Model without stock effects, Public Economics

Model without Stock Effects

In this model it is assumed that extraction cost depends only on extracted quantity. Let us assume that St  is the remaining stock of natural resource at time t, Et is the quantity  of extraction,  Pt  is  the price at which it is sold  in  the market, r is  the discount rate and C(Et) is the total cost of extraction. The problem before us is

Equation 1

45_Model without Stock Effects 1.png

Here we consider the time period t = 1 .... T.

Total profit is ∏, which is the sum of 'revenues minus costs' during all time periods under consideration.

When we extract Et in time period’t’ there is depletion to the stock. Therefore,

Equation 2

 

St = St-1- Et for all t.

 

In addition, final stock that remains cannot be negative. Thus, we impose the constraint

ST  > 0

The Lagrangian function for the above problem is given by

Equation 3

 

321_Model without Stock Effects 2.png

On solving equation 3 the first order condition yields

Equation 4

 

2193_Model without Stock Effects 3.png

As you know Quantitative Methods, λ, in equation 4 is the present value of the shadow price or opportunity cost, and

Equation 5

 

158_Model without Stock Effects 4.png

Equation 6

685_Model without Stock Effects 5.png

An implication of the first order condition equation 5 is that λt = λt=1 for all t, which means the present value  of  shadow price  is constant over time, or it  is time independent.

Let us denote 'resource rent' or royalty to be paid as Rt which is given by

 

Rt= Pt - MCt = λtert from the first order condition.

It can be easily shown that

Equation 7

Rt = λtert  and Rt / Rt = r

 

Where dot over a variable represents its derivative, i.e., instantaneous change  in it. Thus R denotes the change in R, i.e., resource  rent. From equation 7 we find that resource rent rises at the rate ‘r’.  The resource rent  is equivalent to the present value of opportunity cost. Hence, the present value of opportunity cost rises at  the rate equal to market rate of interest (which we had taken as our discount rate).

The model presented at equation 1 can be expressed using discreet time variable.

 

483_Model without Stock Effects 6.png

The first order condition would yield

Equation 8

2209_Model without Stock Effects 7.png

By rearranging terms in equation 8 it is easy to show that

Equation 9

 

2433_Model without Stock Effects 8.png

This yields the same  result that the resource rent increases at  the rate 'r'  over  time.  Moreover,

Equation 10

Pt =MCt + λt (1+r)t

Equation 10 shows that optimal extraction rate is  the one at which the sum of marginal cost and the current value opportunity cost is equal to price.

μ = λT = constant              If  ST >  0  then λ=  0

If  ST =  0  then λ >  0

Opportunity cost is positive only if the resource stock is completely exhausted within the time horizon.

The following results are popularly known as Hotelling's Lemma.

(i)  Current value opportunity cost increases at the constant rate 'r' equal to market rate of interest.

(ii)  Present value of opportunity cost is constant over time.

Posted Date: 12/18/2012 8:33:43 AM | Location : United States







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