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The Neoclassical Growth modelSolow (1956) and Swan (1956)

- Simple dynamic general equilibrium model of growth

Neoclassical Production Function

Output produced using aggregate production functionY = F (K , L ), satisfying:

A1. positive, but diminishing returns

FK >0, FKK<0and FL>0, FLL<0

A2. constant returns to scale (CRS)

Production Function in Intensive Form

- Under CRS, can write production function

- Alternatively, can write in intensive form:
- y = f ( k )
- - where per capitay = Y/L and k = K/L

Exercise: Given that Y=L f(k), show:

FK = f’(k) and FKK= f’’(k)/L .

Competitive Economy

- Representative firm maximises profits and take price as given (perfect competition)
- Inputs paid by their marginal products:
r = FKand w = FL

- inputs (factor payments) exhaust all output:
wL + rK = Y

- general property of CRS functions (Euler’s THM)

- inputs (factor payments) exhaust all output:

A3: The Production Function F(K,L) satisfies the Inada Conditions

Note: As f’(k)=FKhave that

Production Functions satisfying A1, A2 and A3 often called Neo-Classical Production Functions

Technological Progress

= change in the production functionFt

Hicks-Neutral T.P.

Labour augmenting (Harrod-Neutral) T.P.

Capital augmenting (Solow-Neutral) T.P.

A4: Technical progress is labour augmenting

Note: For Cobb-Douglas case three forms of technical progress equivalent:

Under CRS, can rewrite production function in intensive form in terms of effective labour units

- note: drop time subscript to for notational ease
- Exercise: Show that

Model Dynamics

A5: Labour force grows at a constant rate n

A6: Dynamics of capital stock:

- net investment = gross investment - depreciation
- capital depreciates at constant rate

… closing the model

- National Income Identity
- Y = C + I + G + NX
- Assume no government (G = 0) and closed economy (NX = 0)
- Simplifying assumption: households save constant fraction of income with savings rate 0 s 1
- I = S = sY
- Substitute in equation of motion of capital:

Fundamental Equation of Solow-Swan model

Steady State

Definition: Variables of interest grow at constant rate (balanced growth pathor BGP)

- at steady state:

Existence of Steady State

- From previous diagram, existence of a (non-zero) steady state can only be guaranteed for all values of n,g and d if

- satisfied from Inada Conditions (A3).

Transitional Dynamics

- If , then savings/investment exceeds “depreciation”, thus
- If , then savings/investment lower than “depreciation”, thus
- By continuity, concavity, and given that f(k) satisfies the INADA conditions, there must exists an unique

Properties of Steady State

1. In steady state, per capita variables grow at the rate g, and aggregate variables grow at rate(g + n)

Proof:

2. Changes in s, n, or dwill affect the levels of y* and k*, but not the growth rates of these variables.

- Specifically, y* and k* will increase as s increases, and decrease as either n or dincrease

Prediction: In Steady State, GDP per worker will be higher in countries where the rate of investment is high and where the population growth rate is low - but neither factor should explain differences in the growth rate of GDP per worker.

Policies to Promote Growth

- Are we saving enough? Too much or too little?
- What policies may change the savings rate?
- How should we allocate savings between physical and human capital?
- What policies could generate faster technological progress?

Golden Rule

- Definition: (Golden Rule) It is the saving rate that maximises consumption in the steady-state.
- We can use the rule to evaluate if we are saving too much, too little or about right.
- Given we can use
to find .

Golden Rule and Dynamic Inefficiency

- If our savings rate is given by then our savings rate is optimal and
- If then we must be under-saving
- If then we must be over-saving
- Check why this is the case!

Is Golden Rule attained in the US? Is it Dynamically Efficient?

- Let us check: Three Facts about the US Economy
- a)
- The capital stock is about 2.5 times the GDP
- b)
- About 10% of GDP is used to replace depreciating capital
- c)
- OR
- Capital income is 30% of GDP: Note alpha also measures the elasticity of output with respect to capital!

Is Golden Rule attained in the US? Is it Dynamically Efficient?

Since

US real GDP grows on average at 3% per year, i.e.

Hence, US economy is under-saving because

Changes in the savings rate Efficient?

- Suppose that initially the economy is in the steady state:
- If s increases, then
- Capital stock per efficiency unit of labour grows until it reaches a new steady-state
- Along the transition growth in output per capita is higher than g.

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