Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Ch. 5 Dynamic Efficiency and Sustainable Development (Fall 2011) Introduction • First, we will develop conceptual framework for dynamic efficiency. • Efficiency is not the only criterion. • Fairness or justice is another criterion. We will examine the relationship of dynamic efficiency and fairness. Dynamic Efficiency • Time is often a very important factor affecting the use of resources. e.g. pumping exhaustible groundwater now implies less will be available in the future; pollution can accumulate over time. • How do we compare the net benefit in one period vs. another? We use present value. • PV[Bn] = Bn/(1+r)n (present value of a one time net benefit to be received n years from now) Dynamic Efficiency (cont.) • present value of stream of benefits equals: n PV [ B0 ,..., Bn] i 0 Bi (1 r ) i The process of calculating the present value is called discounting. The rate r is referred to as the discount rate. Dynamic Efficiency (cont.) • Dynamic efficiency: allocate resource use over time so that the present value of the net benefits are maximized. • We will start with a two period model: • Assumptions: – fixed supply of resource to allocate over 2 periods – Demand (marginal willingness to pay) function is constant, i.e., the inverse demand function, is given by : P = 8 -0.4q – marginal cost of extraction is constant $2/unit – interest rate is 10% Two Period Model – Note if total supply is greater than 30, the efficient allocation would be 15 units/period, regardless of interest rate. – In this case static efficiency is sufficient. – One period’s consumption does not affect the other period’s consumption. In this sense, they are independent. 9.0 9.0 8.0 8.0 7.0 7.0 6.0 6.0 Price ($/unit) Price ($/unit) Fig. 5.1 The allocation of an abundant depletable resource 5.0 4.0 3.0 5.0 4.0 3.0 MC 2.0 1.0 1.0 0.0 0.0 0 5 10 15 Quantity (a) Period 1 20 MC 2.0 25 0 5 10 15 Quantity (b) Period 2 20 25 Two Period Model – What if available supply is less than 30, say only 20? – How should we allocate across the 2 periods? – Try 15 in period 1 and 5 in period 2. Fig. 5.1 (modified, not shown in text) An arbitrary allocation of an limited depletable resource (q1=15, q2=5) 9.0 9.0 8.0 8.0 P.V. Net Benefits= 0.5*(6)*15=$45 7.0 7.0 P.V.=25/(1.10)=$22.73 6.0 Price ($/unit) 6.0 Price ($/unit) Net Benefits= 0.5*(6+4)*5=$25 5.0 4.0 5.0 4.0 3.0 3.0 2.0 2.0 1.0 1.0 0.0 0.0 0 5 10 15 20 25 0 10 15 Quantity Quantity (a) Period 1 5 P.V. Total Benefits = $67.73 (b) Period 2 20 25 Two Period Model – How do we find the allocation that gives us maximum present value net benefits? We could guess or have a computer search iteratively. – But the best way is use economic logic (and/or math) – Dynamically efficient allocation requires that the present value of the marginal net benefit (PVMNB1) in period one equals the present value of the marginal net benefit in period two (PVMNB2). – Fig. 5.2 shows the PVMNB for each period. Period 1 is read from left to right and period 2 is read from right to left. Note that the intercept for period 1 is $6 ($8-$2), but intercept of period 2 is $5.45 (=$6/1.1) Fig. 5.2 The optimal allocation of a limited depletable resource P.V.MNB1 P.V.MNB2 P.V.NB1 = 0.5*(6+1.905)*10.238 = $40.466 P.V.NB2 = 0.5*(5.45+1.905)*9.762= $35.90 MUC 5.45 1.905 Quantity in Period 1 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 q1=10.238, q2=9.762 4 3 2 1 0 Quantity in Period 2 Two Period Model – The solution set is q1=10.238 and q2=9.762. The Total PV of the net benefits equals $40.47 in period 1 plus $35.90 in period 2, i.e., $76.37. This is the maximum PVNB. Recall for example, that in the q1=15 and q2=5 allocation, PVNB = $67.73 – This solution occurs where the two PVMNB curves cross. The vertical distance where they cross is referred to as the marginal user cost. – marginal user cost (MUC) is the present value of the foregone opportunities at the margin. – in this example the MUC is $1.905, indicating that use of the resource today will reduce the present value of future net benefits by $1.905 Two Period Model – Thus, the PVMNB of the last unit used in period 1 should be worth $1.905. If it is worth more in period 1, than we should consume more in period 1 until the PVMNBs are