Question

Consider the Solow growth model. The production function is given by Y = K αN1−α , with α = 1/3. Depreciation rate δ = 0.05, and saving rate s = 0.25. Labor force grows at the rate n = 0.01. (a) Write down the law of motion for capital per worker. (b) Compute steady state capital per worker. (c) Suppose the economy has initial capital per worker k0 = 4. Describe the dynamics of this economy, i.e., how does capital per worker and output per worker change over time? (d) Suppose the economy has initial capital per worker k0 = 14. Describe the dynamics of this economy, i.e., how do capital per worker and output per worker change over time? (e) What is the growth rate of the stock of capital in the steady state? What about GDP? (f) If savings rate doubles, by how much do steady state output per worker increases (or decreases)? just answer cdef

Answer 1

1. Output and capital per worker grow at the same constant, positive rate in BGP of model. In long run model reaches BGP.

2. Capital-output ratio K Y constant along BGP

3. Interest rate constant in balanced growth path

4. Capital share equals α, labor share equals 1 − α in the model (always, not only along BGP).

Neoclassical (Cobb-Douglas) aggregate production function: Y (t) = F[K(t), L(t)] = K(t) α L(t) 1−α

Y = A Kα L1−α

Constant returns to scale: F(λK, λL) = λ F(K, L) = λA Kα L1−α

Inputs are essential: F (0, 0) = F (K, 0) = F (0, L) = 0

Define x = X L as a per worker variable. Then y = Y L = A KαL1−α L = A µK L ¶a µ L L ¶1−α = A kα • Per worker production function has decreasing returns to scale.

Capital accumulation equation: K˙ = sY − δK •

Important additional assumptions:

1. Constant saving rate (very specific preferences)

2. Constant depreciation rate

Steady state output per worker depends positively on the saving (investment) rate and negatively on the population growth rate and depreciation rate

Why are some countries rich (have high per worker GDP) and others are poor (have low per worker GDP)?

Solow model: if all countries are in their steady states, then:

1. Rich countries have higher saving (investment) rates than poor countries

2. Rich countries have lower population growth rates than poor countries