The statistics office of Pellora wants to know why output grew. It assumes the economy's total output is made by a Cobb-Douglas production function Y = A\,K^{\alpha} L^{1-\alpha} of capital K and labour L, where A is total factor productivity (TFP): how well inputs are turned into output. Taking growth rates of both sides gives, approximately, the growth accounting equation
TFP cannot be observed, so the office measures it as what is left: the Solow residual
All growth rates are year on year: from year t-1 to year t, g_X = X_t / X_{t-1} - 1 (a fraction).
Write growth_accounts(Y, K, L, alpha) that takes the yearly series Y, K, L (same length, years 0, 1, \ldots) and returns a list with one tuple (capital, labour, residual) for each year t = 1, 2, \ldots: the three terms of the equation for growth from t-1 to t. With one year of data, return [].
Examples
Input: Y = [100, 104], K = [300, 309], L = [50, 50.5], alpha = 0.3
Output: [(0.009, 0.007, 0.024)]
Explanation: output grew 4 %, capital 3 %, labour 1 %. Capital explains
0.3 * 3 % = 0.9 points, labour 0.7 * 1 % = 0.7 points; 2.4 points are left over.
Input: Y = [200, 206, 203], K = [600, 624, 630], L = [80, 81, 80], alpha = 0.35
Output: [(0.014, 0.008125, 0.007875), (0.00336538, -0.00802469, -0.00990380)]
Explanation: in year 2 output fell by more than the loss of labour explains,
so measured TFP fell.
Constraints
- answers are compared with a tolerance of
1e-6; do not round
Goals
- Compute year-on-year growth rates of output, capital and labour
- Split output growth into a capital part, a labour part and a residual
- Read the residual as growth in total factor productivity