# The Lewis Labour Surplus Model and Agricultural Productivity Growth

Today I revised for my Growth and Development Economics test next week. One of the topics covered is Theories of Economic Growth and Development. It includes the Basic Growth Model, the Harrod-Domar Growth Model, Solow Growth Model, Lewis Labour Surplus Model and the Neoclassical Two-Sector Model. Fortunately, the Harris-Todaro Model will not be tested in the midterm; only in the final exam. The main textbook for the class is Economics of Development by Perkins, Radelet, Lindauer and Block (2013) and the reading for this section is Chapter 4 and Chapter 16 in case you want to have a look! During revision of the Lewis Labour Surplus Model today I recognised that the textbook mentions population growth and a rise in agricultural productivity as two comparative statics exercises. However, Perkins et al. only show the impact of an increase in population in the Lewis Labour Surplus Model on page 596. So I thought that I’d like to think through the rise in agricultural productivity myself and attempted to replicate an appropriate model!

Let’s start with the agricultural sector. Its output can be derived from the Agricultural Production Function with the only two inputs agricultural labour and land, as well as a measure for agricultural productivity A. It is the usual Cobb-Douglas Production Function with diminishing returns to scale.

In particular, there is a perfectly elastic segment where an additional worker does not add any extra output (food) to total agricultural output. In this segment the marginal product (MP) of agricultural labour must therefore be equal to zero. Also note that we have a decreasing quantity of labour in the agricultural sector as we move from the left to the right and an increasing quantity of labour in the urban sector. This is because we have only these two sectors in the model which employ the total workforce of the economy. So let’s now introduce our rise in agricultural productivity. Mathematically A increases to A’, where A’ > A. Graphically this means that the Agricultural Production Function (panel a) shifts upward. This will increase (1) total agricultural output and (2) the length of the segment at which the marginal product of agricultural labour is equal to zero. In short: a rise in agricultural productivity increases the amount of food produced and reduces the amount of agricultural labour needed to produce the maximum capacity.

Let’s move on to the rural labour market (panel b). In the standard Lewis Labour Surplus Model there is the so-called subsistence wage or minimum wage which is equal to the average product (AP) of agricultural labour.  Hence rural wages are institutionally fixed and not determined in the market as long as there is a labour surplus in the rural sector. So what effect has the increase in agricultural productivity on the rural labour market?

There are two factors at work. Firstly, the average product of agricultural labour increases from AP to AP’. This is intuitive because, on average, the same amount of agricultural labour can now produce more. This drives up the subsistence wage and thereby reduces rural poverty. Secondly, the marginal product will increase because we need less agricultural labour for maximum food output. So each of the now fewer workers in the agricultural sector will increase output by more than before. This is also rather intuitive – despite the maths – because e.g. labour-saving technology such as large tractors makes each of the fewer workers more productive. In panel b this is depicted by the pivot and shift of the MP curve to MP’. MP’ is now flat up until g’ and is steeper than MP. Another important aspect here is the Lewis turning point, i.e. the point where rural wages equal the average product (subsistence wages) and marginal product of labour. To the left of this point rural wages are institutionally fixed; to the right of it wages are determined in the market and equal to the marginal product of agricultural labour. The agricultural productivity increase postpones the Lewis turning point at a higher overall subsistence wage.

One key observation here is that the agricultural productivity increase drives up the amount of surplus labour from g to g’ in panel b, too. This means that a country can shift more labour to the urban sector which unambiguously increases total GDP (due to longer constant agricultural output and increasing industrial output). Furthermore this structural change happens in an environment of less rural poverty which is desirable from a policy makers point of view.

The third panel shows the urban labour market. The labour supply curve is actually taken from the labour supply curve facing the industrial sector in panel b (Perkins et al., 2013). This is just the supply curve of panel b shifted upwards by an amount equal to the rural subsistence wage plus a premium for the costs of migration to the urban sector. Hence the supply curve in the urban sector is congruent to the supply curve in the rural sector. This is a necessary condition for the model to work because otherwise people would have an incentive to migrate. In particular, in the segment where rural surplus labour exists, subsistence wages (AP of agricultural labour) plus the costs of migration will equal urban wages and in the remaining part, the MP of agricultural labour will equal urban wages.

What is the effect of the agricultural productivity increase on the urban labour market? It will actually increase urban wages by the same amount as rural subsistence wages increase. This is quite intuitive because the increase in rural wages must be transmitted into the urban labour market to keep urban labour from migrating to the rural sector. Because the supply curves of panel b and c are congruent the marginal product of urban labour will shift and pivot in the same fashion as in the rural sector once rural labour surplus is exhausted.

In conclusion, the agricultural productivity increase drives up both rural and urban incomes. In addition, there is more surplus labour available to industrial employers (i.e. longer unambiguous GDP growth until g’) but once the pool is exhausted employers face a more inelastic urban labour supply curve and must increase wages by a greater amount to hire the same amount of additional workers compared to before. This is essentially caused by the steeper marginal productivity of agricultural labour. This faster wager growth will make it more costly for employers to hire workers. On the other hand firms might be able to offset these costs by increasing industrial productivity. Also, higher wages will mean that more income available to spend. As rural and urban workers become more wealthy, the proportion of income spent on food will fall (Engel’s law) and more income will be left over for other things. One might then argue that this could increase the demand for goods in the industrial sector. This could offset the increase in wages. However, more plausible to me is that industrial employers will seek to introduce productivity-enhancing and labour-saving technology to offset the wage increases. More on wages and productivity can be found in Gregor Mankiw’s blog post on How are wages and productivity related? (2006) where he explains why real wages and productivity should line up in theory but might actually not line up in the data.

I hope you enjoyed today’s post. Feel free to comment on my analysis and thanks for reading!

Jasse

Mankiw, G. (2006). How are wages and productivity related? [online] Available at: http://gregmankiw.blogspot.co.nz/2006/08/how-are-wages-and-productivity-related.html [Accessed 10/04/2016].

Perkins, D.H., Radelet, S., Lindauer, D.L., and Block, S.A. (2013). Economics of Development, Seventh Edition. New York, NY: W.W. Norton & Company.