Solow Growth Model - Simplified Version

Introduction to the Solow Growth Model

• The Solow growth model is a crucial topic in macroeconomics.
• This video series will cover the Solow growth model, including variations with technological and population growth.
• Future videos will include numerical examples, graphs, and MATLAB simulations.

Simplified Solow Model

• The series starts with a simplified version of the Solow growth model.
• The goal is to develop a strong understanding of the basic model before adding more complex elements.
• The impact of adding elements will be examined.

Key Themes and Questions

• A key theme is endogenous capital accumulation.
• Questions to be addressed:
  • Is capital accumulation sufficient for economic growth?
  • If so, to what extent?
  • If not, what is needed to promote economic growth?

Model Equations

The main equations of the Solow model will be introduced, detailing their real-world relevance and discussing a solution method.

1. Production Function

Yt = A Kt^\alpha L^(1-\alpha)

• Y_t is GDP at time t (endogenous).
• Time (t) is discrete (0, 1, 2, 3, …).
• Variables without a time subscript are exogenous.
• A represents total factor productivity (technology).
• K_t is capital (endogenous).
• L is labor supply (exogenous in this version).
• \alpha is a parameter.

2. Capital Accumulation Equation

K{t+1} = It + (1 - \delta)K_t

• K_{t+1} is capital tomorrow (time t+1).
• I_t is investment today (time t).
• \delta is the depreciation rate.
• (1 - \delta) is the fraction of capital that doesn't depreciate.

3. Resource Constraint

Ct + It = Y_t

• C_t is consumption.
• I_t is investment.
• Y_t is output (GDP).

4. Resource Allocation Equation

It = sYt

• Investment (I) is a constant fraction (s) of output (Y).
• s is the savings rate.

Endogenous and Exogenous Variables

Endogenous Variables:

• Output (Y_t)
• Capital (K_t)
• Investment (I_t)
• Consumption (C_t)

Exogenous Variables:

• Total factor productivity (A)
• Parameter ($\alpha$)
• Labor supply (L)
• Depreciation rate ($\delta$)
• Savings rate (s)
• Initial capital in period 0 (K_0)

Per Capita Terms

It is useful to express the Solow model in per capita terms for simplification.

• GDP per capita: yt = \frac{Yt}{L}
• Capital per capita: kt = \frac{Kt}{L}
• Consumption per capita: ct = \frac{Ct}{L}
• Investment per capita: it = \frac{It}{L}

Rewriting Equations in Per Capita Terms

1. Production Function

• Original: Yt = AKt^\alpha L^(1-\alpha)
• Divide by L: \frac{Yt}{L} = A(\frac{Kt}{L})^\alpha
• Per capita: yt = Akt^\alpha

2. Capital Accumulation Equation

• Original: K{t+1} = It + (1 - \delta)K_t
• Divide by L: \frac{K{t+1}}{L} = \frac{It}{L} + (1 - \delta)\frac{K_t}{L}
• Per capita: k{t+1} = it + (1 - \delta)k_t

3. Resource Constraint

• Original: Ct + It = Y_t
• Divide by L: \frac{Ct}{L} + \frac{It}{L} = \frac{Y_t}{L}
• Per capita: ct + it = y_t

4. Resource Allocation

• Original: It = sYt
• Divide by L: \frac{It}{L} = s\frac{Yt}{L}
• Per capita: it = syt

Solution Method

The first three equations can be simplified into one equation.

• Substitute It = sYt and yt = Akt^\alpha
• Resulting equation: k{t+1} = sAkt^\alpha + (1 - \delta)k_t
• This equation depends only on capital per capita (k).
• Knowing k_0 allows you to find k for all future periods.
• From k, you can recover y, i, and c.

Solving for Capital per Capita

• Given k_0, all other variables can be determined.
• k0 = \frac{K0}{L}
• k1 = sAk0^\alpha + (1 - \delta)k_0
• k2 can be found similarly by replacing k0 with k_1, and so on.

Steady State

Closed-form solution for constant values endogenous variables eventually converge to.

MATLAB Illustration

An example is provided using MATLAB to illustrate how to solve for the sequence of endogenous variables given parameter values.

Steps:

1. Plug in values for exogenous variables.
2. Define capital in the initial period (period 0).
3. Recover per capita variables in period 0.
4. Use a for loop to iteratively solve for the sequence of per capita endogenous variables using the capital accumulation equation:
   • k{j} = s * A * k{j-1}^\alpha + (1 - \delta) * k_{j-1}
5. Find GDP per capita, investment per capita, and consumption per capita.
6. Plot the results.

Observations from the Plot:

• Initial growth in endogenous variables.
• Variables eventually level out.
• Growth occurs for a period, but not forever.

Reasons for Limited Growth

The reasons for the limited growth in the Solow model need to be examined.

• Think about the equations and their features.
• Hint: Consider the production function and the depreciation of capital.

In future videos, additional elements will be added to the model to achieve permanent growth.

Non-Per Capita Variables

Non-per capita variables (K, GDP, C, I) can be recovered by multiplying the per capita variables by L.

Next Steps

• The next video will discuss why the variables eventually flatten out and why there is initial growth.
• A closed-form solution for the steady state will be found.