Federal Communications Commission

 Office of Strategic Planning and Policy Analysis

 445 12th Street, SW 

 Washington, DC 20554

OSP Working Paper Series 

40 Dynamic Pricing and 

Investment from Static 

Proxy Models 

 September 2003

David M. Mandy 

 William W. Sharkey 
 The FCC Office of Strategic Planning and Policy Analysis Working Paper Series 

presents staff analysis and research in various states. These papers are intended to stimulate 

discussion and critical comment within the FCC, as well as outside the agency, on issues in 

telecommunications policy. Titles may include preliminary work and progress reports, as 

well as completed research. The analyses and conclusions in the Working Paper Series are 

those of the authors and do not necessarily reflect the views of other members of the Office 

of Strategic Planning and Policy Analysis, other Commission staff, or the Commission 

itself. Given the preliminary character of some titles, it is advisable to check with authors 

before quoting or referencing these Working Papers in other publications. This document is 

available on the FCC's World Wide Web site at http://www.fcc.gov/osp/. The inside back 

cover contains a partial list of previous titles. 
Dynamic Pricing and Investment from Static 

Proxy Models†

David M. Mandy*

William W. Sharkey**

Abstract 

This paper evaluates the use of static cost proxy models in setting forwardlooking prices, such as the prices set according to the FCC’s TELRIC methodology. First, 

it compares the time paths of prices and depreciation under traditional regulatory 

accounting with the prices and depreciation implied by various versions of TELRIC. 

When TELRIC prices are recomputed at intervals shorter than asset lives, the firm will 

generally not earn the target rate of return. In these cases, a correction factor must be 

applied to the TELRIC price path in order for revenues to exactly recover investment 

cost, including the target rate of return. Next, the paper considers a firm’s cost 

minimizing investment decisions under two different assumptions about asset 

obsolescence. In both scenarios, cost minimizing investment paths and implied utilization 

rates for the firm’s assets are derived under a variety of assumptions about the relevant 

input parameters. Some implications for TELRIC pricing are then derived. 

 

†

 The authors acknowledge helpful discussions with Jay Atkinson during preparation of this paper. We also 

benefited from the comments of Simon Wilkie and Donald Stockdale on an earlier draft. The views 

expressed in this paper, however, are those of the authors alone and do not necessarily reflect the views of 

the Federal Communications Commission, any Commissioners, or other staff. 

*

 Department of Economics, University of Missouri, 118 Professional Building, Columbia, MO 65211; 

MandyD@missouri.edu. 

** Office of Strategic Planning and Policy Analysis, Federal Communications Commission, Washington, 

D.C. 20554; William.Sharkey@fcc.gov. 
Dynamic Pricing and Investment from Static Proxy Models: 

Executive Summary

Since release of the FCC’s Local Competition Order,

1

 incumbent local exchange 

providers have strongly objected both to TELRIC as a methodology and to its specific 

implementation by state regulatory commissions. These objections generally fall into 

three categories. First, incumbent carriers have questioned the theoretical consistency of 

the TELRIC methodology itself. Specifically, they have argued that, when equipment 

prices are falling over time, periodic recalculation of TELRIC prices prevents them from

recovering their investment costs. Second, incumbents have criticized the way in which 

state commissions have implemented TELRIC. They argue either that states have adopted 

unrealistic input values or that the cost models that were adopted contained flawed model 

algorithms.

2

 Finally, some incumbents have argued that any forward-looking pricing 

methodology, including TELRIC, in which the computation of cost is not based on the 

incumbent’s “actual network” as a starting point, is inconsistent with the meaning of the 

1996 Act. 

This paper addresses aspects of each of the above criticisms. Specifically, section 

2 addresses the argument that periodic recalculation of TELRIC prices prevents 

incumbents from recovering the cost of their investment. It shows that, if investment 

costs are falling over time, and the period between TELRIC price adjustments is shorter 

than asset lives, then traditional TELRIC pricing will not permit incumbents to recover 

the cost of their investment. Similarly, when investment costs are rising, TELRIC pricing 

will result in an over-recovery of investment costs. Section 3 examines a carrier’s cost

minimizing investment plan in a dynamic environment with lumpy investment costs. 

While not intended to recommend specific input prices for cost models, this section is 

directly relevant to the selection of forward-looking fill factors for such models. As such, 

it may partially address incumbent’s arguments that state commissions have set 

unrealistically high fill factors.3

Dynamic Pricing of an Already-Determined Investment 

Section 2 is concerned with comparing alternative time paths of prices that will 

allow a firm to exactly recover the cost of network investments when these costs are 

expected to change over time. Incumbent carriers are correct to point out that, when 

TELRIC prices are recomputed at periodic intervals, particular care must be exercised in 

order to ensure that the resulting prices are consistent with the assumptions made about 

asset lives and the allowed rate of return. Indeed, when investment costs are falling over 

 

1 Implementation of the Local Competition Provisions in the Telecommunications Act of 1996, First Report

and Order, 11 FCC Rcd 15499, 15509 (1996). 2

 For example, incumbents argue that proxy models often assume an unrealistic level of sharing of the cost 

of structures (i.e. poles and underground conduit facilities) with other utilities. It is also argued that cost 

minimization procedures in proxy models assume an unattainable level of efficiency that no real world firm

could expect to emulate. The results of this paper are relevant to the setting of one input factor – the firm’s 

utilization rate of existing capacity (also called a fill factor). 3

 The third criticism is also partially addressed at the end of Section 3.3.

 1 
time and TELRIC price reviews are conducted at intervals shorter than expected asset 

lives, the firm will earn less than its target rate of return under traditional 

implementations of TELRIC. Specifically, a TELRIC price is traditionally computed as a 

constant (or levelized) price that, over the life of the asset, will earn sufficient revenue to 

recover the full cost of investment including the target rate of return. When TELRIC 

prices are recomputed at intervals shorter than these asset lifetimes, and investment costs 

fall between price reviews, actual revenues will not fully compensate the firm. The paper 

shows, however, that it is possible to compute a correction factor based on estimates of 

the rate of change of investment cost and the expected time between TELRIC reviews. 

Corrected TELRIC prices rise or fall at the same rate over time as underlying investment 

cost, and allow the firm to earn its target rate of return no matter how frequently price 

reviews are conducted. Moreover, no ad hoc adjustments of assumed asset lifetimes or 

depreciation schedules are needed to implement the suggested correction. 

These points may be illustrated in the following diagrams which represent 

different price paths for a hypothetical investment of $100 in an asset that is assumed to 

have an economic lifetime of 12 years. Figure ES1 compares price paths based on 

traditional regulatory accounting and levelized TELRIC pricing. Both paths assume an 

allowed rate of return of 11.25%, and the traditional price path (adjusted annually) 

assumes straight line depreciation over the 12 year life. Traditional prices fall even under 

straight line depreciation because price in each period reflects operating expense + 

depreciation expense + capital cost. The first two terms are constant by assumption, and 

the capital cost term declines over time at the same rate as the undepreciated balance. The 

levelized TELRIC price is by definition constant over the life of the asset and is set at a 

level such that the present value of revenues equals the investment cost of $100.4

1 2 3 4 5 7 8 9 10 11 12

12

14

16

18

20

22

6 

Figure ES1. Traditional Prices versus a Levelized TELRIC Price

 

4

 Note that, due to levelization, a depreciation schedule does not enter in any way into the TELRIC 

computation in the absence of tax effects. If tax consequences are accounted for, then the TELRIC price is 

set at a level such that the present value of after-tax revenues is equal to the present value of after-tax 

investment costs, where the latter accounts for the deductibility of depreciation expense and an appropriate 

weighted average cost of capital is used. As a result, more accelerated tax depreciation schedules lower the 

levelized TELRIC price. 

 2 
Figure ES1 was constructed without any reference to the rate of change of 

investment costs, since neither the levelized TELRIC price nor the traditional price 

depends on future changes in these costs. Suppose, however, that investment costs are 

expected to fall at a rate of 10% per year. In a competitive market, prices would also be 

expected to fall at a rate of approximately 10% per year, since the price at any given time

would reflect the cost of newly incurred investments made at that time. In order to 

recover their investment costs, potential entrants therefore anticipate setting prices 

according to a declining price schedule, so that prices in early years recover more of the 

investment cost than prices in later years. 

If TELRIC price reviews are conducted periodically, using the current investment 

cost at the time of the review, then TELRIC prices will also decline over time at a rate of

approximately 10%. However, since TELRIC prices were defined to allow the firm to

break even assuming a levelized price schedule for the entire life of the asset, when there 

are periodic reviews and declining investment costs, the TELRIC prices will no longer 

recover the firm’s investment cost including its target rate of return. Figure ES2 compares 

a price schedule that allows competitive firms to earn exactly the target rate of return 

(dotted line) and TELRIC prices (solid line), assuming TELRIC reviews are conducted 

every three years. 

1 2 3 4 5 6 7 8 9 10 11 12

10

15

20

25

Figure ES2. Competitive Prices versus TELRIC Prices 

Reviewed Every Three Years

It is a simple matter, however, to define a correction factor based on: (1) the 

assumed rate of change of investment cost, and (2) the period between TELRIC reviews. 

In the current example, TELRIC prices must be increased by 35% in every year. The 

corrected TELRIC price path is shown in Figure ES3. By construction, corrected 

TELRIC prices recover the firm’s full investment costs, including its required rate of 

return. In addition, TELRIC prices fall at approximately the same rate as the breakeven 

 3 
prices for competitive firms, with differences only in the assumed timing of the price 

changes.5

1 2 3 4 5 7 8 9 10 11 12

10

12.5

15

17.5

20

22.5

25

6

Figure ES3. Competitive Prices versus Corrected TELRIC Prices 

Efficient Investment over Time 

Section 3 of the paper is concerned with deriving cost minimizing investment 

plans in a dynamic environment. This section of the paper seeks to address in part the 

second and third objections raised by incumbents to TELRIC pricing. First, by deriving 

an efficient forward-looking investment plan, the paper is simultaneously solving for an 

efficient utilization rate over time. Since assumed utilization rates are an important input 

to cost proxy models used to estimate TELRIC prices, these results are potentially 

relevant in setting appropriate utilization input rates in a forward-looking context. For 

example, using a cost function derived from outputs of the FCC’s Synthesis Model to 

represent loop investment costs, the paper demonstrates how to compute an efficient 

investment plan for loop plant over a 60 year time horizon. This plan is illustrated by the 

step function in Figure ES4. 

 

5

 Section 2 also demonstrates that when investment costs are increasing over time (e.g. due to general price 

inflation) then periodically reviewed TELRIC prices will lead to an over-recovery of investment cost. In

this case the correction factor will lead to a reduction in TELRIC prices. 

