Optimal Exhaustible Resource Depletion with Endogenous Technical Change (Kamien and Schwartz, 1978) Notes by Arrow Team: Kuan Liu, Daijing Lv, Ibrahim Keita, Alecia Waite

Introduction  Variable List Variable Explanation C(t) The aggregate consumption rate at time t U(C(t)) The instantaneous utility F(K,R) Production function K(t) is the productive capital, R(t) is the exhaustible resource usage y y=R/K, f(y) is the identical productive function S(t) Exhaustible resource remaining at time t m(t) The rate at which the composite good is allotted to R&D z(t) Cumulative effective effort g(m(t)) z’(t)=g(m(t)), the relation between the R&D rate and the growth of cumulative effective effort φ(z) The probability the R&D will be successfully completed by the time cumulative effective effort is z h(z) The completion rate or conditional probability of completion W The maximum value of the discounted utility stream from the time T the new technology becomes available forward λ The marginal values of the capital stock μ The marginal values of natural resource reserves γ Cumulative effective R&D effort Δ Discount rate

 Motivation: imminent exhaustion of some natural resources has prompted calls for conservation through reduction of economic growth.  This paper studies the optimal depletion of nonrenewable resources with endogenous technical change.  The model in this paper is the Ramsey’s optimal growth model. Within the model, there is only one composite good in the economy for consumption.  This multi-purpose good is produced by means of a reproducible resource, capital, and an exhaustible resource and its usage is divided between household consumption, investment in capital, and R&D.  The availability of new technology depends on the proportion of output allotted to R&D and the effective effort put into R&D.  The objective is to maximize the expected discounted stream of utility from consumption.  By achieving the objective, the paper develops a temporal behavior path of household consumption and R&D before the advent of new technology.

The Model  In the model, a single multi-purpose good is produced. Its production rate depends on the productive capital K (t) and the exhaustible resource usage R (t). We assume a production function F (K, R) that is twice differentiable and homogeneous.  We also assume a utility function U, which is increasing, strictly concave, and twice differentiable. So, we have:

 Let C (t) be the aggregate consumption rate at time t. U(C(t)) is the instantaneous utility.  The amount S(t) of exhaustible resource remaining at time t satisfies following conditions:

 If there is no R&D or new technology, then we choose C(t), K(t), S(t) for t ≥ 0 to maximize the discounted utility stream.

Subject to

 However, when technology becomes available at a time T, then we can estimate the maximum W of the discounted utility stream from the time T to be:

 Note that the new technology need not be implemented immediately upon availability.  In this model, technology is a result of successful R&D which requires resources to be diverted from consumption and capital investment. We denote by m(t) the rate at which the composite good is allotted to R&D. The R&D rate and the growth of cumulative effective effort z(t) devoted to the project by time t are related by a bounded, concave, monotone increasing and twice differentiable function g(m(t)). It follows that:

 Since any successful R&D activity is associated with uncertainty, we denoted by ф(z) the probability that the R&D will be successfully completed by the time cumulative effort is z. The function ф is twice continuously differentiable and satisfies:  Let h(z) be the conditional probability of completion. Assume that the R&D is not successful after a cumulative effort z(t), then the probability of completion with increment effort dz is h(z)dz.

 There is a level z0 such that ф(z0) = 1 and a level z1 such that if z ≤ z1, ф(z) = 0 (h(z) = 0).  Once the technology is available at time T, then the discounted utility stream to be maximized through the choice of consumption C(t), R&D rate m(t) and factor productive y(t) is:

Subject to constraints:

 K and S need to be positive at t goes to infinity, otherwise if they become negative or zero, then they cannot increase.

Solutions: Conditions and Interpretations

 Let λ(t), µ(t), and γ(t) be the current value multiplier functions associated with the differential equations governed by the state variables K, S, and z, namely K’, S’, and z’. Define the current value Hamiltonian

Then the first-order conditions are given by:

 The finite horizon transversally conditions hold as the horizon is extended indefinitely. So, we have:  Note that λ, µ, and γ are the marginal values of capital stock, natural resources reserves, and cumulative effective R&D effort respectively. Now, let us interpret the first-order conditions:  The first equation of FOC reduces to: which says that the expected marginal utility of the composite product in consumption should equal its marginal value in investment. The second equation of FOC becomes:

This says that the marginal value of the exhaustible resource should be the same in the use as in reserve. The third and last equation of the FOC reduces to two parts as follow:

 The intuition here is that there is no R&D when its marginal value is below the composite good’s value in other uses.  The temporal behavior of the factor proportion y is governed by the following differential equation and it depends only on the production function.