equal. If it is worth more in period 2, than we should consume more in period 2 until the PVMNBs are equal. – Price in period 1 should be $3.905 = 8-.4*10.238. – Price in period 2 should be $4.095 = 8-.4*9.762. – The actual marginal user cost rises at the rate of interest. $1.905*(1+.1) = $2.095. Two Period Model – Important point: don’t get marginal net benefit and total net benefit confused. – We maximize present value of total net benefit (PVNB) when present value of marginal net benefits are equal (PVMNB1= PVMNB2). – Is this clear? Two Period Model – The undiscounted marginal user cost rises at the rate of interest. Why? – because the present value of the marginal user costs are equal across time periods: – MUC2 = MUC1*(1+r) – e.g. $2.095 = $1.905*(1.10) – What happens if the discount rate increases? – Allocate more to period 1 and less to period 2. The PVMNB curve in period 2 rotates down. (See modified Fig. 5.2) Fig. 5.2 (modified, not shown in book) The optimal allocation of a limited depletable resource P.V.MNB1 5.45 P.V.MNB2 MUC interest rate increases to 50% 1.60 Quantity in Period 1 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Quantity in Period 2 Verify these solutions w/ appendix method. q1=11, q2=9 9.0 9.0 8.0 8.0 7.0 7.0 6.0 6.0 5.0 Price ($/unit) Price ($/unit) Fig. 5.3 The efficient market allocation of a depletable resource: The Constant Marginal Cost Case (a) Period 1 (b) Period 2 P = $3.905 4.0 3.0 5.0 P = $4.095 4.0 3.0 MC MC 2.0 2.0 1.0 1.0 10.238 0.0 9.762 0.0 0 5 10 15 Quantity (a) Period 1 20 25 0 5 10 15 Quantity (b) Period 2 20 25 Defining Intertemporal Fairness • How can we be fair to all generations? • John Rawls, A Theory of Justice: suggests that we pretend we are behind a “veil of ignorance” and had to predetermine the rules we wanted to live by if we did not know when we would be born. – his solution: sustainability criterion: future generations should be left no worse off than current generations. It does not say that present generations can not make themselves better off. – Does dynamic efficiency violate sustainability criterion? Are Efficient Allocations Fair? – Does dynamic efficiency violate sustainability criterion? Not necessarily. – It might seem so since the net benefits in period 1 are $40.466 while they are only $39.512 in period 2. – But if sharing takes place, the first generation could save say $0.466 and invest it at 10% interest, growing to $0.513, when added to $39.512, gives the second generation net benefits of $40.025, more than the first generation receives. – demonstrates that dynamic efficiency can be consistent with sustainability as long as gains are shared among the generations. – See Example 5.1 The Alaska Permanent Fund Applying the Sustainability Criterion – “Hartwick Rule”: easier to apply than Rawls’ sustainability rule – Constant level of consumption can be maintained perpetually if all scarcity rent is invested in capital. – Total capital stock should not decline. – Our endowment consists of natural capital (environment) and physical capital (buildings, machines, etc.) – How easily can these two forms of capital be substituted? • “weak sustainability”: maintenance of total capital • “strong sustainability”: maintenance of natural capital • “environmental sustainability”: maintain physical flow of individual resources – See Example 5.1: Alaska’s permanent fund – See Example 5.2 Nauru: Weak Sustainability in the Extreme Appendix (Chs. 2 & 5) – Pt = a - b qt (called inverse demand equation, i.e., marginal willingness to pay curve) – Total benefit is given by area under demand curve qt Total Benefits (a bq )dq 0 b 2 a qt qt 2 Total Cost c q t Appendix (cont.) – We want to maximize the net benefits of a fixed amount of the resource Q over n time periods: b aq 2 q cq Max Q q q (1 r ) 2 n t i i i i 1 i 1 n i 1 i The first order conditions: a bq c (1 r ) i i 1 0 i 1,..., n n Q 0 q i i 1 Example: a 8, c $2, b 0.4, Q 20, and r 010 . 8 0.4 q1 2 0 8 0.4 q 2 2 110 . q q 1 2 20 0 Appendix (cont.) – solve the above three equations simultaneously: 8 0.4 q1 2 q 1 8 0.4 q 2 2 110 . 5.4545.3636 q 2 20 q 2 6 0.4(20 q 2) 5.4545.3636 q 2 6 8 5.45 .7636 q 2 7.4545 .7636 q 2 q q 2 1 7.4545 / .7636 9.762 20 9.762 10.238 8 0.4 q1 2 8.4 * 10.238 2 $1.905 The End