Assume the production function can be written as: ( )1 dy Lan L hA ?

? ???= ?? ??

where h=human

capital, LAND=the amount of land in the economy, A=productivity, y=income per-capita, and L=population. This economy has no physical capital. Land is in fixed supply. Assume h and A each grow at an exogenous positive rate and the parameter ? in the production function is positive and less than zero.

A. From this equation and the mathematical rules for growth rates we have discussed in class, derive the equation for the growth rate of y in terms of parameters and variables that are growing exogenously.

B. Assume the population growth rate is endogenously given by: 01 L gy ?? = + , where ?0 and ?1 are positive parameters, based on the assumption that higher income per-capita induces a family to have more children. Prove that y eventually reaches a steady state (the level). Carefully explain using the equation(s) and/or graph(s) needed to prove your point. Do you need to make any additional assumptions about the parameters, or functions of the parameters, in the model to ensure a steady state for y obtains?

C. What is the steady state value for y in this model in terms of the parameters and the growth rates of productivity and of human capital? Do you need to make any additional assumptions about the parameters, or functions of the parameters, in the model to ensure a positive steady state for y obtains?

D. What is ss Lg (steady state population growth) in terms of the parameters and the growth rates of productivity and of human capital?

E. If ?1=0 the results in parts B and C are not valid anymore. Under this new assumption, does y reach a steady state or does the growth rate of y reach a non-zero steady state? Clearly show why your claim is true with an equation and/or a graph.

Lint] [hair where h=human capital, LAND=the amount of land in the economyy =productivity, y=income per-capita, and L=population. This economy has no physical capital. Land is in xed supply. Assume h and A each grow at an exogenous positive rate and the parameter E in the production function is positive and less than zero. 5. Assume the production function can be written as: y = [ From this equation and the mathematical rules for growth rates we have discussed in class,r derive the equation for the growth rate of y in terms of parameters and variables that are growing exogenously. Assume the population growth rate is endogenously given by: g1 = 3}, + y. where fin and B1 are positive parameters; based on the assumption that higher income pericapita induces a family to have more children. Prove that y eventually reaches a steady state [the level}. Carefuly explain using the equationts} andfor graphIIs] needed to prove your point. Do you need to ma ke any additional assumptions about the parameters, or functions of the parameters, in the model to ensure a steady state for y obtains? What is the steady state value for y in this model in terms of the para meters and the growth rates of productivity and of human capital? Do you need to make any additional assumptions about the parameters or functions of the parameters, in the model to ensure a positive steady state for y obtains? D. What is g, (steady state population growth) in terms of the parameters and the growth rates of productivity and of human capital? E. If B1=0 the results in parts B and C are not valid anymore. Under this new assumption, does y reach a steady state or does the growth rate of y reach a non-zero steady state? Clearly show why your claim is true with an equation and/or a graph