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Solow Growth Model
Observation: Richer countries have more capital (more machines, factories, etc.)
Is this the cause or the result of their greater income? Two possibilities considered:
Countries have more capital because they save a greater part of their income
Countries have more capital because their income is higher
The whole model is beyond the scope of this class, so we will consider a greatly simplified version
Simplified Solow Growth Model
Consumers: Consume a constant fraction of GDP and own
all the capital in the economy Not modeling:
Unemployment (everyone always works) Lifecycle (no children, students or retirees) Within-country income inequality
Consumers described by one equation:
I = s Y where s, a number between 0 and 1, is the
fraction of output that gets invested.
Simplified Solow Growth Model
Firms: Use the capital to produce output Not modeling:
Labor markets (searching for workers) Finance (borrowing to take on projects) Executive compensation
Firms described by one equation:
Y = A K0.3
where Y is GDP, A is productivity and K is the capital stock
Simplified Solow Growth Model Equilibrium: All output is used either in investment or
consumption (no trade, no government):
Y = C + I
How the stock of capital changes over time:
K’ = I + (1- δ)K where K’ is the capital stock next year,
K is the capital stock this year, I is investment this year, and δ is the depreciation rate
Simplified Solow Growth Model So the entire model is described by four equations: Households: I = s Y Firms: Y = A K0.3 Capital Accumulation: K’ = I + (1- δ)K GDP: Y = C + I Rearranging terms: I = s Y = s A K0.3 K’ = I + (1- δ)K = s A K0.3 + (1- δ)K
How does the capital stock change over time?
K’
K
K’= K
How are capital this year, and capital next year related?
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
The equation above tells you how much capital there will be next year
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Suppose the economy starts with some low capital level K0
K0
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Then the equation says that next year’s capital stock will be K1
K0
K1
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Using the red 45 degree line as a reference, we can find K1 on the horizontal axis.
K0
K1
K1
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Then we can find K2
K0
K1
K1
K2
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Repeating these steps, we can find the capital stock in any future year
K0
K1
K1
K2
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Repeating these steps, we can find the capital stock in any future year
K0
K1
K1
K2
K2
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Repeating these steps, we can find the capital stock in any future year
K0
K1
K1
K2
K2
K3
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Repeating these steps, we can find the capital stock in any future year
K0
K1
K1
K2
K2
K3
K3
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Repeating these steps, we can find the capital stock in any future year
K0
K1
K1
K2
K2
K3
K3
K4
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
Notice that the capital stock is approaching the point where the two lines meet
K0
K1
K1
K2
K2
K3
….
K10
K10
….
How does the capital stock change over time?
K’
K
K’= K
K’ = s A K0.3 + (1- δ)K
The point where the two lines meet is the steady state level of capital. Once the economy is at this level, the capital level does not change.
K*
K*
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