 4 
10 20 30 40 50 60

t

10000

15000

20000

25000

30000

Capacity

Figure ES4: Demand and Efficient Capacity for Loop Plant 

Using these computational results, the paper then derives a price schedule that 

would allow efficient entrants to earn the target rate of return, which is labeled a 

“contestable” price path.6

 A comparison of the contestable price schedule with the price 

schedule implied by TELRIC, as actually implemented, reaches similar conclusions to 

those of section 2. Namely, the periodically reviewed TELRIC prices follow the same

time path as contestable prices but are lower (higher) than contestable prices when the 

underlying investment costs are falling (rising). The case of falling investment costs is 

illustrated in Figure ES5.7

5 10 15 20 25 30

t

2

4

6

Price cccccccccccccccc Month

Figure ES5. Monthly Contestable and Triennially Reviewed Actual TELRIC Prices 

Distribution Portion of Loops: Case of Decreasing Investment Cost 

 

6

 A contestable market is a market in which entry barriers are non-existent. Hence, profits are driven to zero

for all active firms operating in the market. A contestable market generalizes the definition of a competitive 

market since it can apply to markets in which technologies include substantial economies of scale. See

William J. Baumol, John C. Panzar and Robert D. Willig, Contestable Markets and the Theory of Industry 

Structure, New York: Harcourt Brace Jovanovich, (1982). 7

 In Figure ES5, actual TELRIC prices are represented by the lower step function with adjustments made 

every three years. Contestable prices, represented by the upper step function, are assumed to be set at the 

beginning of each year and to be adjusted annually. 

 5 
Section 3 also shows how to compute efficient investment plans based on existing 

capacity levels of incumbent firms. To the extent the initial capacity levels can 

approximate the existing investment base of incumbent carriers, these results may 

reconcile TELRIC price computations with “actual cost” computations favored by 

incumbents. 

 6 
1. Introduction 

In February 1996 the U.S. Congress passed the Telecommunications Act of 1996, which 

significantly amended the Communications Act of 1934. The goals of the 1996 Act were 

to establish a “pro-competitive, deregulatory national policy framework” for local 

telephony competition.8

 Implementation of the 1996 Act by the Federal Communications 

Commission (FCC) included development of a new regulatory pricing standard known as 

“Total Element Long Run Incremental Cost” (TELRIC).9

 TELRIC has been used to 

price unbundled network elements and interconnection,10 and has been widely criticized 

both as a conceptual framework and as actually applied by regulators.11 An underlying 

current in many of these criticisms is the static nature of TELRIC pricing as implemented 

to date, when actual telecommunications industry costs are highly dynamic. 

This paper considers how TELRIC pricing can be correctly implemented in a 

dynamic environment. We do this by exploring first, in Section 2, how prices that are 

periodically reviewed can be calculated to recover the cost of a given investment in 

physical equipment. Second, Section 3 studies how dynamically efficient investment 

plans can be calculated, thereby giving the costs to be recovered via the pricing rules 

discussed in Section 2. Together, Sections 2 and 3 move beyond the criticisms and 

substantially toward theoretically sound TELRIC pricing rules that can actually be 

applied in practice. In contrast, the existing literature is long on criticisms of TELRIC

but short indeed on suggesting rules and calculations thereof that show how TELRIC 

prices as currently implemented might be modified to address some of the criticisms.

12 

Our contribution is to take this next step, with particular emphasis on correctly capturing 

the dynamics of the industry and the sense in which costs and prices should be forwardlooking.

One way to view TELRIC is as a new form of incentive regulation. Traditional 

incentive regulation tries to induce socially efficient behavior from a franchise 

monopolist. The traditional approach was precluded by the 1996 Act, however, which 

nullified the state and local statutes that historically granted franchise monopoly status to 

 

8

 See Joint Managers’ Statement, S. Conf. Rep. No. 104-320, 104th Cong., 2d Sess. 113 (1996) at 1. 9

 See Implementation of the Local Competition Provisions of the Telecommunications Act of 1996

(hereinafter Local Competition Order), 11 FCC Rcd 15499, 15509, and 15812-15922 (1996). 10 The 1996 Act envisioned three alternative methods for competitive entry: construction of new network

facilities, resale of the incumbent firm’s retail services, and leasing of unbundled network elements from

the incumbent. The 1996 Act also required incumbents to provide interconnection. See 47 U.S.C. §§

251(c)(4) and 252(d)(3). 11 See, for example, Hausman (1997); Kahn (1998, 2001); Sidak and Spulber (1998); Alleman (1999);

Kahn, Tardiff, and Weisman (1999); Salinger (1999); Weisman (2000); Mandy (2002); Tardiff (2002); 

Weisman (2002), and Weingarten and Stuck (2003). Although these studies differ in their points of

emphasis, all are critical of at least some aspects of TELRIC in theory or practice. 12 Exceptions are Hausman (1997), Mandy (2002), Tardiff (2002) and Weingarten and Stuck (2003). 

However, even these papers do not calculate prices and costs under the variety of possible assumptions

considered herein, and only Mandy (2002) presents evidence on dynamic and forward-looking price and

cost paths that might form a conceptually sound basis for revisions to TELRIC pricing as currently

practiced. 

 7 
incumbents, and charged the FCC and state regulators with creating rules under which 

new competition would operate.13 Under the “incentive regulation” view, by adopting 

TELRIC, the FCC tried to create a market structure in a highly concentrated industry that 

provides incentives for efficient investment decisions similar to those present in 

competitive industries with many small suppliers. Under this view, TELRIC tries to 

induce socially efficient behavior from firms that may have significant market power, 

thereby sharing the same goal as traditional incentive regulation. But unlike traditional 

incentive regulation, TELRIC must pursue this goal outside of the franchise monopoly 

market structure. TELRIC therefore tries to induce efficient investment decisions, 

including efficient entry and exit, in an industry that, because of high concentration and 

substantial scale economies, might experience very little (efficient) entry and exit without 

such inducement. There is substantial disagreement over whether TELRIC succeeds in 

promoting efficient investment by both entrants and incumbents, or whether it is even 

possible for price regulation to achieve this objective.14 However, even those most 

critical of the TELRIC methodology agree that the 1996 Act “... asks regulators to create 

prices that will induce appropriate new entry.”15 In other words, Congress charged the 

FCC and state regulators with incentive regulating entry and exit, and doing so via price 

regulation.

The strong objections of some writers, as well as incumbent local exchange 

carriers (ILECs), to TELRIC as a methodology and to its specific implementation by state 

regulatory commissions fall into three categories. First, some have questioned the 

theoretical consistency of the TELRIC methodology itself. Specifically, ILECs have 

argued that, when equipment prices are falling over time, periodic recalculation of 

TELRIC prices prevents them from recovering their investment costs. Second, some have 

criticized the way state commissions have implemented TELRIC. They argue either that 

states have adopted unrealistic input values or that the cost models adopted contained 

flawed model algorithms.16 Finally, some have argued that any forward-looking pricing 

methodology, including TELRIC, in which the computation of cost is not based on the 

incumbent’s “actual network” as a starting point, is inconsistent with the meaning of the 

1996 Act. 

This paper examines these claims.

17 Section 2 studies whether TELRIC, as 

implemented, permits cost recovery of investments in physical equipment. We show that 

the claim of potential under-recovery is correct. When investment costs are falling over 

time and TELRIC price reviews are conducted at intervals shorter than expected asset 

 

13 47 U.S.C. § 253. 14 See the references cited in footnote 9. 

15 Justice Breyer (Verizon v. FCC, Dissent: 22), as quoted by Weisman (2002). 16 For example, ILECs argue that proxy models often assume an unrealistic level of sharing of the cost of

structures (i.e. poles and underground conduit facilities) with other utilities. They also argue that cost 

minimization procedures in proxy models assume an unattainable level of efficiency that no real world firm

could expect to emulate. The results of this paper are relevant to the setting of one input factor – the firm’s 

utilization rate of existing capacity (also called a fill factor). 17 We do not, however, evaluate the realism of proxy model algorithms or attempt to recommend specific 

input values for use in proxy model computations. In addition, we do not specifically consider the

conditions under which unbundled network elements should be offered under TELRIC prices. 

 8 
lives, a firm will earn less than its target rate of return under the TELRIC methodology as 

generally implemented. A TELRIC price is usually computed as a constant (or levelized) 

price that, over the life of the asset, will earn sufficient revenue to recover the full cost of

investment including the target rate of return. This paper shows that, when TELRIC 

prices are recomputed at intervals shorter than these asset lifetimes, and investment costs 

fall between price reviews, actual revenues will not fully compensate the firm. The paper 

further shows that it is possible to compute a correction factor based on estimates of the 

rate of change of investment cost and the expected time between TELRIC reviews. 

Corrected TELRIC prices rise or fall at the same rate over time as underlying investment 

cost, and allow the firm to earn its target rate of return no matter how frequently price 

reviews are conducted. Moreover, no ad hoc adjustments of assumed asset lifetimes or 

depreciation schedules are needed to implement the suggested correction. 

Section 3 of the paper partially addresses the second and third objections raised to 

TELRIC pricing. We consider two stylized assumptions about asset replacement. Under 

one scenario, which we call “expected innovation date replacement,” we assume the 

physical life of an asset is effectively infinite, in the sense that technical advances make it 

obsolete before it suffers substantial physical deterioration, so that assets are eventually 

replaced due to technological advances. In the second scenario, which we call “light bulb 

performance,” we assume each asset has a fixed lifetime. In both scenarios, we derive 

cost minimizing investment plans over assumed planning horizons. These results are 

relevant to TELRIC cost studies and cost models generally in that they provide a method 

by which forward-looking capacity utilization factors (an important input in cost models) 

can be derived. Section 3 also shows how to compute efficient investment plans starting 

from existing capacity levels of an incumbent firm. To the extent that initial capacity 

levels can approximate the existing investment base of ILECs, these results may 

reconcile TELRIC price computations with “actual cost” computations favored by ILECs 

and some writers. Based on our derivation of efficient investment plans in the light bulb 

performance scenario, we also compare TELRIC pricing rules, as generally implemented, 

with prices defined by an efficient contestable entry criterion. 

This paper does not address the criticism, raised by some observers and ILECs, 

concerning the appropriate cost of capital to be used in calculations like those performed 

herein. Specifically, some

18 have argued that a competitive environment exposes ILECs 

to more risk than the former franchise monopoly environment. They argue that higher 

risk, including the risk associated with foregone option value when an investment is 

made, must be compensated through higher input values for the firm’s cost of capital. 

The analysis herein makes no judgments regarding these arguments. Throughout, we 

treat the cost of capital as an exogenous input that has been derived via some other 

analysis. 

Both sections 2 and 3 are based on extensive computations using MathematicaTM, 

and details of the relevant programming code are available from the authors upon request. 

Some of the results of section 2 have appeared elsewhere in the academic literature,19

 

18 See Hausman (1997). 19 See, for example, Biglaiser and Riordan (2000), Mandy (2002) and Salinger (1998). 

 9 
though they seem to be less familiar to practitioners of regulation. We view the paper’s 

simulation results as tutorial in nature, and as potential computational aids for regulatory 

commissions faced with setting actual TELRIC prices. Section 3 presents several new 

results on the computation of cost minimizing investment plans and the derivation of 

pricing rules based on these plans. 

2 Dynamic Pricing of an Already-Determined Investment and Usage 

Plan 

The objective of this section is to compare the time paths of prices and depreciation under 

traditional regulatory accounting with the prices and depreciation implied by various 

versions of forward-looking pricing methods such as TELRIC. The analysis in section 2 

assumes a single asset type with a fixed and known life of L time periods. An initial 

investment is made at time t = T ≥ 0 at a cost of FT. This investment installs a fixed 

productive capacity per unit of time normalized for convenience to equal one unit of 

output. The firm's cost of capital is constant and equal to r (or 100r %) and its 

corresponding discount factor is δ = 1/(1+r). The asset may be under-utilized at any 

particular time. The percentage utilization (fill factor) at time t is qt, which is the same as 

the quantity produced and sold given the normalization. 

