# Section 6 Solutions.pdf - Economics 100B Section 6...

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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 and growth rate of labor (which is exogenous) is the growth rate of per capita income and efficiency growth rate.