Welfare Analysis

In this section, the welfare effect is analysed by supposing a central planner, who wishes

This planner is also uncertain about the profitability of the innovation. To avoid confusion, let p t be the planner’s belief that the innovation will decrease the production cost to c, . Each firm’s belief and cost parameters are known by the planner. To see which firm should adopt first from the welfare point of view, we need to calculate the welfare effect for each decision. Let W /’k denote the realised welfare for each stage

i = 1,2, each of LR’s decision j= a, w (a for adopting and w for waiting), and each possible result of the adoption k= g, b (g for success and b for failure). Define

W{J t =CSl' t + , which is the sum of consumer surplus CSf k and the firms’

realised profits With linear demand, we can easily calculate consumer’s surplus (= |( fo /« / output)1), and LR and SR’s realised profits can be derived similarly to the analysis of each firm’s adoption decision. The definition of W," * is explained in detail here and we leave the others for Appendix 2.2. W,“'* is stage l ’s welfare level when LR decides to adopt first («) and the planner thinks the adoption is going to be successful (g),

which is:

11 The welfare effect is restricted to a single industry and the assumption that the income effect is zero, that is, we are not using a general equilibrium approach.

to decide which firm should adopt first if it is to be better for the whole economy ' 1.

>1= Ut.SR

W’“" —"2 [</”.*+<7.v#(i/jt>1)] + {(^ <//j( ciQut}

(2.6)

—

(/‘¡jt is LR's first stage optimal output if it decides to adopt first (see footnote 26 for the

explicit form of ), and <7s»(9«,l) is SR’s reaction function for p = 1. Remember that in this case, the central planner thinks the adoption will be successful, so after LR’s adoption the planner will expect SR to update its prior to p = 1, adopt the technology and set its output as q aSR(q aLK, 1). Hence the consumer surplus will be j [ q a,j, + <7™ (<7^,1)] , which is half of the squared total output in the industry. The second term

{( A - q“iJt - qsK(q1J,A))q‘!j, - } is LR’s realised profit for this case, where the central

planner anticipates LR’s production cost to be c,q“,j ,. The definition for SR ’s realised profit { ( A - q ^ - q l R(qlKA))qsR(qautA ) - c xqaSK(q“lJ,A )\ can be explained in the same way.

For this central planner, LR adopting earlier will be better for the whole economy if:

P„( w r + W"*) + (1 - p ,)(w ; h + W2“h)> P '(W f* + ) + (1 - P ')(W f* + W2wh) (2.7)

By setting p K = p , we can calculate the planner’s cut-off belief p * , when there is no welfare difference for LR to adopt or to wait in the first stage. Unfortunately, due to the complication of function form52, it is not easy to derive any general implication from this cut-off belief. However, with the same numerical example (c,,c2, A) =(1,10,50) as Fig 1, we have Fig 2.

12 The explicit form of /<’ is:

/>,* = ( -lO c ./t + IS A cj - 5 A c - I3 c,c2 + I Icjc + 3(ic2c + 6 c 2 - I9c| - 2 1 c 2 - | I 9 6cJ - 4 2 ( M c J +25/tJc 4 - 12()4c ' + GOIc'jC2 - 156c2c? - 3 6 4 cjc, - 7 1 c Jc 2 + I3 2cc,5 - 5 0 4 c 5c, + 4 2 0 c jc 2 - l(X )8 c3r 5 -3(K I/t! < l C; + l(X)/t2c , c - l6 0 /tc ? c 2 + 3 2 0 Ac,2c + 7 7 0 4 c ,c | - 1 1 0 4 c ,c 2 - I 5 0 4 2c , c + 4 0 4 C jC 4 4 2 0 /tC jC 2 - 574« ¡t f c - 8 K Ocj2c,c + I76 4c2c , c ! - 6 2 0 4 c ,c 2c + l( X ) 4 2cp - I204cj’ + 2 2 5 4 2c |

1

W ''4

(jv lit3^ CO

When a (p,c) pair is located above p * , the central planner will think it optimal for LR to adopt first, and for SR to react to LR’s adoption result afterwards; when a (p,c) pair is below p* , the central planner will think it optimal for LR to wait, and respond optimally after observing SR’s adoption at stage one. Notice that in Fig 2, p ’g lies between p ’ and p ’ , indicating a gap between the planner’s desired equilibrium and the actual equilibrium. When (p,c) is in area A or C, the actual equilibrium coincides with the planner’s desired equilibrium.

Interesting implications come from areas B, and B2. Area B2 is the case when both the actual and desired equilibria are characterised by LR waiting and SR adopting in the first stage. In other words, there is no conflict between LR’s decision and welfare. Area

B, is the situation when the actual equilibrium describes LR to wait and SR to adopt in the first stage, but the desired equilibrium says LR should adopt earlier. In other words, area B, denotes a situation when LR’s optimal decision will cause welfare inefficiency. The intuition for the relatively optimistic attitude of the central planner is because the

expected cost increase from an unsuccessful adoption for the whole society is less than that for the LR firm alone. Hence, the area /¿, illustrates a need for policy intervention in firms’ adoption decisions about this cost uncertain innovation. An example of government interventions in technology adoption is in the water industry, where water price reforms are increasingly used to encourage improvements in irrigation efficiency (Green et al. (1996)).