## _wp1086

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### Major findings and empirical puzzles
- The share of intangible expenditures in overall investment has increased steadily since the early 1980s; by 2000 intangible investment was as large as investment in physical capital.
- Distinguishing price from quantity for intangibles is crucial: part of the rise in intangible expenditures reflects higher unit costs (linked to skilled labor compensation) rather than a purely real increase in intangible quantity.
- Constructing a model-consistent price series for intangibles tied to university-graduate and executive compensation yields an aggregate investment deflator that:
  - falls until the mid-1980s, then rises up to 2000, and declines afterwards;
  - contrasts with the national accounts’ physical investment deflator, which exhibits a downward trend starting in the mid-1950s.
- An extended q-theory with intangible + tangible intermediates produces a smoothly behaving market-value capital stock; Tobin's q predominantly remains above and close to its equilibrium value (avoids prolonged q < 1 episodes observed in tangible-only models).
- Decomposition: the rise in overall capital stock from mid-1980s to late 1990s was mainly driven by intangible accumulation; after 2000 the intangible contribution falls consistently to levels similar to 1990.

### Model structure and key equations (preserved terminology)
- Intangible production technology (as stated): It = Is xhθ (expression preserved in source).
- Intangible unit price under perfect competition (preserved form): pI t = (we t Exec)φ (we t Univ)1−φ θ−1.
- Final investment good tx combines intangible Itx and tangible Ttx via CES:
  - tx = [a t pI t^ρ + (1 − a t) pT t^ρ]^(1/ρ) (presented in source as the CES aggregation with parameter ρ).
  - Baseline: ρ = 0 (CES → Cobb-Douglas / Divisia index); ρ varied in sensitivity analysis.
- Capital accumulation (as in source): k t+1 = (1 − δ) k t + x t.
- Adjustment costs: quadratic homogeneous-of-degree-one cost C(.,.) (equation (16) in source), leading to the investment FOC linking shadow value λ t and acquisition price p x t (equations (15)–(17) in source).
- Hayashi identity used: λ t k t = v t (equation (18)), linking market value of securities v t to installed capital k t.
- Tobin's q definitions (preserved):
  - q t = λ t / p x t (equation (28));
  - q t = v t / (p x t k t) (equation (29)).

### Calibration choices and data definitions (exact values preserved)
- Adjustment-cost parameter: α = 8 (represents doubling/halving time in the calibration rationale).
- Depreciation rate: "The depreciation rate of 2.6 percent per quarter is used by national income accounts for physical capital." (verbatim).
- Baseline elasticity of substitution: ρ = 0 (Cobb-Douglas embodiment baseline); sensitivity analysis includes ρ = 0.5 and ρ = −0.5.
- Initial shares and starting values:
  - Starting share of executives in wage bill: ϕ0 = 0.33 φ (average of 0.47 and 0.21 boundary values cited).
  - Starting share of tangible expenditures in overall investment: a0 = 0.74 a.
- Measurement of v t: market value of net financial claims (equity, bonds, loans and mortgages, short-term paper, trade payables, life insurance and pensions) for non-farm, non-financial corporate sector (1950Q1–2005Q4; Hall (2001) constructs series).
- Price of tangible investment goods: BEA implicit deflator for fixed non-residential investment (Table 7.1, line 32).
- Intangible expenditures data source: Corrado et al. (2006).

### Quantitative findings and behavior of aggregates (preserved descriptions)
- Inferred capital stock k:
  - Smooth increase until the peak of 2000, then falls and flattens out (Figure 8 in source).
  - Allowing for irreversibility does not materially affect the behavior of the capital stock (Appendix D).
- Aggregate investment deflator p x constructed as a share-weighted index including pI and pT (equation (26) in source) exhibits a rise up to 2000 and decline afterwards; this offsets firm-value movements and stabilizes q.
- Contribution of intangibles:
  - Substantial driver of the rise in overall capital stock from mid-1980s until late 1990s.
  - After 2000 the contribution of intangibles falls consistently to levels similar to 1990.
  - Possible interpretation: information-technology arrival and accelerated obsolescence contributed to post-2000 fall in intangible value.
- Implications for Tobin’s q:
  - Extended q-theory: q remains predominantly above and close to equilibrium; avoids extended q < 1 episodes.
  - Hall (2001)’s assumption pI = pT produces a more volatile q and negative implied intangible quantities in some periods.

