Question

The Romer model can be described by 4 equations:
Y_t = A_tL_yt

∆A_t+1 = zA_tL_at

L_yt + L_at = L

L_at = IL

Y is final output, A is ideas/knowledge, L_y is employment in production of final output, L_a is the number of researchers and L is the population.

a) What is the driver of long-run growth in this model?

b) Using the equations in the question, show that the growth rate of output per capita can be written as g = zlL. Explain the intuition for this result.

c) During a business trip abroad, you notice that researchers in the foreign country are much more productive than in your home country. You advise your government how to raise the productivity of research at home. How would you model the effect of this using the Romer model? Show what happens using a chart of GDP per capita over time .

Answer 1

a. The growth model postulated by David Romer, considers Idea and Knowledge as determinant factor in the production and growth process. Romer considered that the returns to scale, i.e. the marginal return from the employed factors for production process in long run, to be increasing in case of Ideas and knowledge. This is the main reason that the Romer model shows a long run growth trend. The knowledge factor or A_t leads toward the constant growth path of the model.

Ideas and knowledge in the model works through laborers, and can be segregated as L_yt or laborers producing output, and L_at laborers producing new ideas_. The equation 3,

L_yt + L_at = L (constant)

exhibits the said division of laborers. This equation acts as the resources constraint for the model.

Equation 1,

Y_t = A_tL_yt,

shows that the output produced using existing knowledge A_t and Laborers contribution L_yt is equal to Y_t, total output.

Equation 2,

∆A_t+1 = zA_tL_at

shows the production of ideas depends on the existing knowledge A_t and the new ideasproduced L_at. A_t is non-rival in use thus features both in output equation and ideas equation, exhibiting increasing returns to scale. Z on the equation represents the constant parameter for producing ideas by laborers.

Finally the forth equation,

L_at = IL

shows the allocation of laborers in the model. In L_yt + L_at = L, say we introduce l representing proportion of workers producing ideas. Thus L_at or the laborers producing ideas is equal to l*L (constant).

b. Now calculating for the per head contribution of knowledge or output per capita, we can use the existing equation 1,

Y_t = A_tL_yt.

Where,

Let, y= Y_t/L

Implies, Y_t/L = A_tL_yt/L =   A_t (1-l);

(1-l) is the proportion of labourers producing output that depends on total stock of knowledge. Hence to trace the growth rate of knowledge would provide us with the growth rate of the model,

(Change in A_t+1)/A_t= zA_tL_at

Or, zA_tL_at = zlL; since L_at = lL

Thus, (Change in A_t+1)/A_{t =} zlL

From the equations and discussion above it is seen that z is the parameter of productivity of laborers producing ideas, l is the proportion of labourers producing ideas and L is the total supply of laborers, that are all constants. Thus the growth rate would depend upon the said factors and,

A_t= A₀ (1+g)^t, and, g= zlL; where, A₀ - Initial knowledge stock

And g is the rate of growth of the knowledge and ideas. Thus the output per capita grows at the rate of growth of knowledge and ideas given by zlL.

c. In the Romer model and given equations, z is the parameter that affects the productivity of laborers in producing ideas. Following the growth rate of knowledge or g= zlL equation, to increase the productivity of researchers at home, the z parameter needs to be increased. Once the productivity parameter z is increased it will lead to the increasing of the productivity of the researchers.

However, if we need to increase growth per capita yet not capable of increasing productivity (z) or the total supply of labor (L), increasing the proportion of the laborer working for generating ideas, l, will also lead to the same result of increasing growth, however not through productivity increase.

In the diagram below, an imaginary GDP from 1990 to 2005 for each five years have been plotted.

Now if we consider the growth path or GDP per capita along the time line we get a straight line from the growth equation of knowledge, g= zlL. An increase in z or l or L will increase the growth of knowledge or g.

Now, in case we increase the z parameter as said in the discussion above, the growth path will become steeper. As in case of the example, in 2000 a policy change have increase z and thus resulting in a steeper growth path; registering a GDP of around 5000 in 2005, as compared to the GDP pre-productivity-change that was 3000 for the 2005.