Solow Growth Model

#economics #macro

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

Solow Growth Model

constant & exogenous saving rate
level of output determined by the population growth rate, saving rate, and technological progress rate.
the economy cannot achieve a permanent growth by accumulating capital.
the long-term economic growth is determined by the technological progress rate.

Assumptions

Production Functions

: output, : capital, : labor, : effectiveness of labor
- : depreciation rate of Capital
- : population growth rate, ; .
- : technological progress rate, ; .
- : saving rate,
: effective labor
- : output per effective labor
- : capital per effective labor
not that the small letter will generally denote capital letter 'per unit of effective labor'.

CRS and Intensive Form

scaling up input times results in scaling up in output.
by letting , we have the intensive form
note that we have the following relationship:
diminishing marginal product: & .
no income, no output: .

Market Equilibrium

CRS condition implies the following equation:
total amount paid to the factors of production equals to total net output.

: Labor's Share, Labor Income
: Capital's Share, price of Capital
: Diminishing Capital.

MPK and Interest rate

The Marginal Product of Capital is, Assume that the capital market is perfectly competitive, and is on equilibrium. Thus the real interest rate is, where is capital depreciation rate.

MPL and Wage

The Marginal Product of Labor is, thus the wage per labor is, the wage per effective labor is,

Closed Economy

ASM:

: no export nor import.
: no government purchase nor tax.

National Income Identity: thus total investment equals to total savings. therefore,

Cobb-Douglas function

since Cobb-Douglas function is CRS:
real interest rate:
real wages:
market clearing:

Dynamics

Dynamics of

In the Closed Economy Closed Economy, from , we have therefore, thus we have

: actual investment, investment per unit of effective labor
: breaking even investment, the amount of investment required to keep in its current level.

Steady State

Pasted image 20240331132952.png

Denote as a Steady-State(SS) investment. i.e. where is stable SS.

if : then , increasing
if : then , decreasing

Properties of SS

at ,

.
.
; output per worker determined solely by technological growth.
.

where the second and third properties follows from:
as at SS, we have and also,

Cobb-Douglas function

let . then at SS, thus therefore at SS

or increases when decrease or increase.
or decreases when increases, since the unit of effective labor increases, i.e.

Impact of Saving rate

If , then

more savings, more capital: , since
more capital, more output: , since $f'(k_{t})>0$.

The impact on other variables:

eventually goes to .
sudden increase in and .

Golden Rule

Consumption per effective labor at SS: by differentiating with , where denotes Golden-rule SS.
therefore, Since , at Golden-rule SS, we have and the real interest rate satisfying Golden-rule is Pasted image 20240331160201.png

Dynamic Inefficiency

Since the saving rate is given exogenously, it is not guaranteed to satisfy Golden-rule

if : over-saving & under-consumption.
- by decreasing the saving rate, the consumption increases, and the total utility increases, in every between.
if : under-saving & over-consumption.
- increasing the saving rate into does not increases the utility in every between.
- if the consumer prefers the current to the future, then might decreases the utility.

Convergence

Speed of Convergence

Every country's GDP converges into its SS determined by . Thus, if the countries' parameters are the same, then those countries will be converge into the similar GDP.

Speed of Convergence

From Dynamics of $K$Dynamics of , .
around , by Taylor's Approximation Taylor's Approximation, we have

: capital's share, i.e. elasticity of output with respect to capital.
: speed of convergence

Half-Lime of Convergence

Since the complete convergence is impossible, we measure the half-life to the distance from S.S.

From , since is a constant, for example, let half-life converging time and the given capital .

put .
calculate , and calculate for .
then if , we have , approximately 18 years.

Growth Accounting

From , the total differentiation is thus which leads to

: elasticity of output with respect to capital.
: elasticity of output with respect to labor.
note that at each .
: Solow residual, TFP(Total Factor Productivity), contribution of technological improvement on the output.

Empirical Application: Baumol (1986)

Question: Do poor countries tend to grow faster than rich countries?
Regression Formula:
- : index of country .
- : log income per person
- : error term
Interpretation:
- : perfect convergence, Higher initial income tends to have lower growth rate.
- : no convergence
Estimations:
- intercept:
- coefficient: , with s.e. .

DeLog(1988) issues two problems in Baumol(1986)

limited sample selection: data from only 16 industrialized countries
measurement error: GDP data in 1870s are not accurate.