### Comparison with prior literature (preserved contrasts)
- Hall (2001):
  - Assumes pI = pT and separate accumulation for tangible and intangible capital; results include negative intangible quantities starting mid-1970s.
  - Under Hall’s setup, implied capital stock shows a pronounced bell-shape around 2000 (massive investment then disinvestment).
- McGrattan and Prescott (2007a, 2007b):
  - Use general-equilibrium approaches and sometimes "guess" pI; do not report pI series in published work.
  - This paper differs by deriving pI from input costs (skilled wages and θ) rather than guessing, and by aggregating intermediates into a single homogeneous capital stock index.
- Quantitative comparison note preserved from source: the extended q-theory reports larger short-run swings in inferred intangible investment (textual comparisons preserved, e.g., rises/falls of 150 percent, 300 percent, 350 percent in cited intervals as reported in the source narrative).

### Sensitivity analysis (Table 2 preserved values)
- Two experiment classes: elasticity of substitution ρ variation and alternative wage series for constructing pI.
- Table 2 — Contribution of intangibles to capital stock (values preserved exactly):

  - 1950-1960: Baseline model 0.17 | Rho = -0.5 0.22 | Rho = 0.5 0.17 | Using only skilled labor 0.06 | Using only the "salary" component 0.13 | Using only the "salary+shares" component 0.07
  - 1960-1970: Baseline model 0.79 | Rho = -0.5 0.76 | Rho = 0.5 0.79 | Using only skilled labor 0.77 | Using only the "salary" component 0.79 | Using only the "salary+shares" component 0.78
  - 1970-1980: Baseline model 0.66 | Rho = -0.5 0.62 | Rho = 0.5 0.65 | Using only skilled labor 0.65 | Using only the "salary" component 0.66 | Using only the "salary+shares" component 0.66
  - 1980-1990: Baseline model 0.24 | Rho = -0.5 0.32 | Rho = 0.5 0.26 | Using only skilled labor 0.32 | Using only the "salary" component 0.26 | Using only the "salary+shares" component 0.29
  - 1990-1995: Baseline model 0.27 | Rho = -0.5 0.41 | Rho = 0.5 0.30 | Using only skilled labor 0.38 | Using only the "salary" component 0.30 | Using only the "salary+shares" component 0.34
  - 1995-2000: Baseline model 0.34 | Rho = -0.5 0.54 | Rho = 0.5 0.42 | Using only skilled labor 0.48 | Using only the "salary" component 0.39 | Using only the "salary+shares" component 0.43
  - 2000-2005: Baseline model 0.30 | Rho = -0.5 0.35 | Rho = 0.5 0.34 | Using only skilled labor 0.35 | Using only the "salary" component 0.32 | Using only the "salary+shares" component 0.35

- Sensitivity summary (preserved conclusions):
  - Varying ρ or wage components changes the quantitative level of p x and the contribution shares but does not alter the qualitative message: national accounts’ physical investment deflator understates acquisition cost of capital in the 1990s because it omits rising intangible prices; the capital series and contribution splits remain qualitatively robust.

### Robustness checks and appendices (preserved points)
- Allowing for strong irreversibility (asymmetric quadratic adjustment costs with α_- up to ten times α_+) does not materially affect the generated capital stock because net investment is almost always positive in the data.
- Data sources and construction details (preserved):
  - CPI: quarterly CPI with base year 1996 (BLS Series ID: CUUR0000SA0).
  - Output per worker: series IDs PRS84006163 and PRS88003163 depending on subperiods.
  - Wages by educational attainment: CPS table A-3; university worker wage constructed as weighted mean of bachelor's and advanced-degree earnings, extended back to 1950 using output per worker growth (per Lemieux (2007) justification).
  - Executive compensation: Frydman and Saks (2007) panel; executive counts extended using OES and BEA series as described in source.
  - Real capital stock and value of securities: from Hall (2001) extended to 2005.
  - Intangible expenditures: Corrado et al. (2006).
- Calibration of initial I_p and θ made to satisfy long-run investment condition and to grow θ at the rate of change of output per worker.

### Conclusions and avenues for future work (preserved recommendations)
- Key messages:
  - Accounting for intangible investment and its time-varying price alters the aggregate investment deflator and stabilizes inferred market-value capital series and Tobin’s q.
  - Movements in skilled-labor and executive compensation are central to understanding the price of intangibles and thereby capital accumulation and firm valuation.
  - The rise in overall capital stock from mid-1980s to late 1990s was mainly driven by intangible accumulation; the contribution declines after 2000.
- Proposed future work (verbatim intent preserved):
  - Use the corrected acquisition cost of capital (including intangibles) to re-examine econometric investment equations and address low R2 and serial correlation issues in investment regressions (Chirinko (1993) referenced).
  - Revisit Hall (2003)'s puzzle that payments to intangibles appear almost nil by recomputing firms' earnings corrected for intangible capitalization to test whether treating intangibles as expenditures causes undervaluation of earnings.

*Source: _wp1086 - References (PDF chapter/section).*

### References .............................................................................................................

