Macroeconomics

Sinking into memories: A general production function in the Solow growth model
Consider a general production function
This is a “neoclassical” production function if there are positive and diminishing returns to K and L; if there are constant returns to scale (CRS); and if it obeys the Inada conditions:
with CRS, we have output per worker of
If we write K/L as k and Y/L as y, then in intensive form:

Sinking into memories: The Cobb-Douglas production function
One simple production function that provides – as many economists believe – a reasonable description of actual economies is the Cobb-Douglas:
where A>0 is the level of technology and α is a constant with 0<α<1. The CD production function can be written in intensive form as
The marginal product can be found from the derivative:

Sinking into memories: Diminishing returns to capital
output per worker, y=f(k)=kα
f(k)
k

Sinking into memories: The economy is saving and investing a constant fraction of income…
gross investment per worker, sf(k)=skα
f(k)
k

Sinking into memories: What is “labor-augmenting technical progress”?
This is technical progress that increases contribution of labor into output!

Sinking into memories: If we take into account “labor-augmenting technical progress” that

Sinking into memories: Production function with technical progress in the intensive form

Sinking into memories: What is break-even investment?

Sinking into memories: Derivation of equilibrium capital per effective worker

Sinking into memories: Equilibrium as a situation of steady-state growth

Sinking into memories: Dynamics of parameters on the steady-state

Sinking into memories: Balanced growth

Sinking into memories: Growth in steady state and outside steady state
In the steady state – when actual investment per “effective worker” = break-even investment - the rate of economic growth will be equal to the sum of rate of population growth and rate of technical progress = n+g.
If “initial” capital stock is less than steady state capital stock, then the rate of economic growth will be more than n+g.

Sinking into memories: Unconditional convergence

Sinking into memories: Conditional convergence

Sinking into memories: The concept of the Golden Rule

Sinking into memories: The Golden Rule – for what?

Sinking into memories: Accounting of growth in Solow model (Part 1)

Sinking into memories: Accounting of growth in Solow model (Part 2)

Sinking into memories: Accounting of growth in Solow model (Part 3)

Accounting of growth in the U.S. economy In the end of the XX century

Accounting of growth among “Asian Tigers” In the end of the XX century

Exercise #1: the condition
The savings rate = 0.3; the rate of population growth = 0.03; the rate of technical progress = 0.02; the depreciation rate = 0.1. The production function is the Cobb-Douglas function with labor-augmenting technical progress, that is: Y = K0.5(LE)0.5
Calculate: Calculate equilibrium capital per effective worker ratio, amount of actual investment and amount of actual consumption.

Exercise #1: the solution: the graph

Exercise #1: the solution: the figures
If Y = K0.5(LE)0.5
Then y = k0.5
2) sy = sk0.5 = (n + g + d)k
0.3k0.5 = (0.03 + 0.02 + 0.1)k
0.3k0.5 = 0.15k ; 2k0.5 = k
k = 4 ; y = 2
3) actual investment = savings = s*y = 0.3*2 = 0.6.
4) actual consumption = y – s = 2 – 0.6 = 1.4.

Exercise #2: the condition
The rate of population growth = 0.04; the rate of technical progress = 0.06; the depreciation rate = 0.08, capital per effective worker ratio = 4. The production function is the Cobb-Douglas function with labor-augmenting technical progress, that is: Y = K0.5(LE)0.5
Calculate: Calculate equilibrium savings rate, amount of actual investment and amount of actual consumption

Exercise #2: the solution:
If Y = K0.5(LE)0.5
Then y = k0.5
2) sy = sk0.5 = (n + g + d)k
s*40.5 = (0.04 + 0.06 + 0.08)*4
s = 0.18*4 : 2 = 0.36 = 36%
3) actual investment = savings = s*y = 0.36*2 = 0.72.
4) actual consumption = y – s = 2 – 0.72 = 1.28.

Exercise #2: the additional question
Is this saving rate – 36% - consistent with the golden rule?

Exercise #2: reply to the additional question
Max c = (1 – s)y
…
If we take ∂c/∂s and make it equal to zero that it implies that s = α or s = 0.5

Exercise #3: the condition
The savings rate = 0.48; the rate of population growth = 0.04; the rate of technical progress = 0.03; the depreciation rate = 0.05. The production function is the Cobb-Douglas function with labor-augmenting technical progress, that is: Y = K0.5(LE)0.5
Calculate: Calculate equilibrium capital per effective worker ratio, amount of actual investment and amount of actual consumption.

Exercise #4: the condition
The rate of population growth = 0.03; the rate of technical progress = 0.02; the depreciation rate = 0.07, capital per effective worker ratio = 36. The production function is the Cobb-Douglas function with labor-augmenting technical progress, that is: Y = K0.5(LE)0.5
Calculate: Calculate equilibrium savings rate, amount of actual investment and amount of actual consumption