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, which is given exogenous.

Assumptions

Production Functions

  • : output, : capital, : labor, : effectiveness of labor
    • : depreciation rate of Capital, constant.
    • : population growth rate, ; .
    • : technological progress rate, ; .
    • : saving rate, , constant.
  • : 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: & .
    • Inada condition: and .
  • 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 of Capital Accumulation

Dynamics of

In the 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.

By dividing the both sides by , the growth rate of capital is determined by where

  • : average production of capital; decreasing as increases.

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

Pasted image 20250331004737.png
Pasted image 20250331004819.png

Properties of SS

at (constant capital per effective labor),

  • ; constant output per effective labor
  • ; total capital and output grows as technological and population growth.
  • ; output per worker determined solely by technological growth if technology advances constantly, permanent economic growth is achieved.
  • .

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.

Impact of Saving rate

If , then

  • more savings, more capital: , since
  • more capital, more output: , since .
    Pasted image 20240331160938.png

The impact on other variables:

  • eventually goes to .
  • sudden increase in and .
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While the increase in saving rate can increase the growth rate temporary, and thus higher level of steady state capital, the growth rate eventually converges to zero. This is because as the incremental in capital decreases the productivity of capital, leading to diminishing growth rate.

Golden Rule

Golden Rule: optimal condition of capital accumulation (saving rate).

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 , .
around , by 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:
    • Pasted image 20240331165531.png
    • intercept:
    • coefficient: , with s.e. .

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

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

CES Production Function

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Table Of Contents
Solow Growth Model
Assumptions
Production Functions
CRS and Intensive Form
Market Equilibrium
MPK and Interest rate
MPL and Wage
Closed Economy
Cobb-Douglas function
Dynamics of Capital Accumulation
Dynamics of
Steady State
Properties of SS
Cobb-Douglas function
Impact of Saving rate
Golden Rule
Dynamic Inefficiency
Convergence
Speed of Convergence
Speed of Convergence
Half-Lime of Convergence
Growth Accounting
Empirical Application: Baumol (1986)
CES Production Function