### _wp1086 - References

### I. Introduction
- Recent empirical findings point to a considerable increase in the share of intangible expenditures in overall investment since the early 80s.
- This paper extends the q-theory of investment to account for:
  - both the existence of intangible investment goods and the movement of their price over time;
  - the embodiment of intangible goods in tangibles;
  - the role of relative price movements in measuring each type of investment good’s contribution to the overall capital stock.
- Uses the model to:
  - derive quantitative implications on the behavior of the capital stock; and
  - measure the contribution of embodied intangible investment goods to the overall capital stock in the U.S.

- Empirical background and motivation:
  - Corrado et al. (2006) construct a data set on U.S. firms' spending on an identified list of intangible inputs consisting of three main categories: computerized information (software and database expenses), innovative property (scientific and non-scientific R&D) and economic competencies (training expenses, advertising expenditures and spending on organizational change and design).
  - They find that the share of intangible investment in overall investment since 1950 has steadily increased over time and that by 2000, investment in intangible capital was as large as investment in physical capital.
  - Other approaches: Hall (2001) infers intangible investment by matching stock-market movements; McGrattan and Prescott (2007a) use a general equilibrium model to generate an intangible series that matches aggregate moments.
  - Key limitation of prior approaches: none separately identify indices for the price and quantity of intangible investment goods. Corrado et al. (2006) and Hall (2001) assume the price of intangibles equals the price of tangible investment goods; McGrattan and Prescott (2007a) guess a price series.

- Price versus quantity distinction:
  - The distinction between the price and the quantity of intangibles is important to determine how much of the increase in overall expenditure on intangibles is attributable to the real contribution of intangibles.
  - Substantial increase in the relative cost of the main input to intangibles production, skilled labor, suggests the relative price of intangibles also rose (references: Katz and Autor (1999) and Lemieux (2007)).
  - Real average compensation of university graduates and executives has been increasing since the early 1980s until it peaked in 2000 and declined somewhat; this should translate into a rise in unit cost of intangibles up to 2000 and a subsequent fall.

- Modeling approach:
  - Develops and implements a generalized q-theory of investment where intangible and tangible goods are optimally combined as intermediates in the production of a final aggregate investment good; that final investment good is what is accumulated over time and used in output production.
  - Changes in the share of expenditures on the two types of investment goods over time reflect exogenous technological change in the final investment good sector.
  - The price of intangible goods is modeled to reflect the costs of producing them, linked to the wages of skilled labor. A price series for intangibles is constructed based upon compensation received by university graduates and executives and estimates of their share in intangibles production.

- Aggregate implications:
  - Accounting for intangible investment yields an aggregate investment deflator whose behavior differs markedly from the physical investment deflator in national income accounts:
    - national accounts’ investment deflator exhibits a downward trend starting in the mid-1950s;
    - the model’s investment deflator that includes intangible investment has a downward trend up to the mid-1980s, then rises up to 2000 and declines afterwards.
  - The model generates a smoothly-behaving series of capital stock with market value that predominantly remains above and close to its acquisition cost, implying Tobin's q fluctuates closely around its equilibrium value — a feature not observed when capital exclusively consists of tangible investment.
  - These implications suggest the mixed econometric success of the q-theory might be a consequence of omission of intangible investment.

- Composition of capital stock:
  - Decomposition shows the rise in overall capital stock from the mid-80s until the late 90s was mainly driven by an increase in intangible investment despite an increase in tangible investment.
  - Once price movements of the two investment goods are accounted for, the relative contribution of accumulated intangible investments to overall capital stock was higher prior to the 1990s and substantially lower during the 1990s than other authors have found.

- Paper roadmap:
  - Section 2 contrasts this approach with others.
  - Section 3 outlines the generalized q-theory and how contributions of each investment good are disentangled.
  - Section 4 discusses calibration, data sources, and technology by which skilled workers (university graduates and executives) produce intangibles.
  - Section 5 presents findings on inferred capital stock, Tobin's q, and changing composition of capital stock.
  - Section 6 compares findings with Hall (2001) and McGrattan and Prescott (2007a).
  - Section 7 reports sensitivity analysis.
  - Section 8 concludes with future research discussion.

### II. Contrast with other approaches
- Traditional q-theory (baseline neoclassical model):
  - Predicts firm investment decision trades off marginal benefit of increasing capital by one unit versus cost of acquiring and installing that unit.
  - Hayashi (1982) showed conditions under which the marginal benefit of increasing capital is identified with the net value of outstanding securities.
  - Empirical anomalies from figures discussed:
    - Large discrepancy between net value of securities and acquisition cost of tangible capital stock suggests a very long adjustment period; Summers (1981) implied an adjustment period of 8 years.
    - Period from the mid-1970s until the mid-1980s features aggregate value of firms below acquisition cost of capital, implying puzzling arbitrage opportunities.

