Caballero and Hammour, Macroeconomics of factor specificity, JPE 1998

Transcription

1 Caballero and Hammour, Macroeconomics of factor specificity, JPE 1998 Notation: Factors 1 and 2 (e.g., labor and capital) that can produce a single consumption good either in autarky or jointly. total employment of factor in autarky, ; total employment of in joint production; production function in autarky, ); -- the -th factor rental price in autarky, implying that is supply elasticity into joint production, i.e., ; Joint production combines units of 1 and units of 2 to produce units of output; (Note that and are small relative to total factor endowment) total number of joint production units; the mass of pre-existing production units formed before the start of the period; and are uncommitted factors; the mass of newly created joint production units, each produces of revenue; ) distribution of (heterogeneous) revenue of preexisting units, ; 0 minimum revenue needed for survival of preexisting unit; degree of specificity (share of factor that cannot be used outside of joint production after it has been invested. If factors separate, only can be used elsewhere); reward of factor in joint production; surplus in joint production; the probability that a unit of factor will be employed in joint production; Efficient equilibrium values are denoted by *.

2 Model: Identities: Total factor supply is fixed to 1, i.e., (1) (2) (3) (4) (5) Given, it follows that. (6) In efficient equilibrium, factors in new joint production are compensated according to their ex ante opportunity cost (as long as ): In the incomplete contracts equilibrium, factors need to produce surplus to compensate for their reduced opportunity cost: Assuming Nash bargaining solution with 50/50 shares, each factor s reward is: (7) (8). (9) In the efficient case, factor participation in joint production requires In order for factor to participate in joint production in the incomplete contract case, we need (10) (11)

3 To make matters simpler, we will assume (unless stated explicitly otherwise) that and. Then, always, 1 = 2 = =. This assumption preserves the most important results of the paper. What happens in equilibrium if? From (8),. Also, from (9), and. Obviously, and That is, the factor that does not experience specificity (factor 2) gets rents, while factor 1 doesn t get any rents in the interior equilibrium. Had factor 1 received rents, it would not have been equilibrium, because more units would have been formed. In other words, factor 1 is the limiting factor, for which the entry condition is binding. Preexisting units are assumed not to have any factor specificity. The probability that a unit of factor will be employed in joint production, λ = x i C/( U i 0 + x i D(y 0 )) (15)

4 1 1 (In the above, it should have been.)

5 Where does this assumption come from? Let x 1 <x 2. Then, the maximum number of joint units formed would be (1/x 2 ) with (1/x 1 1/x 2 )x 1 =1 x 1 /x 2 =U 1 units of factor 1 left over in autarky. Then, p 1 =F 1 (x 1 +(1 x 1 /x 2 ))=(1 x 1 1+x 1 /x 2 ) 1/ηi (x 1 /x 2 ) 1/ηi (if x 1 is small). Also, p 2 =F 2 (0)=1 and y n (x 1 /x 2 ) 1/ηi x 1 + x 2 =x 2 [1+(x 1 /x 2 ) 1+1/ηi ]. (Of course, this proposition makes sense only if we don t assume 2 =0.) Intuitively, factor 1 is appropriated in E and, therefore, it has less incentives to enter E. To induce it to enter E, its opportunity cost p 1 has to be reduced relative to the efficient

6 case. Also, factor 2 gets rents from E. To make these rents possible, its opportunity cost needs to decline (to assure p 2 <w 2 ). For both of these reasons, E i < E i *. This is particularly easy to see, given 2 =0. As we noted before, w 2 > p 2, which means segmentation. That is, some units of factor 2 would like to join E even if w 2 were lower than it is, but they cannot commit to not extracting rent from factor 1 when it comes to splitting the surplus. The number of slots open in joint production is determined by the appropriated factor s free-entry condition, and they are rationed among units of the appropriating factor (factor 2).

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8 Note that we have treated the extent of factor specificity as fixed. But over the long run both institutional arrangements and technology can affect factor specificity. Note that while each factor (say, capital and labor) may want to appropriate rents, there are limits to their desire to do so, because the appropriating factor becomes segmented and the more its winning members win, the more there are losing members of that factor (and the lower their compensation is).