Key Implications Of The Solow Model

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The Solow model is widely considered to be the standard neoclassical economic growth model which serves as the basis for understanding economic growth. I will first introduce the two basic equations that the Solow model is built around, discussing the main assumptions made along the way. I will then present the key equation of the Solow model and discuss its results and implications. I will then address why it is desirable to use log-linearization, and how it can be used to study the dynamics near the steady-state. We will then briefly look into the empirical results of Mankiw et al. (1992) and what they mean for the Solow model. We begin with the aggregate production function of the model. Let us denote output (Y), capital (K), labour (L) and the index of productive efficiency (A). The production function can then be written as:
Y=F(K,AL)
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To the contrary, if a is high then the rate of convergence is low because diminishing returns set in slowly.
The differential equation for output convergence is quite similar. Assuming Cobb-Douglas technology we can show that y=k^a which then implies that y ̇/y=a k ̇/k by taking logs and differentiating with respect to time. Here a is translating a given growth rate of capital into a given growth rate of output. Substituting into the previous differential equation we get a new one for the growth rate of output:

y ̇/y≈-λ[logy-logy^*]

The growth rate in each of these expressions is linear in the gap from the steady-state measured in logarithms.

Solving both differential equations and re-arranging we get: logy(t) =(e^(-λt) )logy(0)+(1-e^(-λt) )logy^* (5) k(t)- k^*=(e^(-λt) )[k(0)-k^*]
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