- Four broad research directions to explain Tobin's q anomalies:
  1. Financial market imperfections and bubble models (LeRoy (2004) survey) — mixed success.
  2. Relax restrictive assumptions of the representative firm framework (e.g., perfect competition, constant returns to scale, quadratic adjustment costs) — some improvements but sometimes marginal (Caballero (1999) review).
  3. Focus on heterogeneity at the plant level rather than aggregate representative firm (Bond and VanReenen (2007) review).
  4. Account for intangible investment goods omitted from previous models — argues departures of net value of securities from acquisition cost of tangible capital reflect accumulation of intangible capital (Hall (2001) proposes this).

- Limitations of Hall (2001) approach (as highlighted by this paper):
  - Hall treats intangible and tangible investment goods as perfect substitutes and assumes equal prices for both; under the framework of this paper, those assumptions are violated.

### Key empirical and modeling observations (selected)
- The list of intangible items used by Corrado et al. (2006) is accepted in the national accounting consensus (see Vosselman (1998)).
- Similar studies for other countries: Belhocine (2009a) (Canada), Marrano and Haskel (2006) (U.K.), Fukao et al. (2007) (Japan) reached similar findings on rising intangibles.
- Observed wage dynamics:
  - Real average compensation of university graduates and executives increased since early 1980s, peaked in 2000, then declined somewhat — used to construct intangible price series.
- Aggregate investment deflator behavior when intangibles included:
  - Downward trend up to mid-1980s, rises up to 2000, declines afterwards — contrasts with physical investment deflator’s long downward trend starting mid-1950s.

### Main contributions of the paper
- Extends q-theory of investment to:
  - incorporate intangible investment goods and their time-varying prices;
  - allow for embodiment of intangible goods in a composite tangible–intangible capital stock;
  - link intangible unit costs to skilled labor compensation.
- Provides:
  - a model-consistent price series for intangibles tied to university-graduate and executive compensation and their production shares;
  - an inferred capital stock whose market value remains close to acquisition cost, yielding a Tobin's q that behaves smoothly around equilibrium;
  - a decomposition of capital stock showing the dominant role of intangible accumulation in the mid-80s to late-90s rise in overall capital stock, with adjusted estimates for the 1990s that are lower than those in previous literature once price movements are accounted for.

*Source: _wp1086 - References (PDF chapter/section).*

### introduction: the share of intangibles in the overall capital expenditures has been increasing

### _wp1086 - introduction: the share of intangibles in the overall capital expenditures has been increasing

### Major findings and empirical puzzles
- The share of intangibles in overall capital expenditures has been increasing over time and the unit cost of intangibles behaves the opposite of the price of physical capital.
- One anomaly documented in Hall's findings is that the quantity of intangibles falls below zero for a decade, starting in the mid-70s.
- Most studies find a substantial size for intangible capital; exceptions include Hall (2003) and Bond and Cummins (2000), who show that returns to physical capital exhaust all payments to capital, leaving no residual to reward intangible capital services.

### Literature comparisons and how this paper departs
- Hall (2001, 2000, 2003): Hall assumes equality between the price of intangibles and tangibles and that intangible and tangible capital stocks evolve separately (implicitly perfect substitutes in Hall (2001)). Hall (2000) links university graduate wages with intangible formation but does not account for changing firm investment structure; Hall (2001) finds negative intangible quantities in the mid-70s.
- Eliades and Weeken (2004): Apply Hall's methodology to the U.K.; find negative intangibles throughout the 70s, the 80s and early 90s, but similar qualitative results to Hall for the late 90s.
- Belhocine (2009b): Applies approach to Canada after 1990; reaches same qualitative conclusions as U.S. and U.K. studies.
- McGrattan and Prescott (2005a, 2005b, 2007a, 2007b):
  - 2005a: Use a general equilibrium approach (no frictions except taxes) to rationalize late 1990s U.S. and U.K. stock market rises; rationalize intangible investment sizes in Corrado et al. (2005) and use tax regulation changes to explain differing U.K./U.S. stock market performance.
  - 2005b: Show that accounting for unmeasured intangible investment in a standard real business cycle model can explain low measured productivity in the early 90s because expensing intangible investment undervalues GDP and biases productivity downwards.
  - 2007a: Extend baseline real business cycle model to allow production of intangibles by the representative agent; feature two stocks of capital (tangible and intangible) to reconcile model predictions with observed increases in hours worked after the 1990s.
  - 2007b: In their solution they "guessed" the price of intangibles to derive a series of intangible investment.
- How this paper departs:
  - Relaxes the assumption of equality between the price of intangibles and tangibles.
  - Allows some degree of substitutability between the two investment goods.
  - Calibrates an extended q-theory model to the aggregate value of securities with a single homogeneous capital stock index embodying both tangible and intangible investment goods.
  - Derives the price of intangibles from the behavior of the cost of its main inputs (rather than guessing).

