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Chapter 20: Regulation and Deregulation |
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Equation 1
Q = f(L,K),
where Q is output, L is units of labor, and K is capital stock. Its profit is
Equation 2
= p(Q)Q - wL - uK = R(L,K) - wL -uK,
where p(Q) is the inverse demand curve, w is the wage rate, u is the user cost of capital, and R(L,K) is the revenue function R(L,K) =
Suppose the board decides the fair rate of return is v, so that, using Equation 2,
Equation 3
------ pkK | = | R(L,K) - wL - uK ------------------------ pkK |  v. |
We can rewrite this constraint as,
Equation 4
| R(L,K) - wL ----------------- pkK | v + | u ----- pk |  s. |
That is, this type of regulation requires that revenues left over after covering labor expenses per unit of capital cannot exceed s, where s is the cost of capital divided by pk plus the fair rate of return. If v were less than 0, then the firm would not produce. We assume in the following that v > 0, so that s > u/pk. [It is not necessary for v to be positive. If v = 0, = 0 and the firm covers costs because the user cost already incorporates a normal rate of return to capital. If v = 0, this solution resembles that of Ramsey pricing in which price equals average cost.]
The regulated firm's objective is to maximize profits, Equation 2, subject to the rate of return restriction implied by Equation 4. The firm's optimal behavior is determined by finding the saddlepoint of the Lagrangian
Equation 5
R(L,K) - wL - uK - 
[R(L,K) - wL - spkK],
where is the Lagrangian multiplier, and the term in brackets is obtained by multiplying Equation 4 through by pkK and rearranging terms.
If the L and K at the saddlepoint of Equation 5 are positive, and the constraint binds, then the Kuhn-Tucker-Lagrange conditions,
Equation 6
RL = w,
Equation 7
RK = u - ((spk - u)/(1 - )),
Equation 8
R - wL - spkK = 0,
Equation 9
> 0,
determine the values of L, K, and . Equation 6 says that the value of the marginal revenue product of labor equals the wage. Equation 7 equates the value of the marginal revenue product of capital to the cost of capital plus an adjustment factor that depends on the fair rate of return (v =
If the constraint does not bind ( = 0), Equations 6 and 7 are the usual profit-maximizing equations of the unregulated monopoly:
Equation 6'
RL = w
Equation 7'
RK = u.
That is, the value of the marginal revenue product of labor equals the wage, and the value of the marginal revenue product of capital equals the per-unit user cost of capital. The nonconstrained ROR, then, is
Equation 10
| s0 = | R0 - wL0 -------------- K0 | . |
For the ROR constraint to matter, s0 must exceed s, which, by assumption, exceeds u/pk.
Takayama (1969) shows that L and K are continuous functions of s. Differentiating Equation 8 with respect to s, we obtain
Equation 11
| dL --- ds | (RL - w) + | dK --- ds | (RK - spk) = pkK. |
Evaluating at s = s0 (the unconstrained, profit-maximizing rate of return) and substituting using Equations 6 and 7, this equation may be rewritten as
Equation 12
| dK --- ds | = | pkK0 ----------- u - s0pk | < 0, |
because s0 > u/pk. Thus, introducing an active fair-rate-of-return constraint (that is, lowering s from s0) must increase capital, which is the Averch-Johnson effect.
Takayama, Akira. 1969. "Behavior of the Firm Under Regulatory Constraint." American Economic Review 59:255-60.
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