OLG with production

Peculiarities of OLG models with endowment economy:

1. CE may not be PO.
2. Indeterminacy of equilibria: continuum of equilibria.
3. Outside asset (fiat money) may have a positive value. Note that fundamental value of fiat money is zero: therefore the fact that the money is valued as a mean of saving in the model makes it a bubble.

Question is: 'Are the properties of OLG models consequence of the absence of production?"

 It turns out that even with a production economy, competitive equilibria can be Pareto inefficient, i.e. Welfare theorem does not hold with OLG models with production.

Endowment

• Agents are endowed with one unit of time when they are young and nothing when they are old.
• Initial old agents are endowed with initial capital stock: K1.

Production

• Inputs: Labor and capital with rental prices of inputs being (wt, rt).
• Assume CRS technology + perfect competition implies zero equilibrium profit.
• Therefore we do not have to specify ownership structure of firms.

Asset

• Saving is possible only as a form of physical capital which is the only asset in the economy.

Timing

• At the beginning of period t, production takes place. Young agents provide their labor and gets labor income w(t). 
• At the end of period t, Young decides how much to consume and how much to save (consumption-saving problem). 
• If he saves s(t) amount, in period t+1, he will get r(t)s(t) as a compensation from a firm. Capital stock is depreciated at the rate of delta.  
• At t+1, the firm uses the saving as a capital input.

 Therefore we have sequential budget constraints:  and .

Competitive Equilibrium: SMCE (defining ADCE for this economy is straightforward.)

 Asset market, or physical capital market clearing condition: demand for capital from firm = supply of capital as a form of saving by young generation.

 This asset market clearing condition can also be interpreted as equalization of gross investment and gross saving. Here saving is the sum of saving by young generation and dissaving by old generation. (*) Note that saving is defined as any difference between income and expenditure.