### Model structure: q-theory extended to include intangibles
- Production of intangibles:
  - "The production of intangibles is governed by the following technology:  = Is xh                                                          (1)" (as stated in the source).
  - Skilled labor is split: amount sl used for production of output and remainder sh used for production of intangibles; skilled labor wage is sw; price of intangible capital denoted Ip and given by "=. s I w p                                                          (2)" (as stated).
- Firms and inputs:
  - Perfect competition in input and output markets.
  - Two types of labor: unskilled (ul, wage uw) used only for output; skilled (sl and sh, wage sw) used for output and intangibles.
  - Output production: "The production of output proceeds according to 1 (,,) us ttt Fk l l  where (.)F is assumed to be homogeneous of degree one."
- Composite investment good and aggregation:
  - Final investment good xt combines intangible goods Itx and tangible investment goods Ttx via CES:
    - "At each period t, the firm combines intangible goods Itx with tangible investment goods Ttx to produce a final investment good tx according to the following constant elasticity of substitution (CES) function   1 =()(1 )() . TI ttttt xaxax    (3)" (as stated).
  - ρ is the elasticity of substitution (assumed ≤ 1); t a is the income share of each input and is allowed to vary over time to capture increasing spending on intangibles relative to tangibles (biased technological change).
  - Baseline calibration sets ρ = 0 (CES → Cobb-Douglas / Divisia index), with sensitivity analysis allowing values >0 and <0.
  - When ρ = 0, xp in equation (11) takes form: "1 = 1 aa tt TI x tt t tt pp p aa            (25)" (as stated).
- Capital accumulation and adjustment costs:
  - Aggregate investment accumulates via: "1 =(1) ttt kkx    (4)" (as stated), where δ is depreciation.
  - Adjustment costs modeled by a quadratic, homogeneous-of-degree-one cost function C(.,.) (equation (16) in source): "2 11 1 (,  )=. 2 t ttt t x Cx kk k        (16)" (as stated).
  - First-order condition (investment equation) relating shadow value λt and acquisition price pxt:
    - "1 =(,) x tt xtt pCxk   (15)" and with quadratic cost leads to "1 1 =(). x t tt t x p k     (17)" (as stated).
  - Hayashi identity used: " = ttt vk (18)" producing equation linking observable pair (vt, pxt) to capital stock via a quadratic (20) and positive-root law of motion (21) as in the source.
- Disentangling contribution of each investment type:
  - Forward substitution of the capital accumulation equation and application of Euler theorem to tx yields relationships (22)–(24) in the source allowing the capital stock to be expressed in efficiency terms and the contribution of each investment type to be recovered, each weighted by relative price.

### Calibration choices and parameter interpretations
- Adjustment-cost parameter α:
  - α = 8. Rationale: represents the time it takes for capital stock to double (halve) when λ doubles (halves); Hall (2001) cites Shapiro (1986) to justify a doubling time parameter of 8 quarters.
- Depreciation rate:
  - "The depreciation rate of 2.6 percent per quarter is used by national income accounts for physical capital." (preserved verbatim).
- Initial condition:
  - Initial capital stock set so that at pre-initial quarter the firm value reflects its quantity of installed capital, i.e., "11 = tt kv " (equivalent to assuming 1 =1 t  ). Recursion is not sensitive to initial conditions for long-run behavior.
- Substitutability / embodiment:
  - Baseline ρ = 0 to capture embodiment of intangibles in tangibles via Cobb-Douglas aggregation; sensitivity analysis will vary ρ above and below 0.
- Price of intangibles:
  - Derived from behavior of the cost of its main inputs (skilled wages and productivity parameter θ), rather than guessed.

### Model objectives and scope
- Main objective: relate aggregate value of securities (firm value) to the value of capital in place within firms and generate a time series for the capital stock and Tobin's q in a model that aggregates tangible and intangible investment into a single homogeneous capital stock index.
- Differences emphasized with other work:
  - Unlike Hall (2001), the model allows the price ratio of intangibles to tangibles to vary over time.
  - Unlike McGrattan and Prescott (2007b), the price of intangibles is derived from input costs instead of being guessed.
  - The model includes executives in skilled labor and adopts a long-term approach that illuminates pre-1990 and post-1990 events.

