In class, we derived several benchmark theories of economic growth.  We did not have time to cover the celebrated “Lewis Model,”sometimes called the “dual sector model.”Arthur Lewis was a Nobel-Prize winning Saint Lucian economist, wrote probably the first growth model that was specifically interested in development and structural transformation,  and Lewis himself helped start the field of development economics. This question is based on an old midterm.

In this question, we’re going to discuss the isolated economy of Lewislandia

The island has two main sectors: agriculture and manufacturing.

In agriculture, the production function is

Y = Aa Tα (La )1 −α                            (1)

where where Aa  is TFP in agriculture, T is land (Terrain), and La  is labor in agriculture. The quantity of land is fixed, because we are assuming that there is no investment or savings for land.

In manufacturing, the production function is

Y = Am Kα (Lm )1 −α                         (2)

where where A is TFP, K is capital, and Lm  is labor in agriculture.  The quantity of capital is fixed, because we are assuming that investment or savings for capital.

1.  The assumption that capital is fixed and unchanging is just to make this problem easy to solve, it’s not correct. In a few sentences, what do you think about the assumption that land is fixed?

2.   In manufacturing, workers get paid their marginal product.  Derive the wage in manufacturing as a function of A,K,Lm , and α

3.   In agriculture, workers get paid their average  product (this is the Lewis twist).  Derive the wage in agriculture as a function of As,T,La , and α

4.  Workers getting paid their marginal product is pretty standard.  But just to make sure we are on the same page, in a sentence or two, why does it make sense?

5.  Workers getting paid their average product is more unusual. But Lewis wasn’t writing down this model for fun or for the love of math, he thought that it captured something true. So in a few sentences, why do you think that workers may get paid their average product?1

6.  In equilibrium, wages are equal in both sectors. Give a short intuitive explanation as to why.

First, let’s assume that T = K and Am  = Aa

7.  Given that the wages are equal in both sectors, what is the relative share of employment in agriculture? That is, solve for La /Lm  as a function of As,T & K, and α . As a hint: not all of these are going to show up in the solution

Now, continue to assume that T = K but don’t assume that Am  = Aa

8.  Given that the wages are equal in both sectors, what is the relative share of employment in agriculture? That is, solve for La /Lm  as a function of As,T & K, and α . As before, not all of these are going to show up in the solution

Now comes the R part.  As always, all graphs should be well-labeled, and if I were to print out your problem set on my cheap black & white laser printer, the graphs should be easy to read.

9.  Make a graph where Aa /Am  is the x-axis and La /Lm  is the y-axis. That is to say, plot how the share of labor in agriculture evolves as manufacturing productivity increases.  This is not technically difficult to do in R, but you may have to spend some time thinking about the right way to set it up.

I very purposefully did not give you numbers to pick for As,Ls,T,K, and α . You should pick numbers for which your graph looks good and conveys the right pattern. You should stick to these values for the rest of the assignment.

You should try to think through how you would do thi before you look at how I would go about it below

First, you want to make a define a data.frame with one column, for Am  that has a bunch of values. seq() is probbaly useful. You get to pick the starting and ending values

Now add a column for Aa  that’s just equal to one.We did something similar on the last problem set if you don’t remember how to do it. We can do this step wince Aa /Am  is what matters, so Aa  can be constant. Since Am  changes in each row so does the ratio. Do try to define values so that Am  is sometimes above Aa and sometimes below.

Now make additional columns for T, K, and L

Now you are cooking with gas:  For each row you have all of the values you need for the equation you solved for in (8). So you can solve for La /Lm , and then make a graph.

10.  Now make a different graph, which is similar except the y-axis is GDP (the sum of agricultural and manufacturing output) and the x-axis is still Aa /Am .

You should try to think through how you would do this before you look at how I would go about it below

To get total GDP, you first need to solve for La  and Lm .  This isn’t so bad: if you know the ratio and you’ve already defined a value of the total labor force, you can get each component. make columns for total labor in each sector.

Now you can plug into the production functions (equations 1 and 2 above) to get a column for output in each sector, add them to get a column for total GDP

11.  The government is considering a policy that introduces a minimum wage in manufacturing, which is above the prevailing wage when Aa  = Am  (how much? to pick something that makes the graph look good). On the same graph, plot GDP as a function of Aa /Am  with and without the minimum wage. Explain in a few sentences what is going on.

You should try to think through how you would do this before you look at how I would go about it below. This question is pretty hard, though once you figure out the steps the coding isn’t too bad

For any given Aa  and Am , you know what the non-policy wage in manufacturing would be.

And, with pencil and paper, you can figure out: if I tell you the wage in manufacturing, what is Lm ?.