Consider the Solow model without technological progress and an economy with the following production function, Y=A[Kα+Gα]1/α where α<1, K is private capital and G is public capital that is used freely and provided by the government. The level of technology A is fixed and assumed to be equal to 1. A. Does this production function feature constant returns to scale? Explain. In order to finance public capital, the government taxes all investment on private capital at the rate 0<τ<1. So, the revenue raised by the government in each period is sKYt(1−τ) where sK is the private savings rate so that sKYt is pre-tax private savings. Public investment towards public capital is a constant fraction sG of total revenue. Then, the accumulation equations for private and public capital, respectively, are, Kt+1−Kt=sKYt(1−τ)−δKt Gt+1−Gt=sG(sKYtτ)−δGt where δ is the common depreciation rate B. Consider a balanced growth path where
Consider the Solow model without technological progress and an economy with the following production function,
Y=A[Kα+Gα]1/α
where α<1, K is private capital and G is public capital that is used freely and provided by the government. The level of technology A is fixed and assumed to be equal to 1.
A. Does this production function feature constant returns to scale? Explain.
In order to finance public capital, the government taxes all investment on private capital at the rate 0<τ<1. So, the revenue raised by the government in each period is sKYt(1−τ) where sK is the private savings rate so that sKYt is pre-tax private savings. Public investment towards public capital is a constant fraction sG of total revenue. Then, the accumulation equations for private and public capital, respectively, are,
Kt+1−Kt=sKYt(1−τ)−δKt
Gt+1−Gt=sG(sKYtτ)−δGt
where δ is the common
B. Consider a balanced growth path where the growth rates of private capital is equal to the growth rate of public capital. What is the ratio of private to public capital along a balanced growth path?
C. Following part (B), along a balanced growth path, express the level of output Yt in terms of private capital Kt.
D. What is the growth rate of output in this economy? How does the growth rate of output depend on parameters such as sK,sG and τ? In what sense is this an endogenous growth model? Explain.
All government revenue that is unused for public capital investment is transferred back to households and consumed together along with any output that is not saved and invested in private capital. So, the level of consumption Ct is,
Ct=(1−sK)Yt+sKYt(1−τ)(1−sG)
E. What is the growth rate of consumption in a balanced growth path? Assuming that sG=1,sK=0.10, and α=0.5, what is the level of
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