# Solow Growth Model

#### Background

The Solow growth model is a basic account of the link between physical capital accumulation, exogenous technological progress, economic growth. The four basic components of the model are: $$Y_t = A K_t^{\alpha} (E_tL_t)^{1-\alpha}$$ $$C_t = (1-s) Y_t$$ $$Y_t = C_t + I_t$$ $$K_{t+1} = I_t + (1-\delta)K_t,$$ where $$Y_t$$ denotes real GDP, $$K_t$$ is the stock of physical capital, $$E_t$$ is labor efficiency, $$C_t$$ is consumption, and $$I_t$$ is investment in new physical capital. The Solow model is frequently formulated using continuous time methods, but here $$t$$ is discrete and increments annually.

The first equation of the model is a standard Codd-Douglas production function and so the parameter $$A$$ denotes total factor productivity and $$\alpha$$ the capital share of national income. The second equation is a consumption function with $$s$$ representing the saving rate. The third equation is the national income accounting identity for a closed economy without government purchases. The final equation is law of motion for the stock of physical capital where $$\delta$$ is the share of the capital stock that falls apart or becomes obsolete each period.

The model has two exogenous variables — population $$L_t$$ and labor efficiency $$E_t$$ — that grow over time at rates $$n$$ and $$g$$. Because of these exogenous growth sources, it is routine to recast the model with all variabels divided by $$E_t\cdot L_t$$ and converted into per effective worker units. The recast model is written as: $$y_t = A k_t^{\alpha}$$ $$c_t = (1-s) y_t$$ $$y_t = c_t + i_t$$ $$k_{t+1} = i_t + (1-\delta-n-g)k_t,$$ where $$y_t = Y_t/E_tL_t$$, etc.

#### Instructions

Use this tool to construct transition paths implied by the Solow growth model. You can use the tool to simulate how the model economy transitions from one steady state to another. You can also use the tool to simulate how the model economy approaches the steady state from an initial stock of capital per effective worker that is not a steady state.

1. Transition between steady states. Set the number of periods to simulate and leave the Start in the steady state box checked. Set all values in the Initial parameter values section. Then make sure that all values in the New parameter values section match those in the initial values section except for the parameter that you want to change. Check the Show original path in red box if you want to also see the original path with no paramter change. Click the Submit button. The parameter change will take place in the fifth period.

2. Transition to the steady state. Set the number of periods to simulate. Uncheck the Start in the steady state box and enter an initial value of capital per effective worker. Set all values in the Initial parameter values section. Then make sure that all values in the New parameter values section match those in the initial values section. Click the Submit button.

Results are displayed in the eight panels at the bottom of the page. Use the dropdown menu in the upper-right corner of each plot to download the image in png format. Click the "Download csv" button to download the data in both figures in a single csv file.

Simulation Initialization

Show original path in red
Initial parameter values
New parameter values