Assume that output is produced using the Cobb-Douglas production function: Y = K^(a)*(AL)^(1-a), with a = 0.3. Also, assume that the depreciation rate is “delta” = 0.3. Finally, assume that “s” (the fraction of output that is saved) is equal to 0.3, population grows at a rate 0.1 and technology grows at a rate 0.3. Assume that the economy is at a stable steady state of capital per effective worker k*. Find the golden rule level of capital per effective worker k*_gold. Where “^” denotes power: K^(a) is K in the power of a, L^(1-a) is L in the power of 1-a

Answer:

Finding the Golden Rule Level of Capital per Effective Worker

The golden rule level of capital per effective worker, k*_gold, can be found using the Cobb-Douglas production function: Y = K^(a)*(AL)^(1-a). With a = 0.3, delta (the depreciation rate) equal to 0.3, s (the fraction of output that is saved) equal to 0.3, population growing at a rate 0.1 and technology growing at a rate 0.3, the golden rule level of capital per effective worker is:

k*_gold = (s/(n+delta+g))^(1/(1-a))

Calculating the Steady State of Capital per Effective Worker

The steady state of capital per effective worker, k*, can be found using the Cobb-Douglas production function: Y = K^(a)*(AL)^(1-a). With a = 0.3, delta (the depreciation rate) equal to 0.3, s (the fraction of output that is saved) equal to 0.3, population growing at a rate 0.1 and technology growing at a rate 0.3, the steady state of capital per effective worker is:

k* = (s/(n+delta))^(1/(1-a))

Calculating the Factor Shares

The factor shares, which are the shares of national income received by capital and labor, can be found using the Cobb-Douglas production function: Y = K^(a)*(AL)^(1-a). With a = 0.3, the factor shares for capital and labor are:

Share of national income received by capital = a = 0.3

Share of national income received by labor = (1-a) = 0.7

Calculating the Steady-State Growth Rate

The steady-state growth rate, which is the rate of growth of output per effective worker in the long-run, can be found using the Cobb-Douglas production function: Y = K^(a)*(AL)^(1-a). With a = 0.3, delta (the depreciation rate) equal to 0.3, s (the fraction of output that is saved) equal to 0.3, population growing at a rate 0.1 and technology growing at a rate 0.3, the steady-state growth rate is:

Steady-state growth rate = g = n + delta + g = 0.1 + 0.3 + 0.3 = 0.7

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