Several assumptions are needed for expositional ease and to focus attention on the 

dynamics of pricing. We assume there is no physical depreciation and no operating cost 

during the life of the asset. We also assume the asset has no salvage value, either positive 

or negative, at the end of its life. Finally, we ignore taxes. All of these realistic features 

can be incorporated without changing the important aspects of the analysis. 

While the analysis is formally concerned with the determination of a time path of

prices that recovers the investment cost of a specific asset (e.g. a switch installed in a 

particular central office), it can be more generally applied to the determination of pricing 

rules for entire classes of network elements (e.g. all switching facilities offered by a 

particular carrier in a particular region). Since the analysis assumes a fixed and known 

asset life of length L, it is necessary that individual network elements priced according to 

the principles derived below be reasonably homogeneous assets with roughly comparable 

asset lives. While TELRIC models are at times employed to estimate the total forward 

looking cost of local exchange service, in order to use the results of this section it is 

necessary to separately compute component price schedules for each class of 

homogeneous assets. The components can then be aggregated, if desired, to estimate the 

forward-looking cost of local exchange service in total. 

It is important to note that the utilization rate qt can be either a traditional fill 

factor as used in a static proxy model, or alternatively a variable rate of utilization of a 

particular asset that might result from an investment made in anticipation of future 

demand growth. Throughout most of section 2 we maintain the assumption of a fixed 

utilization rate. Section 2.8 illustrates how certain price and depreciation schedules may 

 10 
change when a variable utilization rate is assumed, although the substantive conclusions 

reached in earlier sections continue to hold in the more general case. 

2.1 The Economic Depreciation Identity 

There is a fundamental inviolable relationship between prices, utilization, economic 

depreciation, and the firm's discount factor (i.e., the firm's cost of capital). This 

relationship holds by definition, and is derived as follows. Let pt denote the price 

received by the asset owner at time t for the asset’s services, based on the convention that 

all variables are set at the beginning of each period. Then, once the initial investment FT

is made, the value of the asset at any time t ≥ T is V . j j

j t T L

j t t p q − + −

∑ = = 1

δ 20 By definition, 

the economic depreciation of the asset at time t, dt, is equal to 

(1) ( ) ( ) ∑ + −

= +

− + = − + = − − 1

1

1

1 1 T L

j t j j

j t

t t t t t d V V p q δ δ p q for t = T, ... , T+L-1.

Equation (1) can be rewritten as ( ) ( ) ∑ + −

= +

− + = + − 1

1

1 1 T L

j t j j

j t

t t t p q d δ δ p q

+ −1

 so that at 

time t = T+L-1 we have + −1 + −1 = T L d T L T L p q . Recursive substitution then gives 

(2) ( )

t

T L

j t t j

t

T L

j t t j

t q

d r d

q

d d

p

∑ ∑ + −

= +

+ −

= + + = + − =

1

1

1

1

1 δ δ

 for t = T , ... , T+L-1.

By equation (1), for any price path pT , ... , pT+L-1 specified by a regulator or 

imposed by competitive conditions, the economic depreciation at every time in the asset’s 

life is uniquely determined. In this form, the relationship dictates the economic 

depreciation that will occur at time t as a function of the current and future prices that are 

actually received, current and future expected utilization, the cost of capital r = (1 - δ)/δ

allowed by the regulator or required by financial markets, and the life of the asset. 

Alternatively, equation (2) expresses the fact that, if a path of depreciation dT, ... , 

dT+L-1 is specified by the regulator, then there is a unique price path which ensures that 

the chosen depreciation amounts are economic depreciation. This relationship dictates the 

price that must be charged at time t as a function of current and future economic 

depreciation allowed by the regulator, current utilization, the cost of capital allowed by 

the regulator or required by financial markets, and the life of the asset. Equation (2) is 

the traditional (operating expense + depreciation + cost of capital) regulatory pricing 

formula, where operating expense is absent by assumption, and ∑ + −

= +

1

1

T L

j t δ d j r is the 

opportunity cost of carrying the as-yet un-depreciated balance from time t to time t + 1. 

When equations (1) and (2) hold, the firm is by definition earning a rate of return equal to 

its target rate of return r. 

 

20 This expression explicitly depends on the absence of operating costs, salvage value, and taxes. All of

these can be introduced here at the expense of additional complexity. 

 11 
If we assume that a price path and a depreciation schedule are both specified by a 

regulator or a combination of a regulator and competitive conditions, then equations (1) 

and (2) may no longer hold, and the firm’s target rate of return can no longer be 

independently specified by a regulator or by financial markets. The implied rate of return 

from time t to the end of the useful life, however, is given implicitly by rewriting 

expression (2) (taking its present value from time t to the end of the life) as

21

(3) [ ( ) 1 ] 0 1

1 ∑ − − − = 1

∑ + −

=

+ −

= +

− T L

i t

T L

j i i i i j

i t δ p q d δ d for t = T, ... , T+L-1.

Although this final expression cannot be solved explicitly for δ (unless the 

allowed prices and depreciation, and expected utilizations, are solvable functions of 

time), it implicitly dictates the cost of capital (i.e., discount factor) that will be earned 

from time t through time T+L-1 as a function of current and future economic depreciation 

allowed by the regulator, current and future prices allowed by the regulator or imposed 

by competitive conditions, current and future expected utilization, and the life of the 

asset. In particular, this expression determines the cost of capital actually earned, even if 

a regulator uses some other cost of capital in calculating allowed prices and allowed 

economic depreciation. Whether the cost of capital actually earned is consistent with that 

used by a regulator in calculating prices and/or depreciation depends on whether the 

regulator's price and depreciation calculations are consistent with this identity. The cost 

of capital actually earned can be found as a numeric solution to equation (3), using 

MathematicaTM or similar software packages. 

2.2 Three Stylized Scenarios 

For illustration purposes we consider three stylized scenarios. In each scenario we 

assume that an initial investment is made at time T = 0, at a cost of F0 =100. This cost is 

assumed to change at a constant annual rate λ (or 100λ %) in years t > 0. The cost of 

capital is r = 11.25% per year. In the first scenario λ is zero. This is a benchmark case 

that we use to compare traditional regulatory pricing (i.e., equation 2) with TELRIC 

pricing, without the complications created by cost dynamics.22 In the second scenario λ

is negative. Such initial investment cost decreases might be caused, for example, by 

technological improvements embodied in the firm's capital assets. This scenario can be 

thought of as representative of switching assets. In the third scenario λ is positive. Such 

initial investment cost increases might be caused, for example, by general inflation. This 

scenario can be thought of as possibly representative of outside plant. In the remainder of 

this section we let γ = 1 + λ, so the initial investment cost at any time t = T is γ

 TF0. 

 

21 At the time of investment, T, this reduces to ∑ ∑ + −

=

+ −

=

− = T L 1 1

j T

T L

j T j j j

j T δ p q d . 

22 We note that under traditional regulatory accounting, prices are based on the traditional interpretation of 

economic depreciation (i.e., equation (2)) applied to an aggregation of all the firm’s assets. Since under

historical cost accounting, assets of different vintages are included in this calculation, the specific price and 

depreciation schedules discussed below are unlikely to hold in the aggregate. 

 12 
2.3 Benchmark Case: Traditional Revenue Recovery versus TELRIC 

For the benchmark case, in addition to assuming that λ = 0, we use an economic life L = 

12 and a utilization rate qt = 0.8 for all t = 0, ..., L-1.

2.3.1 Traditional Price Path Based on Straight Line Depreciation 

Under traditional historical cost-based regulatory accounting, both investments and 

operating expenses for the regulated firm are recorded on an annual basis.23 Price in each 

year for the life of a given asset is then computed as the sum of operating expense plus 

capital cost, where capital costs are equal to the sum of depreciation and a return on undepreciated assets. If straight-line economic depreciation is assumed then dt = F0 / L for t 

= 0, ..., L-1. Hence, (since we are ignoring operating expenses) the price that must be 

charged at time t to implement straight-line depreciation, while allowing cost of capital r 

= (1−δ)/δ, is found by substituting the expression for straight-line depreciation into 

equation (2). This price path is illustrated in Figure 1. 

1 2 3 4 5 7 8 9 10 11 12

12

14

16

18

20

22

6

Figure 1. Traditional Prices under Straight Line Depreciation: 

Case of Constant Investment Cost 

Assuming straight-line depreciation, the price decreases over time because the as-yet undepreciated balance decreases over time, and consequently the cost of capital applied to 

that balance decreases over time. 

2.3.2 Levelized TELRIC Price for the Life of the Asset 

While traditional pricing is fundamentally backward-looking, TELRIC pricing is 

designed to be forward-looking. In addition, TELRIC prices can be defined in various 

ways, depending on the extent to which regulators incorporate expectations about the 

future. A TELRIC price is defined so as to recover the initial investment cost F0 when the 

 

23 This representation of traditional regulatory accounting is admittedly a caricature. Rate reviews do not 

necessarily occur annually, and in some cases explicit levelization of rates is imposed. 

 13 
utilization path is qj, provided the price remains in effect for the life of the asset. The 

basic formula for the level TELRIC price, pTELRIC0, that recovers an initial investment 

cost F0 made at time 0 over an asset life L and with discount factor δ, is: 

(4) 

∑ −

=

= 1

0

0

0 L

j j

j

q

F pTelric

δ . 

Figure 2 shows a comparison of the levelized TELRIC price with traditional prices based 

on straight-line depreciation. 

Equation (4) differs in two ways from TELRIC calculations used in practice. 

First, many existing cost models, such as the FCC’s Synthesis Model, include 

assumptions on depreciation schedules to derive the numerator of (4).24 This is 

superfluous since the present value of the actual investment is known to be F0 (it has to 

be known before it can be depreciated and then expressed in present value). Actual 

economic depreciation when the level TELRIC price (4) is received over the life of the 

asset is given by (1) evaluated at the price (4), even if some other depreciation schedule is 

used to derive the numerator F0. Second, variable utilization rates are not assumed in the 

FCC’s Synthesis Model. The impact of variable utilization rates is studied in section 2.8 

below.

1 2 3 4 5 7 8 9 10 11 12

12

14

16

18

20

22

6

Figure 2. Levelized TELRIC versus Traditional Prices: 

Case of Constant Investment Cost 

The levelized TELRIC price implies a path of economic depreciation given by 

equation (1) above. The resulting depreciation schedule is necessarily rising, given a 

constant utilization rate, since the first term in equation (1) is constant while the second 

term is increasing in t. Figure 3 shows that levelized TELRIC pricing substantially 

 

24 In particular, depreciation schedules are used to derive the capital carrying cost for each year in an asset’s 

life. However, the resulting annual charge factor used by the model uses only the present value of this 

schedule. 

 14 
decelerates economic depreciation compared with traditional pricing based on straightline depreciation. 

