4 ANTICIPATED LEARNING 674 Anticipated LearningThis chapter explains how anticipated learning affects optimal decisions. Weconsider t wo scenarios: complete learning, and a generalization, partial learn-ing. With complete learning, a planner knows, at the time of making a firstdecision, that she will learn the value of a unknown parameter before havingto make a second decision. Our objective is to understand how this antici-pated learning changes the first period decision, relative to the case where theplanner does not expect to learn the value of the parameter before having tomake the second decision. With partial learning, the planner acquires infor-mation about the unknown parameter, without learning its exact value. Hereour objective is to compare the effect, on first period decisions, of differentlevels of learning.Chapter 3 studies the problem of a planner who choses an optimal consump-tion path facing uncertainty about the size of the resource stoc k , or in thealternative interpretation, about the damage caused by GHG emissions. Thisoptimal plan does not incorporate the possibilit y that the planner might learnabout the unkno w n parameter as time evolves. That neglect is rational if theplanner either knows that she will learn the total resource stock only aftermaking her final decision, or if she has to commit to decisions before observ-ing the stock/damage. Most of the following analysis as well as extensionscan be found in Lange & Treich (2008) and Eeckhoudt, Gollier & Treich(2005). Lange & Treich (2008) also extend this model to analyze effects ofambiguous uncertainty. Cite Gollier, Jullien and Treich) provide the analyisof the model of partial learning.4.1 Complete learningHere we analyze the situation where the decision maker anticipates learningthe value of the uncertain parameter after the first period but before hersecond period choice. In the resource extraction model w e can think of thischange as an earlier resolution of uncertainty, i.e. uncertainty resolves atthe beginning of the second period rather than at the beginning of the thirdperiod. In the GHG model we can think of moving to a more flexible decisionprocess that allows the social planner to react to observations.Theroleofuncertaintyintheclimatechangedebateissometimesmodeled4 ANTICIPATED LEARNING 68using the setting in Chapter 3, where there is uncertainty but not anticipatedlearning. It matters not only that the decision mak er learns, but also thatshe anticipates this in making early decisions. Without the anticipation oflearning, she would solve the same problem as in Chapter 3 in the first period.Then, when she learns about the actual size of the resource stock or thedamage parameter, she will make a new plan for second period consumption.In contrast, in the setting here, the decision mak er takes into account herreaction to the resolution of uncertainty in the second period when she picksfirst period consumption.Climate policy will unfold over a significant period of time, at least severaldecades. During this period, we are likely to get information both aboutabatemen t costs and about climate-related damages. It is important thatfuture decision makers take their new information into account, and thatcurrent decision makers understand that future decision makers will do so.The formal difference in the setting with anticipated learning is that theplanner maximizes over the second period consumption before she take s theexpected value. She understands that by the time she picks 2she will knowthe realization  of˜. A t period 1 she therefore takes the expectation overwelfare giv en an optimal adjustmen t of 2to the realization of˜.Atperiod1 the optimization problem ismax1Emax2(12˜)=max1[(1)+Emax2(2)+(˜ − 1− 2)] We solve this problem recursively. The first order condition for 2given 1and a realization  is0(2)=0(˜ − 1− 2)  (69)By solving this equation we find a function 2(1). Because 0is strictlymonotonic, it is optimal to perfectly smooth consumption over periods 2 and3 in the resource model, respectively to pick 2such that marginal damageequals marginal benefits, i.e.2=  − 1− 2⇔ 2= − 124 ANTICIPATED LEARNING 69We substitute this decision rule into the first period objective, take the ex-pected value, and then maximize with respect to 1using the period-2 firstorder condition. The resulting first period first order condition is0(1)=−E½0³2(1˜)´21+ 0³˜ − 1− 2(1˜)´³1+21´¾⇔ 0(1)=E0Ø − 12! (70)We obtain the second line by using equation (69) to cancel equal terms.Comparing the first order conditions here with those of Chapter 3 enablesus to determine h ow the anticipation of learning (or the early resolution ofuncertaint y ) changes consumption (extraction/emissions) in the first period.We denote the optimal consumption levels under uncertainty without learn-ing as 1and 2. We intitially assume that the decision maker is pruden t,i.e. 000 0. By equation (62) we know that the optimal decisions satisfy0(1)=0(2)=E0(˜ − 1− 2)  (71)The first order condition under uncertainty without learning is0(1) − E 0(˜ − 1− 2)= (72)0(1) − E∙120(˜ − 1− 2)+120(2)¸=0The assumed convexity of 0impliesE∙120(˜ − 1− 2)+120(2)¸ E 0µ12(˜ − 1− 2)+122¶=E0Ø − 12!The only way to satisfy the first order condition (70) for the learning sce-nario is by increasing 1above the level 1; this change decreases 0(1)andincreases E 0(˜−12) until both of them are equal for some 11.It is worth remembering ho w this argument implicitly uses the second ordercondition. The second order condition for maximization implies that the4 ANTICIPATED LEARNING 70graph of the first order condition under uncertaint y without learning (thesolid graph in Figure 1), as a function of 1,hasanegativeslope,atleastin the neighborhood of the equilibrium. Moving from the scenario with un-certainty and no learning to the scenario with learning causes this graph toshift upward, thus increasing the intersection of the graph