Model with stock effects, Public Economics

Model with Stock Effects

Extraction costs could depend upon the remaining stock in several possible ways. When most of the original stock becomes depleted, total and marginal extraction costs will increase based upon physical difficulties of extracting remaining quantities. Lower stock may also increase costs independent of the extraction rate, as reductions in remaining stock of water in an aquifer or intense mining leads to subsidence of overlying land. In addition, in earlier phases of resource extraction "learning by doing" could decrease costs as experience with a deposit is gained. If that experience comes about through extracting the resource, then costs can be modeled as declining endogenously with reductions in remaining stock. Using discrete time model, the Lagrangian function can be expressed as

2449_Model with Stock Effects 1.png

and the first order conditions are

Equation 1

126_Model with Stock Effects 2.png

1479_Model with Stock Effects 3.png


Based on the first order condition above we find that

Equation 2

2197_Model with Stock Effects 4.png

The third first order condition gives us

Equation 3

μST = 0

And implies that λTST = 0

By combining equation 2 and 3 we find that when λt = 0

342_Model with Stock Effects 5.png

By re-arranging terms in the above, we find that

Equation 4

2051_Model with Stock Effects 6.png

The equation 2 implies that whether  the opportunity cost increases or decreases over time depends on the sign of

∂C / ∂St.

There are two possibilities

If ∂C / ∂St < 0 (normal case), then the current value opportunity cost may increase or decrease. However, even if it increases, its growth rate is lower than 'r'.

If ∂C / ∂St > 0 (learning through experience), then the current value opportunity cost growth rate is faster than ‘r’ and the present value increases over time.

The second term on  the  right hand side of  equation 2 is  the present value of  future cost increase due  to one unit additional extraction at time 't'. The essence of this model is, opportunity cost exists even if the resource is ultimately not depleted. This is in sharp contrast to the results of the model without stock effects where positive opportunity cost  exists  only if  the stock  is  completely depleted within the time horizon.












Posted Date: 12/19/2012 12:05:17 AM | Location : United States

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