Endogenous Growth Model

#economics #macro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Kang, Minwook.

Romer's Endogenous Growth Model

Unlike Solow or RCK, is treated as endogenous
: effectiveness of labor, knowledge, or technology.
Knowledge or technological progress is from R&D

Nature of Knowledge

Non-rival: multiple usages of knowledge is possible
Exclusive: one can restrict the use of other people's using
Increasing Returns of Scale: more input returns more technological progress, while requiring high initial cost.
Imperfect Competition: there exists some monopoly in developing the knowledge.

Assumptions

Model setting:

Improvements in technology is determined by the combination of labor, capital and technology.
R&D and goods production function is general Cobb-Douglas function, where the sum of exponents or not necessarily 1.
saving rates(), R&D fraction() are exogenous.

Goods Production Function

: capital stock used in R&D
: labor used in good-producing
, : exogenous population growth rate

R&D Production Function

: fraction of capital stock used in R&D
: fraction of labor used in R&D
: shift parameter.
, : where is not always (not assuming CRS function)
: effect of the existing stock of knowledge on new R&D
- : past discoveries make the future discoveries easier.
- : greater stock of knowledge, harder to make new discoveries.

Dynamics of

: zero capital depreciation rate
: exogenous saving rate

Model Without Capital

Simplified Model:

where and is assumed.

Dynamics of

From the R&D production function, therefore, thus we divide the case into , , and .

Case 1:

From

If at , we have thus the economy is on a balanced growth path with .
Pasted image 20240401211241.png

Note that the change in does not affect S.S. of , while affecting temporally through
inlL
inlL

Case 2:

From this implies that is always positive for all possible . thus the economy shows ever-increasing growth.
Now the increase in has immediate effect on , and the more rapidly rises through .
Pasted image 20240403131004.png

Case 3:

From the growth of knowledge is proportional to the knowledge stock.

if , then , thus we have similar result to Case 1 $ theta<1$Case 1: .
if , then is growing over time, thus we have similar result to Case 2 $ theta>1$Case 2: .

General Model

Remark:

Dynamics of and

Dynamics of

Combining the equation of and dynamic equation of , thus the growth rate of is taking logs of both sides and differentiating with respect to time, we have since is always positive,

if , then is rising
if , then is falling

Dynamics of

Similarly, from we have thus the growth rate of is

if , then rising.
if , then falling.

Case 1:

Since at the point , we have thus we have Pasted image 20240403153619.png

Case 2: and

Since while and , at the point , we have thus we have therefore, regardless of the starting point, the dynamics of and is always 45 degree line.

Pasted image 20240403153632.png