Question

Consider a version of the Solow model with a diminishing productive resource such as oil, described in section 7.2 of the text. In this case the production function is given by Yₜ =Kₜα(AₜLt )βEₜϵ where Eₜ is the amount of oil used in production in period t. Note that the stock of oil remaining in period t+1 is then given by Rₜ+₁ =Rt −Eₜ , where Et =sERt . We also assume that α+β+ϵ=1. a) Show that the growth in GDP per capita in this case is given by

gʸ= β/β+ϵ g- ϵ/β+ϵ n - ϵ/β+ϵ sₑ

Answer 1

To determine the growth rate of GDP per capita in the Solow model with a diminishing resource such as oil, the production function is analyzed, and the growth rate formula incorporates productivity growth, population growth rate, and oil usage rate, weighted by their respective elasticities.

Step-by-step explanation:

To address the student's question regarding the growth in GDP per capita using the revised Solow model with a diminishing productive resource like oil, we begin with the production function: Yt = Ktα(AtLt)βEtε where Yt is output or GDP at time t, Kt is the capital stock, At is the level of technology, Lt is labor, and Et is the amount of oil used.

The oil left, Rt+1, is given by Rt minus the oil used, where Et equals sERt, and sE is the fraction of remaining oil used. Given the assumption that α+β+ε=1, the growth rate of GDP per capita (gy) can be derived and is determined by the factors of productivity growth (g), population growth rate (n), and the oil usage rate (se). The formula for gy reflects the contributions of these factors proportionally weighted by their elasticities: gy = β/(β+ε)g - ε/(β+ε)n - ε/(β+ε)se.