
The Harrod-Domar model assumes that:
Y = cK, i.e. that income has a linear relationship to capital.
c is called the marginal product of capital.
sY = S = I This is a fundamental assumption of the model,
that savings = investment = the rate of output.
∆K = I - ∂K i.e. that change in stock equals Investment less stock depreciation.
Since Y=cK, so ∆Y = c∆K = c(I - ∂K) = csY -c∂K = csY - c∂ (Y/c) => ∆Y/Y = cs - ∂
The texts I found defined Y=f(K), and set f(0) = 0 [that Income is a function of capital and that there must be some]. Then they argue that dY/dK = constant, c But this implies directly that Y=cK + €, because f(0) = 0, so €=0 and c = Y/K. It makes a nonsense of the first assumptive declaration [bothering to create a general f(K)] unless this f(K) is to undergo future revision]. The feature dY/dK = Y/K exhibits something called constant returns to scale. The linear relationship says that the elasticity of output is unity. Or so economists say: I think they make some very simple modelling sound — and read — as difficult.
An alternative argument says Y=cK => lgY = lg c + lg K = lg K “because c is constant”: you should try to explain this. Once satisfied, lg Y = lg K =>dY/Y = dK/K => dY/dK = Y/K. Like we need to use logs?
Also ∆K/K = I/K - ∂ = sY/K - ∂ => ∆Y/Y = sc-∂ Like I said, they’re just making it look hard.
Opinion: the model started off being a reasonable descriptor of observed economics, Taking the differential and calling it linear is probably a simplification to make the maths accessible, but that reduces the whole thing to linear relationships (at worst, by summation, a quadratic), which keeps the whole thing within GCSE maths and requires no calculus, no logs — and their use is merely froth — unless the model is going to be revisited with the assumptions changed.
∆Y/Y is the output growth rate: sc-∂ is the savings rate times the marginal product of capital minus the depreciation rate. To achieve growth under this model we must increase s or c reduce ∂.
The result shown does not explain the initial comment that the national saving rate must equal the capital output ratio multiplied by the rate of growth of the labour force.
s= ( ∆Y/Y +∂) / c
Maths issue here over modelling. This is pretty well defined under MEI and can be found in Edexcel in the Stats work. In principle:
Make some assumptions relating to an identified topic for modelling.
Develop a mathematical model based on those assumptions
Develop the analysis to produce some results
Go compare those results against reality
Review your assumptions [= Refine the model]
In practice, there is a phase when the assumptions are made pretty fiercely restrictive so that any analysis can take place within the scope of the modeller. For exam purposes (back when there was A-level coursework) one used to force students to make at least one cycle of adjusting assumptions, which I will have written about about in Mechanics here.
this is still under construction.... I could do with some feedback, even guidance
DJS 20110201
No feedback or help at 20130525, when I find myself looking at this yet again
See also FM revision 2, which poses a number of questions that undergraduates who may not have done so much maths at school might dare to tackle. I expect answers to embrace the wider issues not covered in Maths A-levels.
DJS 20100928
Decimalisation was 19710215; I remember it well.
What would happen to the alternative argument if the log base was itself c?
I found, today, output from the office of Budget Responsibility, the OBR. This document includes charts 2.10 and 2.11, which do not copy. 2.11 is the derivative, in the maths sense, of 2.10, which makes it of interest to put in front of sixth formers, indeed anyone struggling with imagining applications of calculus. To copy similar tables, go to the OBR policy measures database and look for excel files.
Similarly, in trying to research about air transport demand, this document indicates that its tables are to be found online. I had no success finding them. However, economic students will be keen to explore the use of elasticity of demand, including some actual values, found around page 18.
Down at around page 130 there are details of the models used. a grasp of S2 Statistics would help, along with A-level Economics.
On the same day, I found some modelling as used by the DfT (UK transport ministry). See here for links.
DJS 20161120
I find it difficult to reconstruct the formtting that should apply to what follows. I leave it here until I can decipher it, 20171113.
112
108
104
100
96
March 2011 March 2012 March 2013 March 2014 March 2015 March 2016 Latest
June 2010
92
Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1
2008 2009 2010 2011 2012 2013 2014 2015 2016
Source: ONS, OBR
