ECON 448: Week 4-
1. CRS Production Function and Diminishing Returns for Each Factor
1. A production function is a simple mathematical description of how various inputs (labor, capital, ...) are combined to produce an output.
2. Cobb-Douglas Production Function
Y = F(K, P) = AKαPβ
3. Derive the condition for the above production function to be DRS, IRS, and CRS.
4. Under the CRS assumption, if the amount of capital, K, is fixed, does the output increase proportionally to the amount of workers employed (P)?
2. Growth Models: Harrod-Domar v. Solow
Harrod-Domar:
s/θ ≈ g∗ + n + δ
Solow (with population growth):
(1 + n)kt+1 = (1 - δ)kt + syt
In the steady state,
k∗/y∗ = s/n + δ
1. The Harrod-Domar model assumes a constant capital-output ratio (K/Y = θ), while the Solow model does not. In other words, Harrod-Domar considers θ as exogenous, but Solow model endogenizes the capital-output ratio.
2. Harrod-Domar model predicts an economy can grow at a positive rate which is determined by some parameters in the economy (s, n, δ, θ). In contrast, the (longterm) growth rate of an economy is always zero in a simple version of Solow model.
3. Assume that s = 12%, θ = 4, n = 1%, and δ = 2%. If the saving rate is doubled, what is the growth rate predicted by Harrod-Domar? By Solow?
4. In a more sophisticated version of Solow model with labor-augmented technology, an economy can grow at a positive rate, but the rate is equal to the rate of technical progress and exogenous. That is, the (long-term) growth rate is not affected by any other element in the model, except for the technical progress which is exogenously given.
3. Convergence: Unconditional v. Conditional
1. One implication of the Solow model is convergence of per capita incomes across countries, provided that all the parameters are the same across countries. (why?)
2. Unconditional convergence is clearly rejected by the data, while conditional convergence is supported by empirical evidence.