*Source: _wp1086 - introduction: the share of intangibles in the overall capital expenditures has been increasing*

### 0.026 Hall (2001)

### 0.026 Hall (2001)

### B. Varying Parameters — Measurement of vt and data scope
- vt is defined as the present value of payouts to securities' holders and is measured as the market value of net financial claims (financial liabilities minus financial assets), including equity, bonds, loans and mortgages, short-term paper, trade payables, life insurance and pensions.
- For all t, λt k t = vt, so the value of securities equals the value of the installed capital stock.
- Data: national balance sheet account at market value from 1950Q1 to 2005Q4; equity at market value, other liabilities at book value converted to market value by Hall (2001).
- Sector focus: non-farm, non-financial corporate sector (owns around 90 percent of the non-residential fixed capital stock).
- Appendix B contains full data sources and manipulations.

### Intangibles: production, unit cost, and links to skilled labor
- Intangibles produced by a class of skilled workers: university graduates (Univ h) create raw intangibles, executives (Exec h) create organizational designs. Combination:
  - It = φExec,t θExec,t = φUniv,t θUniv,t  (expressed in text as 1()() . tt Is ExecUniv ttttt xh h h φφ θθ   )
- Unit cost of an intangible capital good under perfect competition:
  - pI t = (we t Exec)φ (we t Univ)1−φ θ−1 (expression preserved in original form)
- φ allowed to vary to reflect executives' weight in intangible production wage bill.
- Executive compensation components used: Sal + Bonus; Sal + Bonus + Shares; Sal + Bonus + Shares + Options (Figure 4).
- Because NIPA records only 20 percent of identified intangible expenditures (Corrado et al. (2006): software, mineral exploration, architectural and design services), TT px equals private fixed investment from NIPA minus these recorded intangibles.

### Constructed prices and shares: behavior over time
- Price of intangibles pI rises at a constant rate from 1950 until mid-80s; growth rate accelerates up to 2000 driven by CEO wage rise and share in the wage bill; growth falls afterwards.
- Share of intangible expenditure in overall investment, a, exhibits a striking upward trend beginning in the 1950s; after 2000 firms' investment in intangibles surpasses tangible investment (Figure 6).
- Aggregate investment price index px constructed as a shared-weighted index:
  - px t = (a t pI t)α (px t T)1−α  (presented in the original formulaic form as equation (26))
- Behavior of constructed aggregate investment deflator contrasts with NIPA physical investment deflator: model deflator falls until mid-1980s, then rises until 2000, peaks, and then falls (Figure 7). Physical investment deflator from national accounts shows a secular downward trend beginning mid-1950s.

### V. Quantitative findings and results — Behavior of capital stock and contributions
A. Behavior of the stock of capital
- Inferred capital stock k smoothly increases until the peak of 2000, then falls and flattens out (Figure 8).
- Allowing for irreversible investment does not affect the behavior of the capital stock series (Appendix D).

B. Contribution of intangibles and tangibles to capital stock
- Intangible investment made a substantial contribution to the rise in overall capital stock from mid-80s until late 90s.
- After 2000, the contribution of intangibles falls consistently to levels similar to 1990 (Figure 9).
- Interpretation: arrival of information technology and related obsolescence may have accelerated the fall in intangible value post-2000.

C. Implications for Tobin's q
- Tobin's q defined as q t = λ t / p x t (equation (28)); with λ t k t = v t, q t = v t / (p x t k t) (equation (29)).
- Extended q-theory produces a Tobin's q that predominantly remains above and close to its equilibrium value; q does not fall below one for extended periods (Figure 10).
- Hall (2001) produces a more volatile q because he assumes pI = pT (single investment price) and lacks the rise-then-fall in pI that offsets firm value movements.

### VI. Comparison with existing literature (Hall (2001); McGrattan and Prescott (2007a))
- Hall (2001) assumes pI = pT and two separate capital accumulation equations (one per investment good), leading to value of the firm V t = pT t kT t + pI t kI t (equation (31) in text).
- McGrattan and Prescott (2007a) use V t = pT t kT t + pI t kI t + ... (their setup implies pT is the numeraire and pI is guessed as part of general equilibrium; pI series not reported).
- In this paper, the two investment goods are intermediate and yield a single aggregate price deflator attached to one overall stock of capital (equation (34) in text).
- Embodiment hypothesis consequences: when TI pp =, the capital stock can be written as k = kT + kI + ... (equation (36)).
- Comparison of findings:
  - Hall's implied capital stock has a pronounced bell-shape around 2000 (Figure 11); Hall's assumption that acquisition cost equals tangible deflator (falling) implies massive investment then disinvestment.
  - Extended model: acquisition cost skewed toward intangibles whose price rises from the 80s and falls post-2000, yielding a much less pronounced bell-shape.
  - Hall's model implies negative quantity of intangible capital from mid-70s until mid-80s; extended model avoids this anomaly.
  - Comparison with McGrattan and Prescott (2007a) is limited by their not reporting pI and not reporting intangible investment series. Qualitatively, McGrattan and Prescott find intangibles falling in early 90s; this paper finds intangible investment rising in that period. Quantitatively, the extended q-theory reports larger swings: e.g., rise of intangible investment by 150 percent between 1990 and 1992 and fall by 150 percent between 1992 and 1994; then rise by 300 percent until 1999 and fall by 350 percent afterwards (textual comparisons preserved).