1 2 3 4 5 7 8 9 10 11 12

6

8

10

12

14

6

Figure 3. Depreciation under Levelized TELRIC and Traditional Pricing: 

Case of Constant Investment Cost 

Whenever the firm’s price schedule and depreciation schedule are chosen to 

satisfy equations (1) and (2), the firm’s earned rate of return from the time the asset is 

acquired is necessarily equal to the target cost of capital used by the regulator in 

computing these schedules. This can be verified by computing explicit numerical 

solutions of equation (3). For any t, 0 ≤ t ≤ L, if TELRIC prices are set according to 

equation (4) and the depreciation schedule satisfies equation (1) for these prices, then the 

earned rate of return from time t to the end of the asset’s life is equal to 11.25%. The 

same result holds when the depreciation schedule is straight line and prices are set to 

satisfy equation (2) for these depreciation values. In addition, even though the levelized 

TELRIC price and straight line depreciation do not jointly satisfy equations (1) and (2), 

together they imply that the firm earns the target rate of return over the entire life of the 

asset (i.e. from t = 0 to the end of the asset’s life). 

2.4 Levelization in Cost Proxy Models and the Impact of Changing

Investment Cost 

The illustrations in the previous section show that the levelization technique used to 

calculate TELRIC prices dramatically changes economic depreciation, but nonetheless 

can produce prices that exactly recover the initial cost of an asset and a target rate of 

return. 

When initial investment cost is changing, however, this consistency of TELRIC 

depends on how the TELRIC calculation is applied over time. If TELRIC prices are 

subject to periodic reviews that are shorter than asset lives, the prices will not generally 

recover exactly the initial cost of the asset and the cost of capital used in deriving the 

TELRIC prices. By “periodic reviews” we mean that the TELRIC price is reset based on 

a then-current initial investment cost. 

 15 
In particular, when investment cost is changing at the constant rate λ = γ - 1, the 

cost of investment at any time t is γ

 t F0. Assuming the utilization path is invariant to the 

date of installation, levelized TELRIC prices for an investment made at any time t = T ≥ 0 

are then given by: 

(5) 

∑ −

=

= 1

0

0

L

j j

j

T

T

q

F pTelric

δ

γ . 

Comparison of equations (4) and (5) reveals that, if investment cost is constant (so 

that γ = 1), then the levelized TELRIC price pTELRICT is equal to pTELRIC0 for any T. 

Otherwise, a periodic review at time t that bases the levelized TELRIC price on thencurrent initial investment cost will change the levelized TELRIC price to pTELRICT. 

 

Sections 2.5 and 2.6 consider the impact of TELRIC rate setting methodologies 

when the firm does not receive the initial TELRIC price for the entire life of the asset.

Specifically, we focus on a three-year review period, which is significantly shorter than 

the assumed life of most network elements. In this case, the consistency properties found 

in section 2.3 no longer hold. In section 2.7, however, we show that consistency can be 

restored by applying a straightforward correction factor based on a price path that 

recovers the target revenue and the time path of utilization rates. When the correction 

factor is applied to the TELRIC price path of interest, corrected prices exactly recover the 

initial cost of the asset and a target rate of return. 

In section 2.7 we also derive a contestable pricing rule. In the absence of entry 

barriers, the contestable price path would represent a constraint on the regulated firm’s

prices that is in many ways similar to the constraint implied by periodic TELRIC 

reviews. The same correction factor discussed above can guarantee that the firm earns its 

target cost of capital in these situations even when its prices must closely follow a time

path dictated by external competitive conditions.

2.5 Decreasing Initial Investment Cost 

Consider now the second scenario, in which an investment is made at time T = 0 and 

initial investment cost is decreasing at the rate λ. We use values that are intended to be 

representative of end-office switching, including an asset life of L = 16 years, a constant 

utilization rate of qt =0.94, and an annual -10.98% rate of decrease in initial investment 

cost. Therefore γ = 0.8902.25

25 These values are derived from a study done as part of an FCC order which established input values for

the cost proxy model used in that proceeding (Federal-State Joint Board on Universal Service, ForwardLooking Mechanism for High Cost Support for Non-Rural LECs, Tenth Report and Order, CC Docket Nos.

96-45, 97-160, FCC 99-304 (1999), Appendix C). See Mandy (2002) for details of the derivation. The 

assumed rate of change of switching investment cost derived from that report is used for illustrative 

purposes only, and no attempt has been made to update or verify those findings. 

 16 
Figure 4 compares TELRIC prices reviewed every three years with traditional 

prices based on straight-line depreciation. As Figure 4 indicates, TELRIC prices subject 

to three-year reviews understate traditional prices in every year. Note that neither

traditional prices nor TELRIC prices computed at each review period account for the rate 

of decline of investment cost in future years. While this does not create a problem in the 

case of traditional pricing, since it is designed to recover the historical cost of the asset 

over its lifetime, it creates a problem for TELRIC computations subject to a three-year 

review. More specifically, the TELRIC computation ignores the impact of future cost 

reviews on prices during the life of the asset, instead incorrectly assuming the initial 

levelized TELRIC price will continue to be received for the entire life of the asset. Thus, 

while the TELRIC price computed in year 0 was computed to fully compensate the firm

for its investment costs made in year 0, it does so only if maintained throughout the life 

of the asset. If prices are reduced in future years due to the lower investment costs at 

future TELRIC cost reviews, then the resulting price schedule no longer fully 

compensates the firm.

1 2 3 4 5 6 7 9 10 11 12 13 14 15 16

2.5

5

7.5

10

12.5

15

8

Figure 4. Traditional and Triennially Reviewed TELRIC Prices: 

Case of Decreasing Investment Cost 

Figure 5 compares TELRIC economic depreciation and straight-line depreciation. 

While in the benchmark analysis, depreciation under TELRIC pricing was increasing 

over time (see Figure 3), the economic depreciation identity (1) now reveals a secularly 

decreasing depreciation schedule under TELRIC that increases only within each review 

cycle. This is because the periodic reviews with decreasing investment cost produce 

TELRIC prices that fall fast enough in this example to more than offset the decelerated 

depreciation caused by levelization. 

 17 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

4

5

6

7

Figure 5. Straight Line and Implied Depreciation under Triennial TELRIC 

Reviews: Case of Decreasing Investment Cost 

If a regulator chooses to impose TELRIC prices, subject to a three year review 

and straight line economic depreciation, the earned rate of return over the life of the asset 

is substantially below the regulator’s assumed level, in this case approximately 1.15%

instead of the target rate of 11.25%.26

2.6 Increasing Initial Investment Cost 

Consider now the third scenario, in which an investment is again made at time T = 0, and 

initial investment cost is increasing at the rate λ. We use values that are intended to be 

representative of loops, including an asset life of L = 30 years, a utilization rate of qt = 

0.70 (assumed to represent average utilization over the life of the asset) and an annual 

increase in the cost of investment (due to inflation) equal to 3%, so that γ = 1.03.

27

 

Figure 6 compares TELRIC prices that are reviewed every three years with 

traditional prices based on straight-line depreciation. The TELRIC prices now increase, 

understating traditional prices early in the life of the asset but substantially overstating 

traditional prices during most of its life.

26 If TELRIC reviews are conducted annually instead of on a three-year basis then under the present 

assumptions the earned rate of return is negative. In this case, the firm does not recover even the initial cost 

of the asset. 27 These values are used for illustrative purposes only, and do not represent values that have been verified 

as reasonable for loop investments. 

 18 
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

5

10

15

20

25

30

Figure 6. Traditional and Triennially Reviewed TELRIC Prices: 

Case of Increasing Investment Cost 

The economic depreciation implied by TELRIC, as shown in Figure 7, is now 

severely decelerated, to the point that it is negative in the early years after the investment. 

This is because the deceleration caused by levelization is reinforced by the decelerating 

effects of rising prices. Rising prices decelerate economic depreciation for two reasons. 

First, other things constant, economic depreciation is higher when prices are higher. 

Recall that economic depreciation is simply the change in the value of the asset from one 

period to the next. As equation (1) indicates, however, the drop in the value of the asset is 

equal to the revenue received in the period.28 Thus, if prices rise, revenue will rise, which 

will increase depreciation. 

 

Second, as time passes discounting takes a progressively lighter toll on the present 

value of the as-yet unrealized relatively high late-life prices. Even with only 3 percent 

growth and triennial reviews, prices are so high at the end of thirty years compared to the 

initial prices (over twice as high) that the value-enhancing effect of moving them one 

year closer in time can be larger than the revenue actually received in an early year. 

Hence value can actually increase over time. An increase in value that occurs without 

additional investment is negative economic depreciation. This second factor is 

responsible for the negative economic depreciation observed in Figure 7. 

28 Prior to time t, the revenue pt qt is in the asset’s future and therefore contributes to its value. After time t, 

however, the revenue pt qt is no longer in the asset’s future, so the asset must lose value pt qt at time t.

 19 
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0

5

10

15

20

Figure 7. Straight Line and Implied Depreciation under Triennial TELRIC 

Reviews: Case of Increasing Investment Cost 

Finally, if the triennially reviewed TELRIC prices and straight line economic 

depreciation are both imposed by the regulator, the earned cost of capital over the life of

the asset is equal to 14.18%, which is substantially above the target rate of 11.25%. 

2.7 TELRIC Corrections and Contestable Price Paths 

As mentioned in section 2.4, a correction factor can be applied to ensure that the 

earned cost of capital is equal the target rate used when calculating TELRIC prices and 

depreciation. The correction factor is simply the ratio of desired revenue to revenue 

resulting from the TELRIC price path, or any other price path in which the “pattern” of 

prices is desired, both expressed as present values at the investment time T. Corrected

prices are then computed by multiplying the “pattern” price in each time period by the 

correction factor. Formally, the correction factor for an investment made at time T is 

defined as follows: 

(6) ( )

( ) . Re

1

0

1

0

∑

∑ −

= + +

−

= + +

= L

j T j T j

j

L

j T j T j

j

T

pPattern q

p venue q

correction

δ

δ

2.7.1 Decreasing Initial Investment Cost 

We first consider the case of falling investment cost as specified by the parameter values 

in section 2.5. We can correct the TELRIC prices that are reviewed every three years so 

that they deliver the same revenue as the traditional prices.29 A numerical computation of 

equation (6) reveals that the correction factor in this case is 1.5014. Thus, given the 

parameter values assumed in scenario 2 and a three year review process, TELRIC 

switching prices should be increased by approximately 50% in order to allow the 

 

29 Since only the present value of total revenues are relevant in (6), the same correction would result for any 

Revenue and Pattern price paths having the same present value revenues.

 20 
incumbent firm to recoup its costs and earn the target rate of return on a forward looking 

basis. 

Figure 8 illustrates both corrected (solid line) and uncorrected (dashed line) 

TELRIC prices compared to traditional prices (dotted line). The corrected TELRIC prices 

begin above but quickly fall below traditional prices, reflect the difference in three year 

versus one-year review, and have the same pattern as the uncorrected TELRIC prices.

1 2 3 4 5 6 7 9 10 11 12 13 14 15 16

5

10

15

20

8

Figure 8. Traditional versus Corrected and Uncorrected Triennially Reviewed 

TELRIC Prices: Case of Decreasing Investment Cost 

By construction, the corrected TELRIC prices also deliver the regulator's target cost of

capital, equal to 11.25% in the present case. 

