# Section 6.pdf - Economics 100B Section 6 September 2020 1...

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Economics 100B, Section 6 September 2020 1 Review Last section, we introduced the Solow-Swan model. This model builds upon the Cobb-Douglas production function, Y = AK α L 1 - α , and introduces a set of additional conditions: We assume that the economy produces a single good (output). All output goes towards either consumption or investment: Y = C + I . In other words, G = 0 and the economy is closed ( X = M = 0 ). The amount of consumption and investment both depend on the saving rate, s Capital stock each period increases due to investment and decreases due to depreciation (i.e. machines and equipment wearing-off) which occurs at constant rate δ 0 . So we can write capital accumulation equation: dK dt = I - δ 0 × K = s × Y - δ 0 × K = sAK α L 1 - α - δ 0 K We will sometimes substitute total factor productivity A with labor efficiency E , where E ( t ) = A ( t ) 1 1 - α We defined normalized capital, κ , which is a function of capital, labor, and efficiency. κ = K EL Using this information, we came up with our per-capita capital accumulation equation: dt = α - ( g E + g L + δ ) κ 1
2 Steady State and Long-Run Growth In the Solow-Swan model, we have what’s called a "steady state equilibrium when dt = 0 : dt = 0 κ = constant We can solve for the value of κ at equilibrium using our capital accumulation equation: dt = 0 α - ( g E + g L + δ ) κ = 0 α = ( g E + g L + δ ) κ κ 1 - α = s g E + g L + δ κ = ( s g E + g L + δ ) 1 1 - α Thus, κ ( t ) is constant at equilibrium and is always equal to κ * . 2.1 Per-Capita Ratios Now, we want to think about the implications of our steady-state equilibrium. First, let us consider the growth rate of output per worker (or per-capita income). We know that κ is constant when we’re at our steady state equilibrium. Thus, our per capita income is: Y ( t ) L ( t ) = ( κ * ) α ( t ) E ( t ) Or, plugging in our equation for κ * : Y ( t ) L ( t ) = ( s g E + g L + δ ) α 1 - α E ( t ) We know that g ( κ ) = 0 in equilibrium. Thus, in steady-state equilibrium, growth of per capita income equals the growth rate of efficiency. g Y L = g E Next, let’s tackle growth rate of capital. Once again, remember that κ is constant. Recall that κ = K EL , and we can plug in our equations for K ( t ) = K (0) e g k t , E ( t ) = E (0) e g Y t , and L ( t ) = L (0) e g L t . This gives us: κ * = K (0) e g k t E (0) L (0) e ( g Y + g L ) t κ * = K (0) E (0) L (0) e ( g K - g E - g L ) t 2
Since we know that κ is constant at equilibrium, g K - g E - g L = 0 g Y L = g E = g K - g L Hence, in the steady state, the growth rate of the capital-labor stock ratio is the same as the growth rate of efficiency and per capita income. Alternatively, the difference between growth rate of capital