### VII. Sensitivity analysis
- Two experiments:
  1. Vary elasticity of substitution ρ from baseline ρ = 0 to ρ = 0.5 (more substitutability / less complementarity) and ρ = −0.5 (more complementarity / less substitutability).
  2. Use different wage series for intangible price calculation: only real wage of university graduates; only "salary" component of executive compensation; only "salary+shares" component.
- Impact on contribution of intangibles to capital stock summarized in Table 2 (values preserved exactly):

  Periods | Baseline model | Rho = -0.5 | Rho = 0.5 | Using only skilled labor | Using only the "salary" component | Using only the "salary+shares" component
  - 1950-1960: 0.17 | 0.22 | 0.17 | 0.06 | 0.13 | 0.07
  - 1960-1970: 0.79 | 0.76 | 0.79 | 0.77 | 0.79 | 0.78
  - 1970-1980: 0.66 | 0.62 | 0.65 | 0.65 | 0.66 | 0.66
  - 1980-1990: 0.24 | 0.32 | 0.26 | 0.32 | 0.26 | 0.29
  - 1990-1995: 0.27 | 0.41 | 0.30 | 0.38 | 0.30 | 0.34
  - 1995-2000: 0.34 | 0.54 | 0.42 | 0.48 | 0.39 | 0.43
  - 2000-2005: 0.30 | 0.35 | 0.34 | 0.35 | 0.32 | 0.35

A. Elasticity of substitution experiments
- ρ = 0.5: px behaves similarly to baseline but with less pronounced rise and fall.
- ρ = −0.5: px behaves almost opposite to ρ = 0.5 case.
- Generated capital and contributions are not significantly affected; Table 2 shows discrepancies.

B. Wage series experiments
- Using only skilled labor wage: because θ grows faster than skilled wages, pI falls monotonically; pX remains almost always higher than pT because fall in pI is less steep than fall in pT before the 80s and after mid-90s. Qualitative contribution results remain same though quantitatively resemble Hall's findings.
- Using successive components of executive compensation (removing components one at a time): composite investment price series retains same overall behavior but rise up to 2000 and subsequent fall are less pronounced. Downplaying executive compensation makes pX closer to pT and implied capital series closer to Hall.

- Overall sensitivity conclusion: changing ρ or wage-series alters px quantitatively but not the qualitative message — national investment deflator understates price of capital goods in the 90s due to rising intangible contribution; capital series and contribution splits are not qualitatively affected.

### VIII. Conclusion — key messages and avenues for future work
Key findings:
- Extending q-theory to account for intangible investment yields an aggregate investment price index whose trend contrasts with national accounts' investment deflator because intangibles' price movements are excluded from NIPA.
- The extended model produces a smoothly-behaving capital stock whose market value predominantly remains above book value (Tobin's q mostly above one), unlike empirical measures when capital is only tangible.
- Rise in overall capital stock from mid-80s to late 90s was driven mainly by intangible investment; contribution of intangibles fell after 2000.
- Movements in the price of intangibles (linked to skilled labor and executive compensation) are crucial for understanding capital accumulation and firm valuation.

Proposed future work:
- Use the corrected acquisition cost of capital (including intangibles) to re-examine econometric investment equations and address low R2 and serial correlation issues in investment regressions (Chirinko (1993) referenced).
- Revisit Hall (2003)'s puzzle that payment to intangibles appears almost nil by recomputing firms' earnings corrected for intangible capitalization to test whether treatment of intangibles as expenditures causes undervaluation of earnings.

*Source: _wp1086 - 0.026 Hall (2001), extracted from the provided content unit.*

### References

### _wp1086 - Appendix summaries

### A. Derivation of First-Order Conditions
- Firm's dynamic optimization problem is set up with a Lagrangian; first-order conditions equate lifetime return to increasing capital by one unit with its marginal cost (price of a unit of capital plus marginal adjustment cost of installing this unit).
- The first-order conditions include:
  - Optimal investment condition determining the investment amount.
  - Two conditions equating real wage to marginal product of labor (usual labor market equilibrium).
  - A dynamic equilibrium equation for the Lagrangian multiplier λ with its continuation value.
  - The investment technology constraint recast in the first-order conditions.
- Bubbles in the shadow price of capital are ruled out; i.e., lim_{T→∞} 1/(1+r)^{T+1} q_{t+T} = 0.
- Interpretation: λ is the shadow price of an additional unit of capital; the first equation equates its lifetime return to its marginal cost.