2.7.2 Increasing Initial Investment Cost 

These same comparisons between traditional and TELRIC prices can be made when 

investment cost is increasing, using the parameter values in section 2.6. The TELRIC 

correction factor is now 0.802743. That is, assuming the parameter values in scenario 3 

and a three year review period, TELRIC loop prices should be reduced by approximately 

20% in order to allow the incumbent firm to recoup its cost and earn the target rate of 

return on a forward looking basis. The corrected TELRIC prices, again shown as a solid 

line in Figure 9, begin below but rise above traditional prices after less than one-third of 

the life, again reflect the difference between three-year versus one-year reviews, and 

again have the same pattern as the uncorrected TELRIC prices.

 21 
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

5

10

15

20

25

30

Figure 9. Traditional versus Corrected and Uncorrected Triennially Reviewed 

TELRIC Prices: Case of Increasing Investment Cost 

2.7.3 Contestable Price Paths 

So far we have compared TELRIC prices, both uncorrected and corrected, with 

traditional regulatory prices. We have not considered whether those prices emulate or 

equal equilibrium prices that would result in a competitive market with potential entry in 

every time period. Competitive markets generally provide efficient entry and investment 

signals to both incumbents and new entrants. Regulated prices that do not at least emulate 

competitive equilibrium prices may therefore fail to give efficient investment incentives 

to incumbents and new entrants. Thus, if one wants TELRIC prices to provide efficient 

entry and investment signals, it is desirable that they emulate competitive equilibrium

prices. This section addresses this issue. 

Competitive equilibrium prices are usually defined as prices that equate aggregate 

supply and aggregate demand when buyers and sellers all act as price-takers. Such prices 

do not generally exist when production (in the present environment, capacity acquisition) 

has increasing returns to scale. Although we have not yet considered the relationship 

between the productive capacity installed at time T, which is normalized in the present 

section at 1, and its cost , many telecommunications assets have substantial 

increasing returns in capacity acquisition. We explicitly consider increasing returns cost 

structures in section 3. Hence, in that context we cannot define true competitive 

equilibrium prices. 

F0

T γ

However, the important feature of competition that provides incentives for 

efficient entry and exit is that economic profit is zero for efficient incumbents and 

efficient potential entrants no matter when entry is contemplated. Prices with this 

property are well-defined even when capacity acquisition has increasing returns to scale. 

For lack of a better term, we call such prices “contestable,” although it is important to 

realize that we are requiring social cost efficiency as part of this definition. 

 22 
As with the TELRIC prices defined in (5), we continue to assume: 1) the 

utilization path is invariant to the date of installation, 2) contestable prices are set at the 

beginning of each year and remain constant during the year, and 3) is the efficient 

cost of a capacity unit installed at time

F0

T γ

t = T. Then, if the target discount factor δ reflects 

a competitive rate of return, contestable prices are defined by the breakeven condition 

0

1

0 pContestable q FT L

j T j j

j

∑ δ = γ −

= + for T = 0,1,2, ... 

The solution to this equation is:

30

(7) 

∑ ( ) −

=

= 1

0

0

L

j j

j

T

T

q

F pContestable

δγ

γ for T = 0, ..., L-1.

Compared with equation (5), the contestable price at time T accounts for the fact that 

future prices will be different when costs are not constant (i.e., γ ≠ 1), whereas the 

TELRIC price does not anticipate future price changes. 

Assuming utilization is constant, equation (7) shows that the contestable prices 

decrease over time when investment cost is decreasing (i.e., γ < 1) and increase over time 

when investment cost is increasing (i.e., γ > 1). These are the same secular trends 

displayed by TELRIC prices that are reviewed every three years. Moreover, as we have 

already seen, the corrected TELRIC prices yield the target rate of return. Hence the 

triennially reviewed corrected TELRIC prices indeed emulate contestable prices. Figures 

10 and 11 present the comparisons for scenarios 2 and 3, respectively. The only 

significant difference is due to the timing of three-year TELRIC reviews versus the 

(assumed) annual adjustment of contestable prices. 

 

30 See also Mandy 2002 Theorem 1, wherein utilization is assumed to be unity and constant returns to scale 

is assumed.

 23 
1 2 3 4 5 6 7 9 10 11 12 13 14 15 16

5

10

15

20

8

Figure 10. Contestable versus Corrected Triennially Reviewed TELRIC Prices: 

Case of Decreasing Investment Cost 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

12.5

15

17.5

20

22.5

25

27.5

Figure 11. Contestable versus Corrected Triennially Reviewed TELRIC Prices: 

Case of Increasing Investment Cost 

2.8 Variable Utilization Rates 

All computations up to this point have assumed a constant utilization rate. If demand is 

growing, a cost-minimizing firm may make investments in anticipation of future growth, 

in which case utilization rates will increase as the asset ages. All of the definitions of 

previous sections are valid under these conditions, although the specific time paths of 

price and depreciation will change when utilization rates are not constant.

However, when utilization rates are not constant, a discrepancy emerges between 

the TELRIC pricing rule (5) and the TELRIC prices actually calculated by cost proxy 

models, such as the FCC’s synthesis model. Specifically, the Synthesis Model uses 

average utilization to calculate the TELRIC price, rather than anticipating that utilization 

will change over the life of the asset. In addition, the size of the investment F0 is assumed 

 24 
to be sufficient to satisfy current demand divided the utilization factor. These issues will 

be explored further in the following section.31

In this section, we assume increasing investment cost using the same parameter 

values as in section 2.6, except that we assume a growing utilization rate given by qt = 

.40 + .55 t / (L-1) for t = 0, ... , L-1. This utilization path models loops that are used at 

40% of capacity when installed but whose usage increases linearly to 95% at the end of 

the thirty year life. 

Figure 12 below compares three separate price paths: traditional prices (solid 

line), contestable prices (dotted line), and TELRIC prices reviewed every three years 

(dashed line). The traditional prices still fall, as when the utilization rate is constant, but 

now they begin much higher and decrease faster. This is because the traditional rule only 

uses utilization in period t to calculate the price in period t, so the price must drop to 

offset growing quantity. This effect reinforces the downward slope caused by straightline depreciation. In contrast, the contestable and TELRIC prices still increase over time 

because they take account of the entire utilization path to calculate the price at each time 

(see equations 5 and 7). Hence their dynamics are driven entirely by the growth rate of 

initial investment cost. Both paths are higher than under scenario 3, where utilization 

rates are constant, reflecting the fact that average utilization over the life of the asset is

about the same as scenario 3 but revenues are delayed because utilization is initially low. 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

10

20

30

40

Figure 12. Traditional, Contestable, and Triennially Reviewed TELRIC Prices: 

Case of Increasing Investment Cost and Increasing Utilization 

As before, the TELRIC prices can be corrected so that they yield the same rate of 

return as the traditional and contestable prices. The correction factor in this case is 

0.761538, which is not much different from scenario 3. The corrected TELRIC prices 

 

31 In the present section, the function q has served both as a utilization rate and as a proxy for potentially 

varying demand conditions. When an efficient investment plan requires repeated investments made over a 

planning horizon, there is no longer a simple relationship between utilization rates and aggregate demand. 

In section 3, these issues are treated in a more general framework in which demand is the exogenous

variable and utilization rates are determined through cost minimization. 

 25 
(solid line) again track the contestable prices (dotted line), as shown in Figure 13, and as 

expected, the corrected prices yield the target rate of return of 11.25%. 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

15

20

25

30

Figure 13. Contestable and Corrected Triennially Reviewed TELRIC Prices: 

Case of Increasing Investment Cost and Increasing Utilization 

We note in conclusion that these results demonstrate that demand growth as well 

as changing investment cost must be considered when setting a time path of prices.32 

Generally, a firm’s utilization rate at any point in time is determined by a solution to a 

cost minimization problem. The cost minimum determines the time and level of 

investments needed to satisfy demand at every future point in time. It may very well 

involve intentional under-utilization that varies over time. We consider some of these 

issues in the next section in a framework in which efficient investment decisions can be 

determined when the rate of demand growth is explicitly modeled, and a more general 

investment cost function is introduced.

3. Efficient Dynamic Investment 

Section 2 was concerned with the time path of TELRIC prices that could be used to 

recover the cost of an arbitrary investment F0 that has an arbitrary utilization path qt. 

There, it was assumed the investment cost and utilization path had already been 

determined. 

Actual investment cost and utilization of assets subject to TELRIC pricing result 

from decisions made by incumbent suppliers. An efficient supplier makes such decisions 

to minimize the cost of serving an evolving expected demand, given the expected costs of 

raw inputs, the expected evolution of those costs, the expected performance of the assets 

over time, the cost of capital and other input prices. These are dynamic cost-minimizing 

investment decisions. 

 

32 While the asset’s utilization rate has been allowed to vary in the present section, the aggregate level of 

capacity has remained constant at a normalized value of one unit. 

 26 
The prices discussed in section 2 can be expected to provide incentives for 

efficient entry and exit provided they are based on efficient investments. Accordingly, in 

this section we characterize, in some cases, the dynamic cost minimizing investment 

decisions of an efficient firm. Under our stated assumptions, these investment decisions 

can be used in the corrected TELRIC pricing rules of section 2 to provide incentives for 

efficient entry and exit. Interest here centers on the efficient investment plan and its 

utilization path, which was part of the analysis throughout section 2 but only allowed to 

vary in section 2.8. In the present section, we introduce explicit demand growth and input 

cost dynamics, and then derive the investment plan and utilization path as the solution to 

a cost minimization problem for the firm. 

We do this under two stylized scenarios about the performance of assets over 

time. The scenarios differ in the assumption made about asset obsolescence. In scenario 

1, which we label “expected innovation date” replacement, we assume that an asset’s 

physical life is so long relative to the pace of innovation that the physical life is not a 

constraint on the firm’s use of the asset. In other words, assets are replaced only when 

technical progress either reduces operating cost or improves quality and performance of 

replacement assets enough to make old assets economically obsolete, even though old 

assets still function as they did when new. This scenario is broadly representative of 

switching assets. 

In scenario 2, which we label “light bulb” replacement, we assume that an asset 

has a fixed and known physical life equal to L years. During this life there is no 

deterioration of an asset’s productivity, but at the end of its life the asset suddenly and 

permanently ceases to be productive. Here, an asset’s physical life is the constraint on 

the firm’s use of that asset. Replacement of such assets occurs when they reach a certain 

age, rather than at a certain date irrespective of age as in scenario 1. Scenario 2 is 

broadly representative of loop assets.33

In spite of the rudimentary nature of these computations, they may suggest both a 

plausible range of utilization factors for efficient telecommunications firms in a forwardlooking environment, and perhaps more importantly the way these factors depend on the 

assumed underlying parameters.34

3.1 Maintained Assumptions 

As in section 2, an initial investment is made at time t = T1 ≥ 0. Subsequent investments 

are made at times T2, T3, ..., which are numbered in chronological order. Each 

investment installs a fixed productive capacity per unit of time that depends on the size of 

the investment. Generally, we denote by F(I,t) the investment required at time t to install 

productive capacity I per unit of time. For example, the investment F0 studied in section 

 

33 More general interpretations of this scenario are possible, in which the asset life is fixed but only the 

expected life of the asset is known to the regulator (and possibly to the firm). We do not pursue them here. 34 While we do not intend for the results of this section to be directly applied to input selections for cost 

proxy models, such as the Synthesis Model, we believe this discussion can inform the process of choosing 

those inputs. 

