Econ 208 Week 2

Supply

Production Function-   Y = F(K,L)

Constant Returns to Scale-    zY = F(zK,zL)

Relative to size of Labour Force- z=1/L    Y/L = F(K/L,1)     y = f(k) - output per worker as a function of capital per worker.

Slope = MPK (marginal product of capital - how much extra output one worker produces when given one extra unit of capital)

MPK = f(k+1) - f(k)

Production function becomes flatter as k increases.
Shows diminishing MPK.

Demand

Output per worker is divded between consumption per worker and investment per worker. y=c+i 

We assume that people save a fraction of their income (s) and consume (1-s).     c=(1-s)y

Substitute (1-s)y for c.    y = (1-s)y + i

Rearrange     i=sy        Investment = Saving

Captial Stock - Investment

Two forces influence the capital stock: investment and deprecitation.
i = sy substistute production function y=f(k)   i = sf(k) investment expressed as capital per worker

For any value of k, the amount of output is determined by the production function f(k), and the allocation of that output between consumption and saving is determined by the saving sate (s).

Output is f(k)
Investment is sf(k)
Consumption is f(k) - sf(k)

Captial Stock - Depreciation (  )
We assume a fraction of the captital stock wears out each year.

E.g

If captial last 25 years then the depreciation rate is 4% per year (   = 0.04).

Impact of investment and depreciation on the capital stock:   k = i -    k
Substitute i = sf(k)

    k  = sf(k) -    k

The higher amounts of captail stock the higher amounts of output and investment but also greater amount of depreciation.

k* - is where investment and depreciation meet. At this point the capital stock wont change because the two forces are acting on it, balance. At k*,    k=0.
So the capital stock k and output f(k) are steady over time. k* is the steady state of capital. 

An economy not at steady state will go there.
k* is the long-run equilibrium of the economy.

A Saving Rate Increase

• The economy starts in a steady state with saving rate S1 and capital stock k*1.
  
• When the saving rate increases from S1 to S2, the sf(k) curve shifts upwards.
• Investment now exceeds depreciation
• The captial stock will gradullay rise until the economy reaches the new steady state k*2
• k*2 has a higher capital stock and a higher level of output. 

The saving rate is a key deteminant of the steady-state capital stock. If S is high = large capital stock and high level of output. If S is low = small capital stock and low level of output.

The population and labour force grow at n per period. Population growth decreases k.

Change in capital stock per worker is:         k = i - (   +n)k              

(   +n)k is break-even investment- the amount of investment needed to keep k constant.

Substitute i=sf(k)          k = sf(k) - (   +n)k

The steady state of capital is k*. If k is less than k*, investment is greater than break-even invesment so k rises. If k is more than k*, invesment is less than break-even investment so k falls.

At the steady state, investment replaces the depreciated capital and provides the new workers with capital.

Technological progress causes the efficiency of labour to grow at constant rate g. It has similar effects as population growth.

   k = sf(k) - (    + n + g)k     break-even investment now includes three terms.

k* is the steady state.

Policy makers can influence the saving rate and therefore k*. When choosing a steady state, the goal is to maximise the level of consumption. 

The Golden Rule level of captial is the value of k that maxmises consumption.    k*gold.

• Determine the steady-state consumption per worker   c*
  
• Consumption is output minus investment     y=c+i     c*=y-i.
• Substitute f(k*) for y     c* = f(k*) - i
• Because in the steady state, the capital stock isnt changing investment is equal to depreciation. Sub    k* for i.      c* = f(k*) -    k*.    

The graph shows steady state output and steady state depreciaiton as a function of steady state  capial stock. Steady state consumption is the gap between output and depreciation. k*gold is where there is the biggest gap. At k* MPK=