### B. Data Sources and Definitions
- Price deflator and CPI:
  - Deflator: quarterly CPI with base year 1996.
  - CPI series: Bureau of Labor Statistics Series ID: CUUR0000SA0.
  - Monthly CPI averaged over three consecutive months to obtain quarterly equivalents.
- Output per worker series:
  - 1950–1957: output per person of the business sector (Series ID: PRS84006163).
  - 1957–1975: output per person of the non-financial corporate sector (Series ID: PRS88003163).
- Wages by educational attainment:
  - Source: Current Population Survey, table A-3 “Mean Earnings of Workers 18 Years and Over, by Educational Attainment, Race, Hispanic Origin, and Sex: 1975 to 2005”.
  - Five educational levels: not a high school graduate; high school graduate; some college/associate degree; bachelor's degree; advanced degree.
  - Earnings definition: total income for work as an employee during the income year, including wages, salary, armed forces pay, commissions, tips, piece-rate payments, and cash bonuses, before deductions.
  - Wage of university workers: weighted mean of bachelor's and advanced degree earnings (multiply mean earnings by number of workers for each degree, sum, divide by total number with bachelor's and advanced degree).
  - Data extended from 1975 back to 1950 by applying the growth of output per worker (assumes mean earnings of university graduates grew at same rate as productivity per worker from 1950 to 1975; supported empirically, see Lemieux (2007)).
- Wage bill of skilled labor:
  - 1975–2005: mean earnings of university graduates multiplied by number of university graduates.
  - Extended back to 1950 using economy-wide compensation of employees (BEA Table 1.10, line 2 “gross domestic income by type of income”).
- Executive compensation:
  - Source: Frydman and Saks (2007); comprehensive panel dataset 1936–2003 tracking individual officers in largest firms.
  - Sample: top three officers in 102 firms (largest 50 publicly traded corporations ranked by sales in 1940, 1960, 1990); authors conclude representativeness for largest 300 publicly-traded corporations.
  - Wage bill of executives: average CEO compensation from Frydman and Saks (2007) multiplied by number of chief executives.
- Number of chief executives:
  - OES Survey (BLS) occupational title “chief executive” occupational code 11-1011; available 1998–2005.
  - 1983–1997: OES used different classification; closest definition used is “management, business, and financial operations occupations” (Series ID: LNU02032202); growth rate of this occupation used to extend chief executive counts back to 1983.
  - 1950–1982: extended backward using growth rate of employment in the private sector (BEA Table 6.5 A-D “full-time equivalent employees by industry”).
- Price of tangible investment goods:
  - BEA implicit deflator for fixed non-residential investment, Table 7.1 “quantity and price indexes for GDP quarterly”, line 32.
- Real capital stock and value of securities:
  - Taken from Hall (2001) and extended to 2005.
- Intangible expenditures:
  - Taken from Corrado et al. (2006).

### C. Calibration of Starting Values
- Initial calibration approach:
  - Initial value of I_p is chosen so that the firm's investment condition (equation (19) in the text) holds in the long run (sample-average values used for variables).
  - Once 0_I_p is found, 0_θ is deduced and made to grow at the rate of change of output per worker; 0_I_p is also used to calculate starting value of x_p.
- Empirical starting values for shares:
  - Empirical starting value of the share of executives in overall wage bill: 0 = 0.47 φ.
  - Corrado et al. (2006) calibrate this value to 0 = 0.21 φ (based on executives spending 20 percent of their time in activities involving organizational change and design).
  - Author chooses starting value 0 = 0.33 φ as an average of the two boundary values; results are not sensitive to departures within this range.
  - Starting value of a (share of tangible expenditures in overall investment expenditures): 0 = 0.74 a.

### D. Allowing for Irreversibility in Investment
- Irreversibility modeled via piece-wise quadratic adjustment-cost function:
  - Adjustment-cost function C_t(x_t, k_t) specified with different curvature parameters for upward and downward adjustments:
    - If x_t/k_t - 1 >= 0: use α_+ > 0 and quadratic form with α_+.
    - If x_t/k_t - 1 < 0: use α_- < 0 and quadratic form with α_-.
  - Interpretation: α_+ and α_- represent the time it takes for the capital stock to double (halve) when λ doubles (halves).
- Calibration and robustness:
  - Main text sets α_+ = 8.
  - Allowing α_- to be up to ten times higher than α_+ (introducing strong asymmetry/irreversibility) does not affect the generated capital stock.
  - Explanation: gross investment is almost always above depreciation in the data (net investment almost always positive); when net investment is negative its magnitude is quite small, so aggregate evidence on irreversibility is limited.

*Appendix material and data source descriptions are drawn from the document text.*

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