 27 
2 is given by F0 =F(1,0) since it was installed at time zero and its capacity was 

normalized to one. The investment times Tk and the productive capacity Ik installed at 

each investment time are chosen to minimize cost. 

We assume a simple multiplicatively separable structure for F, and that F changes 

over time at the constant continuous rate λ. Hence where f(I) is the 

investment required to install productive capacity I at time 0. Depending on its shape, the 

investment cost function f(I) may reflect different returns to scale in the activity of

capacity installation. It will be parameterized below for each of the two scenarios to 

reflect the increasing returns typical of switch and loop installation. 

( , ) ( ) ,

t F I t f I eλ =

The firm's cost of capital is still constant and equal to r (or 100r %), which we 

continue to assume is 11.25% per year, implying a continuous rate of r = 0.1066. Also, 

as in section 2, we continue to assume there is no physical depreciation, operating cost, or 

salvage value; and we ignore taxes. Again, all of these realistic features can be 

incorporated without changing the important aspects of the analysis. 

Demand begins at time t = 0 at an initial level Q(0) and grows at the constant 

continuous rate η ≥ 0. Hence, demand at time t ≥ 0 isQ( ) (0) .

t t Q eη =

Efficient dynamic investment decisions in this environment must balance two 

competing incentives. First, the time value of money gives an incentive to delay 

investments. Second, increasing returns in capacity acquisition gives an incentive to 

acquire capacity in large amounts, when it is acquired, thereby driving down the average 

cost of a unit of capacity. The first incentive implies that an efficient firm will never

acquire capacity if it already possesses unused capacity.35 The second incentive implies 

that, when acquiring capacity, an efficient firm will install more than is needed to satisfy

current demand. At each investment date, some additional capacity will be acquired in 

anticipation of future demand growth. Thus, the second incentive is to accelerate the rate 

at which capacity is acquired before it is ultimately used. The strength of this second 

incentive depends on the extent of increasing returns. If capacity acquisition has constant 

returns then the second incentive disappears and efficient new investments are made only 

to accommodate current new demand or replacement of currently obsolete capacity. 

To establish benchmarks, we assume capacity at time t = 0 is at the lowest 

possible level, so new investment is required at time t = 0 in order to satisfy demand in 

the immediate future. In the expected innovation date scenario this means capacity is 

Q(0) at time t = 0. In the light bulb scenario this means capacity is zero at time t = 0. 

The latter is sometimes known as a “green field” approach, and is close to the approach 

taken in most TELRIC cost studies.36 However, we also investigate efficient investment 

plans that begin with excess capacity. The results which assume initial excess capacity (in 

both scenarios) indicate how an efficient plan changes when an incumbent’s existing 

 

35 Our results do not attempt to estimate the efficient level of so-called “administrative fill” – i.e. the 

minimum level of excess capacity that a firm requires to handle customer turnover, unexpected outages, or

internal maintenance procedures. If the firm requires a certain level of administrative fill, then it would

invest in new capacity when this minimum level of excess capacity was reached. 

 28 
plant is taken into account. This alternative approach is closer in spirit to traditional 

incremental costing methodologies that have long been used in regulatory analysis. 

3.2 Expected Innovation Date Replacement 

In this scenario, there is a future expected innovation date D > 0 at which all assets 

acquired before that date become obsolete. At date D all past decisions about the 

particular asset type under study become irrelevant to the firm’s subsequent investment 

decisions.37 Hence the cost minimization problem starting at time t = 0 has a finite time

horizon that lasts only until time t = D. Since increasing returns implies that optimal 

investment occurs in discrete amounts, the cost minimizing investment plan will involve 

a finite number n of investment times. 

The firm begins at time t = 0 with an initial capacity I0 ≥ Q(0) that was acquired at 

some time prior to t = 0. Since all assets installed before date D remain productive until 

date D, the productive capacity at any time t in the relevant time horizon is 

Moreover, since it is cost-minimizing to delay investments until all existing capacity is in 

use, the efficient investment undertaken at time T

. ∑ =0, <

n

j T t j j

I

k will be to install exactly enough 

capacity to satisfy demand growth through the next investment time Tk+1. That is, 

( ) , 1

0 1 ∑ −= = + − kj k k j I Q T I for k=1, ..., n,

where Tn+1 is defined to be D (i.e., the last investment must install exactly enough 

capacity to satisfy demand growth through the end of the relevant time horizon), and the 

first investment time is determined by I0 = Q(T1). Therefore Ik = Q(Tk+1) – Q(Tk) for k = 

1, ..., n.38 This expression is a demand constraint on the cost-minimizing relationship 

between investment capacities and investment times. 

For a fixed number of investment times n, the cost minimization problem is to 

choose the investment times and capacities at each time to minimize the present value of 

the investment cost, subject to the demand constraint. Using the 

demand constraint and the assumed functional forms for investment cost and demand, the 

cost minimization problem is 

( , , ∑ =1

− n

k k k

rT

e F I T k )

 36 TELRIC differs from the “green field” approach, however, in that it assumes that loop and switching 

facilities are designed based on the location of existing wire centers. 37 This description is admittedly a caricature. However, the majority of certain types of assets are indeed

replaced within very short time periods relative to the length of time the assets are used. We believe the 

fixed (expected) innovation date scenario is a useful conceptual and modeling tool even though, like all 

models, it is stylized compared with the real world. 38 We note that by assumption demand growth is monotonically increasing, so that this expression is well

defined.

 29 
(8) Min ( ) ( ) ( ) 0 [ ], 1

1

=

− − + n

k

r Tk Tk Tk e f Q e e

λ η η

∑ { ,..., } T2 Tn

where ( ) ( ) ( )

η

ln 0 ln 0

1

I − Q T = and T . n+1 = D

Once equation (8) is parameterized with input values, the cost-minimizing 

investment plan can be found as a numeric solution to (8) for a given n, nested within a 

numeric search over n. Since the scenario under consideration is replacement at an 

“expected innovation date,” which is representative of switching assets, we parameterize 

(8) with input data on the cost of switching equipment that was developed for the 

Synthesis Model during the FCC’s universal service proceedings. These inputs are based 

on regression analysis of switching contracts supplied to the FCC during that proceeding, 

which were used to estimate the initial investment cost function for autonomous endoffice digital switches f ( )I = 486,000 + 87I, where I is the number of lines on the 

switch.39 As in section 2.5, our benchmark analysis assumes that this cost decreases at an 

annual rate of -10.98% per year, implying a continuous rate of λ = -0.1163, and the 

innovation date is D = 16. To establish a benchmark, we further assume that demand 

grows at the continuous rate η =0.02 (approximately 2% per year), initial demand is 

Q(0)=20,000 lines, and there is no initial excess capacity (i.e., I0 = Q(0), so T1 = 0). 

The benchmark efficient investment plan is shown in Figure 15. It involves an 

initial investment in switching of 3,365 lines at a cost of $778,728, followed by one 

subsequent investment after 7.8 years of 4,178 lines at cost $343,925. The minimized 

present value of cost is $928,878. The percentage of excess capacity in the efficient 

investment plan is shown in Figure 16. The initial investment installs excess capacity of

3,365/23,365 = 14.4%, which is gradually used by growing demand until reaching zero at 

the time of the second investment. The second investment then installs excess capacity of 

4,178/27,543=15.2%. The average utilization rate (1 – average excess capacity) is 91.5%. 

 

39 Strictly speaking, this cost function may not be an appropriate cost function for the current analysis since 

it estimates the full stand alone cost of a fully configured switch, and not the cost of additional line cards 

and other growth additions. In the absence of alternative public data on switching costs, we use it for 

illustrative purposes only. In addition, we have not attempted to verify that it is an accurate representation

of current switching costs. 

 30 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

t 20000

21000

22000

23000

24000

25000

26000

27000

Capacity

Figure 15. Demand and Efficient Capacity, Benchmark Case 

Switching Scenario 

2.5 5 7.5 10 12.5 15

t

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Percent Excess Capacity

Figure 16. Percent Excess Capacity, Benchmark Case 

Switching Scenario

Table 1 summarizes the results for various parameterizations. The benchmark 

parameterization is in row 1. Subsequent rows report only those parameters that differ 

from the benchmark. The second row, which may be interpreted as an incremental cost 

calculation, shows the optimal investment plan if initial demand is only 80% of initial 

capacity. This initial excess capacity is adequate to satisfy demand until year 11.2, at

which time efficient investment installs exactly enough capacity to satisfy demand 

through the end of the horizon. Costs are much lower than in the benchmark plan, 

because the initial excess capacity both reduces the need for new investment and 

forestalls the time of that investment. The third row examines the effect of a change in 

the cost of capital. It shows that the optimal plan places heavier emphasis on delaying 

investment costs rather than exploiting scale economies when the cost of capital is higher. 

This delaying tactic, along with the pure discounting effects of a lower discount factor, 

reduces the present value of the plan compared with the benchmark even though the total 

cash outlay is higher. The fourth row shows how a 50% delay in the innovation date 

affects the optimal investment plan. The second investment is accelerated slightly and a 

 31 
large third investment is added. The total cost in current dollars does not increase much 

because the large third investment is discounted over 15 years and reflects 15 years of

falling investment costs. Finally, the last row shows the effect of a slower decline in 

equipment prices. Investment is delayed relative to the benchmark because later costs are 

relatively higher, and total cost increases. 

Table 1: Efficient Investment Plans in Scenario 1 (Switching) 

Efficient Investment Plan 

Parameters

Times Amounts Costs

Present Value

of 

Minimum Cost

Benchmark

r = 10.66% 

λ = -11.63% 

D = 16 

η = 2% 

Q(0) = 20,000 

I0 =20,000

T1 = 0 

T2 = 7.8 

I1 = 3,365

I2 = 4,178

F(I1,T1) =

$778,728 

F(I2,T2) =

$343,925

$928,878 

Q(0) = 16,000 T1 = 11.2 I1 = 2,034 F(I1,T1) =

$181,116 $55,135 

r =15% 

T1 = 0 

T2 = 6.7 

T3 = 12.6 

I1 = 2,859

I2 = 2,870 

I3 = 1,813 

F(I1,T1) =

$734,763 

F(I2,T2) =

$338,236 

F(I3,T3) =

 $148,795 

$881,415 

D =24 

T1 = 0 

T2 = 7.6 

T3 = 15.2 

I1 = 3,305 

I2 = 3,821 

I3 = 5,196 

F(I1,T1) =

$773,538 

F(I2,T2) =

$336,304 

F(I3,T3) =

$159,442 

$953,794 

λ = -5% 

T1 = 0 

T2 = 8.9 

I1 = 3,894 

I2 = 3,645 

F(I1,T1) =

$824,759 

F(I2,T2) =

$515,010 

$1,024,310 

 32 
The results in Table 1 cannot be directly utilized as inputs to the TELRIC pricing 

rules discussed in section 2 because the replacement regime in section 2 was the “fixed 

life” scenario (i.e. assets last exactly L years) rather than the “expected innovation date” 

scenario in Table 1 (where assets are retired on a given date D regardless of age). 

