Goodwin's Growth Cycle Model

Consider an economy where at any time t, the aggregate output is given by

Y_t =min{_t L_t, K_t}

where _t is the productivity of labor L_t, and = 1 pins down capital output ratio K_t/Y_t, when K_{t t}L_t. Assumption that _tL_t = K_t.

Assume that the growth rate of labor productivity is 10% ( = 0'1) while the population grows at 5% ( = 0'05). The percentage change in the wage rate is derived from the Phillips curve and is equal to:

g_w = _t/3 - 1/5

The stock of capital changes over time according to the following equation:

g_K = s_t -

where _t is the share of capital income, the marginal propensity to save of capitalists is equal to 75% (s = 0'75) and the depreciation rate is 10% ( = 0'1)

(a) State the equation for the rate of employment () and calculate the percentage change/rate of growth in the rate of employment (g) as a function of the growth rate of output (g_y).

(b) Calculate the growth rate of output (g_y) as a function of the worker's labor share (_t) at time t. Plug this into (a) to find one of the differential equations of the model. (Hint: Use the equation for g_K.)

(c) Calculate the worker's labor share (_t) and calculate the growth rate of the labor income share (g) as a function of the employment rate at time t (_t).

(d) Derive the steady-state of the model where the labor share and the rate of employment remain unchanged.

(e) Draw a phase diagram showing how the employment rate and the labor share of output evolve towards the steady-state. Explain it intutively.