MACROECONOMICS CHAPTER 8
Capital Accumulation as a Source of Growth
The Solow Model: Summary Designed to show how:
● Growth in the capital stock
● Growth in the labor force
● Advances in technology
Interact and affect a nation’s total output of goods and
services.
For now, we assume that:
● labor force → fixed
● available technology → fixed
Supply and Demand:
● The Supply of Goods→ Production Function
● The Demand for Goods→ Consumption
Function
The Supply of Goods and the Production Function The supply of goods in the Solow model:
● Is based on the production function with constant
returns to scale: zY =F(zK , zL)
Relevance of the Labor Force:
● We can use constant returns to scale to analyze
all quantities in the economy relative to the size of
the Labor Force (L)
● Make z=1/ L
● We get Y / L=F ( K / L ,1)
○ Y/L → amount of output per worker
○ K/L → amount of capital per worker
○ Y/L is a function of K/L
○ 1 → a constant and can be ignored
● This implies that the size of the economy (as
measured by the number of workers) does not
affect the relationship between the amount of
output per worker and capital per worker.
New, “per-worker” Terms:
● y=Y /L (Output per worker)
● k =K / L (Capital per worker)
● Production function: y=f ( k)=F(k ,1)
Graph:
● Shows how the k determines y.
● The slope: MPK (marginal product of capital)
○ If k increases by 1 unit, y increases by
MPK units.
○ The production function becomes
flatter as k increases→ diminishing
marginal product of capital.
, MACROECONOMICS CHAPTER 8
Capital Accumulation as a Source of Growth
MPK Mathematically:
● MPK =f ( k +1)−f (k )
The Demand for Goods & the Consumption Function The demand for goods in the Solow model:
● Comes from Consumption & Investment
● So Output per worker (y) is divided between
○ consumption per worker (c)
○ investment per worker (i)
National Income Accounts Identity (Per-Worker
Version).
● y=c+i
○ No government purchases (G)
○ No net exports (N)→ we assume a
closed economy
Consumption Function:
● The model assumes that each year people:
○ save a fraction (s) of their income
○ consume a fraction (1 – s)
● c=(1−s) y
● The saving rate (s)→ a number between 0
and 1
Consumption Function In Terms Of Investment:
● Update national income accounts identity
○ y=(1−s) y+ i
○ Simplify
○ i=sy
● This shows that investment = saving
○ Thus, the rate of saving (s) is the same as
the fraction of output devoted to
investment (i).
Capital Accumulation as a Source of Growth
The Solow Model: Summary Designed to show how:
● Growth in the capital stock
● Growth in the labor force
● Advances in technology
Interact and affect a nation’s total output of goods and
services.
For now, we assume that:
● labor force → fixed
● available technology → fixed
Supply and Demand:
● The Supply of Goods→ Production Function
● The Demand for Goods→ Consumption
Function
The Supply of Goods and the Production Function The supply of goods in the Solow model:
● Is based on the production function with constant
returns to scale: zY =F(zK , zL)
Relevance of the Labor Force:
● We can use constant returns to scale to analyze
all quantities in the economy relative to the size of
the Labor Force (L)
● Make z=1/ L
● We get Y / L=F ( K / L ,1)
○ Y/L → amount of output per worker
○ K/L → amount of capital per worker
○ Y/L is a function of K/L
○ 1 → a constant and can be ignored
● This implies that the size of the economy (as
measured by the number of workers) does not
affect the relationship between the amount of
output per worker and capital per worker.
New, “per-worker” Terms:
● y=Y /L (Output per worker)
● k =K / L (Capital per worker)
● Production function: y=f ( k)=F(k ,1)
Graph:
● Shows how the k determines y.
● The slope: MPK (marginal product of capital)
○ If k increases by 1 unit, y increases by
MPK units.
○ The production function becomes
flatter as k increases→ diminishing
marginal product of capital.
, MACROECONOMICS CHAPTER 8
Capital Accumulation as a Source of Growth
MPK Mathematically:
● MPK =f ( k +1)−f (k )
The Demand for Goods & the Consumption Function The demand for goods in the Solow model:
● Comes from Consumption & Investment
● So Output per worker (y) is divided between
○ consumption per worker (c)
○ investment per worker (i)
National Income Accounts Identity (Per-Worker
Version).
● y=c+i
○ No government purchases (G)
○ No net exports (N)→ we assume a
closed economy
Consumption Function:
● The model assumes that each year people:
○ save a fraction (s) of their income
○ consume a fraction (1 – s)
● c=(1−s) y
● The saving rate (s)→ a number between 0
and 1
Consumption Function In Terms Of Investment:
● Update national income accounts identity
○ y=(1−s) y+ i
○ Simplify
○ i=sy
● This shows that investment = saving
○ Thus, the rate of saving (s) is the same as
the fraction of output devoted to
investment (i).