Alternative TELRIC rules, as yet unexplored in the literature, are needed for assets whose 

obsolescence is best described by the expected innovation story. In the next two sections 

we explore how efficient investment plans that are derived for the “fixed life” scenario

can be directly used in TELRIC pricing rules. 

3.3 Light Bulb Replacement 

In this scenario, there is a fixed useful life L > 0 for each asset. Since there is no date at 

which the regime is expected to end, the cost minimization problem starting at time t = 0

has an infinite time horizon. 

The firm begins at time t = 0 with initial capacity that was acquired at some

time(s) prior to t = 0. This initial capacity may consist of different vintages. For 

notational convenience, denote those capacities, in order of age, by I-(b-1), I-(b-2), ..., I0 and 

their installation times by T-(b-1), T-(b-2), ..., T0, where b ≥ 1 is the number past investments 

that are still productive at time t = 0 (if there are no productive investments left at time t 

= 0 then we set b = 1, T0 = 0, and I0 =0). The productive capacity at any time t ≥ 0 is the 

sum of those capacities that are not older than L. So the existing productive capacity at 

an investment time Tk ≥ 0 is ( ) ( )

−

=− − ≤ +

1

1

k

j b k j j ∑ χ T T L I , where χ(C) = 1 if condition C is 

true and 0 otherwise. 

As in the “expected innovation date” replacement scenario, it is cost-minimizing 

to delay investments until all existing capacity is in use, so the efficient investment 

undertaken at time Tk must install exactly enough capacity to satisfy demand growth 

through the next investment time Tk+1. That is, 

( ) ( ) ( ) , 1

1 1 ∑ 1

−

=− − = + − + ≤ +

k

j b k k k j j I Q T χ T T L I for k=1, 2, ...,

where the first investment time is the smallest value of T1 such that 

 These expressions are the demand constraints on the 

cost-minimizing relationship between investment capacities and investment times. 

( ) ( ( ) . 0

1 1 ∑ 1 =− − ≥ ≤ +j b j j Q T χ T T L)I

The cost minimization problem is to choose the investment times and capacities at 

each time to minimize the present value of the investment cost, 

( ) ( ) , ( ), ∑ ∑ 1 1

∞

=

∞

=

− − = j j j

r T

j j

rT

e F I T e f I j λ j 

subject to the demand constraints. In general, this is an infinite horizon optimization 

problem that is non-recursive, and therefore it cannot be solved with standard techniques 

 33 
such as those discussed in Stokey and Lucas (1989). In the examples below, we solve it 

for finite planning horizons where the times of investment are restricted to integer 

values.40 

This “light bulb” replacement scenario is representative of loops, so we 

parameterize it with publicly available data41 on loop costs from the state of Maryland. 

With these data, a run of the Synthesis Model was completed in which inputs 

representing utilization factors were set equal to 100%.42 Disaggregated loop investment 

costs, representing distribution facilities, feeder facilities, and loop electronics were then 

extracted from the model outputs for each of the 189 wire centers in the state. A set of 

loop cost functions were derived from regressions of investment costs on the number of 

access lines and wire center area. For distribution investments, the results are as follows. 

:ParameterTable →

Estimate SE TStat PValue

1 55974.3 218400. 0.256292 0.798013

Lines 276.35 7.95594 34.735 0.

Lines2 −0.00112738 0.0000614206 −18.3551 2.34459× 10−43

Area 49270.6 4745.78 10.382 0.

Area2 −77.0684 23.8266 −3.23455 0.0014457

Area Lines 0.717975 0.0796032 9.01942 2.22045× 10−16

,

RSquared → 0.962218, AdjustedRSquared → 0.961185, EstimatedVariance → 1.2873× 1012,

ANOVATable →

DF SumOfSq MeanSq FRatio PValue

Model 5 5.9995× 1015 1.1999× 1015 932.108 0.

Error 183 2.35576× 1014 1.2873× 1012

Total 188 6.23508× 1015

>

By varying both the number of lines and wire center area, it is possible to 

represent loop investment cost under different conditions of scale and density. Using an 

area equal to the mean area of the Maryland sample data (52 square miles) resulted in a 

distribution cost function 

(9) f(I) = 2.4 * 106

 + 314 I – 0.001 I2, 

where investment I represents the number of access lines. This cost function implies that 

average cost is steeply declining over the range of reasonable outputs as illustrated in 

 

40 Subject to these restrictions, the algorithm we use searches for a global cost minimum of investment

times for any given number of investments, nested within a search over the number of investments. There is 

no guarantee that the resulting solution is a global cost minimum, although the solutions we find are below 

the theoretical upper bound described in section 3.4. A numerical solution to this problem has been 

implemented in Mathematica. 41 The Maryland data used in this simulation are based on customer locations created by randomly 

distributing the population of each Census block within the block boundaries. These data are available on 

the internet at http://www.fcc.gov/wcb/tapd/hcpm/, and while they are less accurate than those actually 

used in the Synthesis Model, they are capable of capturing the economies of density and scale that

characterize loop investment costs over a large and diverse geographic area. 42 Therefore, the resulting cost function f(I) represents the cost of deploying exactly I loops instead of the 

cost of deploying loop plant to satisfy a current demand for Q < I loops, as in the standard operation of the 

Synthesis Model.

 34 
Figure 17 below. Average investment cost functions for feeder and electronics exhibit 

similar, though less pronounced, scale economies, as also illustrated in Figure 17. 

20000 30000 40000 50000

I

100

200

300

400

500

fHILêI

Distribution

Feeder 

Electronics 

Figure 17. Average Cost of Distribution, Feeder and Electronics Investment 

Estimated for Maryland

As in section 2.6, our benchmark analysis assumes that cost increases at a 

continuous rate of λ = .03 (approximately 3% per year) and that the useful life is L = 30. 

As in the previous section, we assume that demand grows at the continuous rate η =0.02. 

Moreover, to establish a benchmark, we assume a planning horizon equal to twice the 

asset life (60 years), initial loop demand equal to Q(0) = 10,000 lines, and no initial 

capacity (i.e., all units of initial capacity are assumed to expire at time t = 0). For the 

benchmark case, the cost minimizing capacity path is illustrated in Figure 18 and the 

implied levels of excess capacity are shown in Figure 19. The average utilization rate 

associated with this capacity expansion sequence is 86.7%. 

10 20 30 40 50 60

t 10000

15000

20000

25000

30000

Capacity

Figure 18. Demand and Efficient Capacity, Benchmark Case 

Loop Scenario 

 35 
10 20 30 40 50 60

t

0.05

0.1

0.15

0.2

0.25

0.3

Percent Excess Capacity

Figure 19. Percent Excess Capacity, Benchmark Case 

Loop Scenario

Table 2 presents results when some of the benchmark assumptions are modified.43

As the cost of capital rises, it becomes more expensive to have units of unused capacity 

and the present value of future investment costs is reduced. As a result, there is a 

reduction in the size of the initial investment and a corresponding acceleration in the time

that subsequent investments are made. When investment costs are assumed constant, 

instead of increasing at the benchmark rate of 3% per year, the size of the initial 

investment is again reduced modestly. If the life of the asset is 15 years, substantially less 

than in the benchmark case, it is efficient to invest as infrequently as possible under the 

light bulb replacement scenario. The cost minimizing investment plan then consists of

investments made at 15 year intervals with no intermediate investments. 

If there is existing capacity from previous investments at time t = 0, then the time 

of initial investment is delayed until demand growth exhausts the initial capacity. In this 

situation, the cost minimizing investment plan of an efficient firm should take account of 

these initial investments. However, the timing of subsequent investment decisions will 

not, in general, have a simple relationship to the time that existing investments were 

made.

44 The last row of Table 2 and Figure 20 describe the efficient investment program 

for one possible set of initial conditions. 

 

43 In this table only we use the notation Fj

 = F(Ij , Tj). 44 Note that the optimal investment plan illustrated in Figure 20 has existing assets retired at times t = 10, 

15, 25, and even as late as year 55, without any corresponding new investments at those times. Thus, initial 

capacities can have a substantial impact on the firm’s capital stock even after many years have elapsed. 

 36 
Table 2: Efficient Investment Plans in Scenario 2 (Loops) 

Efficient Investment Plan 

Parameters

Times Amounts Costs

Present Value

of 

Minimum Cost

Benchmark

r = 11.25% 

λ = 3% 

L = 30 

η = 2% 

Q(0) = 10,000 

I0 = 0 

T0 = 0 

T1 = 0 

T2 = 19 

T3 = 30 

T4 = 49 

I1 = 14,283 

I2 = 3,476 

I3 = 22,395 

I4 = 9,772 

F1 =$6,659,870 

F2 =$6,113,410 

F3 =$21,527,700 

F4

= $22,844,800 

$8,468,400 

r = 15% 

T1 = 0 

T2 = 17 

T3 = 30 

T4 = 46 

I1 = 13,728 

I2 = 4,031 

I3 = 20,348 

I4 = 11,819 

F1 =$6,503,410 

F2 =$6,042,270 

F3 =$20,208,700 

F4

= $23,213,000 

$7,407,570 

λ = 0 

T1 = 0 

T2 = 17 

T3 = 30 

T4 = 47 

I1 = 13,728 

I2 = 4,031 

I3 = 20,836 

I4 = 11,331 

F1 =$6,503,410 

F2 =$3,655,670 

F3 =$8,456,020 

F4

= $5,819,390 

$7,484,330 

L= 15 

T1 = 0 

T2 = 15 

T3 = 30 

T4 = 45 

I1 = 13,195 

I2 = 17,758 

I3 = 23,901 

I4 = 32,167 

F1 =$6,352,370 

F2 =$11,879,000 

F3 =$22,483,400 

F4

= $42,858,400 

$10,024,400 

I0 = 2,000 

I-1 = 3,000 

I-2 = 6,000 

T0 = -5 

T-1 = -15 

T-2 = -20 

T1 = 5 

T2 = 24 

T3 = 35 

T4 = 53 

I1 = 13,769 

I2 = 5,838 

I3 = 22,166 

I4 = 10,002 

F1 =$7,552,710 

F2 =$8,542,720 

F3 =$24,786,600 

F4

= $26,032,000 

$5,778,770 

 37 
10 20 30 40 50 60

10000 t

15000

20000

25000

30000

35000

Capacity

Figure 20. Demand and Efficient Capacity, Loop Scenario 

With Initial Installed Capacity at Time t = 0 

3.4 Implications for TELRIC Pricing 

Let us now consider some implications of the “light bulb” efficient investment 

results for TELRIC pricing. TELRIC pricing rules have traditionally been defined in an 

environment in which little or no account is taken of market dynamics. In particular, 

current TELRIC pricing rules are generally defined without reference to either expected 

changes in future investment cost or expected changes in future demand (or utilization 

rates). In section 2.7 we illustrated how a simple multiplicative correction factor could be 

defined to take proper account of changing investment cost. Under corrected TELRIC 

pricing, a regulated firm would exactly recover its forward looking investment cost, even 

if TELRIC price reviews occur at intervals shorter than assumed asset lifetimes. In 

section 2.8, it was similarly demonstrated that varying utilization rates, which could 

result from cost minimizing decisions made in the face of growing demand, could be 

easily incorporated into the TELRIC pricing rule. 

