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.
∆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
this is still under construction.... I could do with some feedback, even guidance
DJS 20110201
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, OBRDJS 20100928
Decimalisation was 19710215,
What would happen to the alternative argument if the log base was itself c?