This shows that the ratio of exhaustible resource to capital in production falls over time. In the same way, the temporal behavior of consumption is given by:

 So, consumption rises and falls depending on the right hand side being positive or negative respectively. Moreover, by rearranging the above equation, we get: -U’’C/ U’ + δ = f(y) – y f’(y) – g(m)h(z)  The left hand side is how much the decision maker is willing to accept in order to postpone consumption. Beginning the search for a new technology ---find a point where R&D begins

 Rewrite the criterion for choice of the R&D rate m

 =0 (i) and (ii) and

So, for s.t , from deduction we get

 Thus, there will be aat which and R&D begins. Suppose (the marginal value of cumulative effective R&D effort is greater than 0), the innovation would have positive value.  R&D may, but need not begin immediately at =0. Eventually, however, the diminishing supply of exhaustible resource will exert pressure to begin the search for an alternate technology. That is , time t0 will be attained

While the search goes on From differentiating (33i) and substitution, we get (42)

 Since g’>0 and g”<0, the R&D rate (m’) increases( decreases) when the right side of (42) is positive (negative)  We have three positive control variables to study: y, C, m  Y is declining  C and m need further investigation

Behaviors of C and m

 Donate by tm the moment R&D begins. Then,

 Behavior of C (38)

 If h(z)=0, sign =sign  If at s, then C’(t)<0 for all t>s  If h(z)>0 and , then C’(t)<(>)0 as m(t)>(>0)m0(t), where m0(t) is implicitly defined by

 Differentiate (44) totally with respect to t:

We can see that the critical value m0 decreases through time so that C’=0 locus moves down in Figure 1  When m’=0, C and m satisfies (46) Assume ; we get

 Since left side of (46) takes non- negative values; the rage of C is

 Left side of (46) is increasing function of m, right side of (46) is concave function of C, we get value of C that maximizes G(C) is  Behavior of m  If h(z)=0, then with strict inequality in case m(t)>0  If , then  If , then as  Differentiate (46) with respect to t, we get

We can get locus increases over time. The locus moves up

 C-coordinate of its peak remains fixed at C*. The intercepts of the curve on the m=0 axis satisfy (57) Left side of (57) decreases over time; the smaller root C of (57) decreases over time while the larger root grows. Shows in Figure 2, in which Ci, are the intercepts of the locus on the axis at time ti. Thus we obtain Figure 3. Combined and Temporal Patterns

In this section, we make assumptions about y, the ratio of resource usage to capital, in order to get 2-dimensional cross-section of just m (effort) and c (consumption).

 Could have done this with z instead of m if we knew that g was a non-monotonic function; actually, this would be a useful simplification.

 What is ? Marginal product of capital

 (Discounted marginal) shadow value of capital is increasing when and decreasing when (Aside: what does the shadow value do? The shadow value rations use of k between time periods. It does this by insuring that, at the margin, resource has the same discounted value at each period).

A high discount rate leads to higher consumption today, lower consumption tomorrow.

 When the discount rate is “high,” selfish parents don’t want to save trees for children.

 Thus, we increase consumption today, but necessarily, this decreases consumption in the future (“resource solvency”).

 This happens at all t, so C is decreasing over time.

 We also increase effort devoted to R&D because we know that the resource is exhaustible and h(z) = 0, which means we have not yet successfully completed the project. This is maintained so long as h(z) = 0. After R&D progresses to give us a positive probability of eminent completion, we invest less effort.

 In Figure 6, we look at the case that discount rate high today. Then, we are going to consume the most today. Necessarily, must decrease consumption in future: “resource solvency.”

 Another situation is that depicted in Figure 5: discount rate projected to be high at some time in the future. Call that time .

 Then, we delay our effort until that’s the case. We know that at that time we will have higher consumption than today.

 Note that we must have a continuous expected consumption path after today, so we will increase our consumption until we get to that point.

 Then, after that point, because we have consumed beyond our means, it is necessary that, because stock cannot be negative at the end of the planning period, we must decrease consumption in future periods. With a low discount rate, more consumption saved for later.

 Benevolent parents save oil reserves for their children.

 We want to postpone consumption of resources for future generations. Can see from figure 7 that c’ is falling over time and m’ is rising over time.

 Which effect stronger? This determines optimal path in c-m space.

 Possibilities are illustrated in figure 8 (c’=0 falls more important) and figure 9 (Stop when T or is attained. is our maximum effort. It is the effort with which we are SURE that the technology will be available. If these not attained, m’ = 0 locus may overtake the path, and then C and m will both fall.)

 Recall that our rate effort spent on R&D increases until we have a positive probability of completion of the project (which requires effort), and then decreases. So, m increases and then decreases.

 Figures 10 and 11 (curve that says = 0) show what happens to m and c over time in this case.  Consumption peaks before R&D effort

 Case of Delayed R & D (Fig 11, curve that says “>0”):

 In delayed R & D case, we see that eventually, diminishing supply of exhaustible resource will exert pressure to begin search for alternate technology, that is, time to will be attained.

 We see that if effort delayed, effort will decrease. This is due to the fact that the m’ curve is not stable!

If we have a low discount rate but the m’ curve moves faster than the c’ curve, then effort peaks before consumption.  Fig 12: Case in which rising optimal path in the C-M plane meets the m’ = 0 locus before intersecting the c’=0 locus. This causes c and m to both rise after (the time the effort m is invested) until the moment the path meets the m’ =0 locus (denoted ). Thereafter, the path falls, and turns inward when we meet the c’ = 0 locus. So, both m and c fall.

 In this case, effort peaks before consumption. The intuition here:

 We quickly attain the point at which m gives us a positive value of completing the project (). So, we decrease m.

 We further increase our consumption because we can, leading to a decrease in the future.

K(t) R(t)

m(t)

S(t) Z(t)

h(z) Ф(z)