Sections 3.2 and 3.3 have focused on another aspect of efficient investment 

planning in a dynamic environment. In these sections, a cost minimizing investment 

program was explicitly calculated in a variety of circumstances. In order to complete our 

analysis, it only remains to demonstrate how these cost minimizing investment decisions 

can be used as inputs into a fully efficient forward looking pricing regime. In the case of 

assets having light bulb performance characteristics this process is relatively 

straightforward. The TELRIC prices for a sequence of investments made as part of an 

optimal investment plan now depend on the demand conditions at the time at which the 

investment occurs.45 With this modification, equation (5) of section 2 is repeated using 

notation of the current section as equation (10): 

 

45 In section 2, utilization rates, which play the same role as aggregate demand in section 3, were generally 

assumed to be constant. In the case of growing utilization rates discussed in section 2.8, the utilization rate

path was assumed to be based on the time of investment rather than calendar time. 

 38 
(10) ( )

( ); ,

1

∑ 0

−

= + = Lj k

j

k k

T

Q T j

F I T pTelric k

δ

where F is the investment cost function; I1, ... , In is a sequence of cost minimizing 

investments; T1, ... , Tn is a sequence of investment times; and Q(.) is aggregate demand. 

In contrast, TELRIC prices as actually calculated in practice are usually based on 

a somewhat different set of assumptions. Actual TELRIC prices assume that a single 

investment is made at the time of a costing study. The size of the initial investment is 

determined by current demand divided by an exogenously determined utilization rate. In 

the current notation of section 3, where q is an assumed utilization rate and T is the time

of the TELRIC study, these prices are defined in equation (11): 

(11) ( )∑ −

=





 





= 1

0

, ( )

L

j

j T

Q T

T

q

Q T F

pActualTelric

δ

In section 2.7 we derived a contestable price path and argued that TELRIC prices 

ought to emulate these prices when the regulated firm faces potential competition from

rival carriers. Like the TELRIC prices considered in section 2.7, the contestable prices 

assumed an investment in a single asset and a utilization path that is invariant to the time 

of investment. Also like the TELRIC prices in (10), the contestable price definition now 

depends on demand conditions at the time at which the investment occurs. Moreover, 

unlike TELRIC prices, which are designed to recover only the cost of an asset installed at 

the time of a TELRIC cost study, contestable prices must make efficient entry yield zero 

economic profit over the entire future of the entrant, at every potential entry time. Hence 

the contestable price path in now defined by 

(12) ( ) ( ) 1

pContestable Q t PV T T H

t T t

t T

∑ = + −

=

− δ for T = 0, 1, ...,

 

where is the present value cost of a sequence of 

cost minimizing investments I

∑ ( =

− = n

k k k

T T T PV T F I T T T k

1

( ) ( ) δ ( ), ( ))

1(T), ... , In(T) made at times T1(T), ... , Tn(T) for a de novo

entrant at time T = T1(T), and H is the planning horizon used to calculate the cost 

minimum.

Solving equation (12) for the contestable price path gives 

(13) ( ) ( ) ( ) ( 1) R T

Q T

PV T PV T pContestableT + − + = δ for T = 0, 1, ... , H – 1,

where 

 39 
(14) ( )

( ) ( ) Q T

pContestable Q T H R T T H

H + = δ +

is a remainder that arises because the finite horizon H is used to estimate the cost

minimum while the actual costing and pricing problem has an infinite horizon. This 

remainder must be estimated in order to calculate (13), which means we must have an 

estimate of pContestableT+H. As the planning horizon H becomes very long the 

remainder approaches zero, so one estimate is to simply ignore the remainder in 

calculating (13). Alternatively, under our stated assumptions it can be shown that 

(15) (1 ( (1 )) ) ( )

(1 ) ( ( ), )

Q T

F Q T L T pContestableT L δγ η

δγ

− +

− + ≤ for T = 0, 1, ...,

which provides an upper bound on the contestable price path that can be used in (14) to 

place an upper bound on the remainder.46

Figure 21 compares the price paths (11) and (13) for the Benchmark case reported 

in Table 2 (with investment costs rising at 3% per year), and Figure 22 shows these price 

paths under the same assumptions except that investment costs decline at the rate of 5% 

per year. The actual TELRIC path (11) is assumed to be subject to TELRIC price reviews 

every three years (T increments by 3), and the assumed utilization rate is the expected

utilization rate in the efficient investment plan. The contestable price path assumes that 

price reviews (by potential entrants) are conducted annually during every year of the 60 

year planning horizon and ignores the remainder.47

As was the case in section 2, TELRIC prices are less than contestable prices when 

investment costs are falling and greater than contestable prices when investment costs are 

rising.48 Also as in section 2, the dynamic pattern of both TELRIC prices and contestable 

prices follows the dynamic pattern of investment costs, and the TELRIC prices could be 

corrected to yield the same rate of return as the contestable prices. If investment costs do 

not change over time (i.e. λ = 0) then both TELRIC and contestable prices fall over time

due to growing demand and economies of scale in the investment cost function. 

 

46 The upper bound in (15) is based on a simple feasible and recursive investment plan in which the firm

invests once every L periods, investing enough at each investment date to satisfy demand until the newly 

acquired assets expire. An investment plan based on these assumptions is representative of traditional 

network planning by ILEC engineers. See, e.g. “Telcordia Notes on the Networks,” p. 12-2, which states:

“This [distribution] plant is designed to meet the greatest expected customer demand in an area for the life 

of the plant.” Equation (15) also assumes non-decreasing returns in capacity acquisition over the range of 

acquired capacities, and that 0 ≤ δγ (1+η) < 1, which holds in all of our examples. 47 Only the first 30 years are illustrated. The monthly remainder is inconsequential, as an estimate based on

the upper bound (15) does not exceed 23 cents per month during the 30 illustrated years in Figure 21. When 

investment costs fall by 5% per year, as in Figure 22, the remainder term never exceeds 0.16 cents per 

month. 48 TELRIC prices also understate contestable prices when investment costs are constant. The magnitude of 

the assumed utilization rate, or fill factor, also influences the relative magnitude of the two price schedules. 

 40 
5 10 15 20 25 30

t

5

6

7

8

9

Price ccccccccccccccc Month

Figure 21. Monthly Contestable and Triennially Reviewed Actual TELRIC Prices 

Distribution Portion of Loops: Case of Increasing Investment Cost 

5 10 15 20 25 30

t

2

4

6

Price cccccccccccccccc Month

Figure 22. Monthly Contestable and Triennially Reviewed Actual TELRIC Prices 

Distribution Portion of Loops: Case of Decreasing Investment Cost 

The upper bound on prices given in (15) can be used to verify that the numerical 

dynamic cost minimum discussed in this section indeed gives important information 

about prices that is missed by a more crude estimate of costs. Figure 23 plots the upper 

bound (15) for the benchmark loop scenario along with the contestable price path from 

Figure 21. We see that the contestable prices based on dynamic cost minimization are 

substantially below the theoretical upper bound. As the theoretical upper bound is 

representative of traditional ILEC network planning (see footnote 44), Figure 23 shows 

that explicit consideration of dynamically efficient costs can result in substantially lower 

cost estimates than those based on traditional network planning. 

 41 
5 10 15 20 25 30

t

5

6

7

8

9

10

11

Price ccccccccccccccc Month

Figure 23. Monthly Minimum Cost Contestable & Theoretical Upper Bound Prices 

Distribution Portion of Loops: Case of Increasing Investment Cost 

Finally, we note that the results of this section suggest that alternative definitions 

of TELRIC pricing may be possible. Such definitions could capture some of the dynamic 

investment behavior considered herein. For example, the contestable price (13) could be 

used as a definition of a “dynamic” TELRIC price. These prices do not depend on 

embedded costs, so they are entirely forward-looking, but do have the feature that they 

are dynamically consistent: A firm that invests efficiently from time zero forward, and 

serves the whole demand included in the price calculation either directly or through 

unbundled elements, will exactly recover costs incurred at or after time zero with these 

prices, including the rate of return used in calculating the prices. Such prices can 

therefore be regarded as incentive regulation of investment decisions of entrant and 

incumbent firms. A price path calculated in this way would explicitly abandon 

levelization (even between TELRIC price reviews) and would be based on cost 

minimizing dynamic investment decisions at every point in time. 

With these prices, a firm that has existing capacity at time zero will recover the 

cost of its existing capacity as well, provided 1) the existing capacity was obtained as part 

of a dynamically efficient investment plan, 2) all current parameter estimates (including 

the expected rate of change of asset acquisition costs) are the same as they were when the 

existing capacity was installed, and 3) past prices were calculated according to the 

contestable rule. Some adjustment to the prices may be required if any of these 

properties do not hold. For example, the cost minima could be calculated with an 

estimate of existing capacity used as initial values, as in Figure 20, and then some

addition could be made to the contestable prices to recover some costs associated with

existing capacity at time zero.49 This would smooth the transition to fully forwardlooking prices by providing a way to recover some embedded costs that were incurred 

 

49 This approach would not necessarily require the use of historical depreciation. For example, the same

cost function (or cost model) used to estimate future investment costs could be used with past parameter 

values to estimate the cost of efficient past investments. The resulting contestable price path could be

compared to prices that actually prevailed to determine how much of efficiently incurred embedded costs 

have not yet been recovered and how much will be recovered during the remaining life of the assets. 

 42 
under a previous regulatory compact. Details of possible additions for embedded costs, 

that preserve incentives for efficient entry and exit, are left for future research. 

4. Concluding Comments 

The objective of this paper has been to evaluate the use of static cost proxy models in 

setting forward-looking prices such as the prices set according to TELRIC. Section 2 

compared the time paths of prices and depreciation under traditional regulatory 

accounting with the prices and depreciation implied by various versions of TELRIC. 

Under levelized TELRIC pricing, where prices are assumed to remain constant for the 

entire economic life of an asset, both prices and implied depreciation schedules follow 

significantly different trajectories than under traditional regulation. Under both pricing 

rules, however, the firm earns revenues that fully recover investment costs and a target 

rate of return. 

When TELRIC prices are recomputed at intervals shorter than asset lives and 

investment costs change over time, however, the firm will generally not earn the target

rate of return. In these cases, a correction factor must be applied to the TELRIC price 

path in order for revenues to exactly recover investment cost, including the target rate of

return. When investment costs are falling by 11% per year (as is assumed for switching 

assets in the FCC Synthesis Model), the TELRIC correction factor is approximately 50%. 

That is, switching prices should be increased by 50% from those suggested by Synthesis 

Model runs. When investment costs are rising at the rate of inflation, assumed to equal 

3% per year, the TELRIC correction factor is -20%. In this case, TELRIC prices should 

be reduced by 20% from those suggested by model outputs. 

Section 3 of the paper considered a firm’s cost minimizing investment decisions 

over time. Two different assumptions about asset lives were analyzed. Under one 

scenario, characterized as “light bulb” performance, assets have a fixed and known 

physical life. In a second scenario the assets last forever, but due to technological change 

it is assumed that they will be replaced by a new generation of assets regardless of the 

date at which the original assets were put into service. In both scenarios the optimal 

investment path and the implied utilization rates were derived under a variety of 

assumptions about the relevant input parameters. TELRIC pricing rules, as well as a 

contestable pricing rule, were considered. 

 43 
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