Problem 440243 · medium · Economics · Level 04 Strategy and Growth

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Solow model · technological progress · effective workers · balanced growth · simulation

In the plain Solow model income per worker stops growing at the steady state. In the workshops of Quill it keeps growing, because techniques improve: each worker's efficiency E_t rises by a fraction g a year,

E_t = E_0\,(1 + g)^t

Total output is Y = K^{\alpha} (E\,L)^{1-\alpha} with 0 < \alpha < 1: technology multiplies labour (it is labour-augmenting), as if each worker counted as E effective workers. Measured per effective worker, capital is \tilde{k} = K / (E L) and output is \tilde{y} = \tilde{k}^{\alpha}, and capital per effective worker follows

\tilde{k}_{t+1} = \tilde{k}_t + s\,\tilde{y}_t - (\delta + n + g)\,\tilde{k}_t

with saving rate s, depreciation delta and workforce growth n (all fractions: 2 % is 0.02). Output per worker is y_t = E_t\,\tilde{y}_t.

Write quill_growth(alpha, s, delta, n, g, k0, E0, T), where k0 is \tilde{k}_0, that returns a tuple (k_star, y_tilde_star, y_T, growth_T):

  • k_star: the steady state of \tilde{k}, the level at which \tilde{k}_{t+1} = \tilde{k}_t (positive);
  • y_tilde_star: output per effective worker there, (\tilde{k}^*)^{\alpha};
  • y_T: output per worker in year T, y_T = E_T\,\tilde{y}_T, simulating \tilde{k} from k0;
  • growth_T: the growth rate of output per worker from year T-1 to year T, y_T / y_{T-1} - 1.

Examples

Input:  alpha = 0.5, s = 0.2, delta = 0.04, n = 0.04, g = 0.02, k0 = 4, E0 = 1, T = 10
Output: (4.0, 2.0, 2.43798884, 0.02)
Explanation: k0 is already the steady state, so y~ (output per effective worker)
stays 2 and output per worker is 2 * 1.02 ** 10: it grows at g = 2 % a year.

Input:  alpha = 0.5, s = 0.2, delta = 0.04, n = 0.04, g = 0.02, k0 = 1, E0 = 1, T = 1
Output: (4.0, 2.0, 1.06978503, 0.06978503)
Explanation: k~ (capital per effective worker) goes from 1 to 1 + 0.2 - 0.1 = 1.1,
so y~1 = sqrt(1.1) = 1.04881 and y1 = 1.02 * 1.04881. Below the steady state,
output per worker grows faster than g.

Constraints

  • answers are compared with a tolerance of 1e-6; do not round

Goals

  • Work in units of effective workers, where the Solow model has a steady state
  • Convert output per effective worker back into output per worker
  • See that on the balanced growth path output per worker grows at the rate of technical progress
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