solow
introduction
The Solow Model is an economic model of production that incorporates the incorporates the idea of capital accumulation. Based on the Cobb-Douglas production function, the Solow Model describes production as follows: \[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:
- \(\bar{A}\): total factor productivity (TFP)
- \(\alpha\): capital's share of output—usually \(1/3\) based on empirical data
In this simple model, the following statements describe the economy:
- Output is either saved or consumed; in other words, savings equals investment
- Capital accumulates according to investment \(I_t\) and depreciation \(\bar{d}\), beginning with \(K_0\)
- Labor \(L_t\) is time-independent
- A savings rate \(\bar{s}\) describes the invested portion of total output
Including the production function, these four ideas encapsulate the Solow Model:
- \(C_t + I_t = Y_t\)
- \(\Delta K_{t+1} = I_t - \bar{d} K_t\)
- \(L_t = \bar{L}\)
- \(I_t = \bar{s} Y_t\)
solving the model
Visualizing the model, namely output as a function of capital, provides helpful intuition before solving it.
Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that \(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L_t}^{\frac{2}{3}}\). Utilizing this simplification and its graphical representation below, output is clearly characterized by the cube root of capital:
conclusions
hello conclusions