1. Introduction
The AI buildout expands capital expenditure and contracts employment at the same time. Commentary treats the two as separate stories, one about investment and the other about labor markets. In a Leontief–Solow price system they are one event: a change of technique in which the capital stock coefficients of a sector rise while its direct labor coefficient falls. The price system then answers a definite question. Under what condition does this joint change lower prices, and what does it do to the rate of return, to the wage share, and to the sensitivity of prices to interest rates?
This note sets out the algebra. The exposition follows the convention of the EdgarStat blog: a definitional identity, one or more structural equations, and a reduced form. The last section translates the sectoral price equation into the firm-level regression estimated on Compustat data in Silva (2026), so that the structural alternatives become testable restrictions on a single reduced form.
2. Notation
Let n be the number of sectors. Vectors are columns; a prime denotes transposition.
- A is the n \times n matrix of intermediate input coefficients: a(ij) is the quantity of good j consumed per unit output of i. Under this convention the unit intermediate cost of sector i is (Ap)(i).
- B is the n \times n matrix of capital stock coefficients: b(ij) is the stock of good j held per unit output of i.
- h is the n-vector of direct labor hours per unit output. The letter avoids confusion with the identity matrix I and the scalar 1; the observable employment level of sector j is L(j) = h(j)\,x(j), with x the output vector.
- p is the price vector, w the money wage per hour, r the rate of return, and \delta the depreciation rate, taken uniform across capital goods for simplicity.
- \Delta denotes a finite one-period change; \lambda_{\max}(\cdot) the dominant (Perron–Frobenius) eigenvalue of a nonnegative matrix.
3. Labor Values
Marx’s labor values solve the identity “value equals value transferred plus living labor”:
\quad
(1)\qquad
v = Av + h,
\qquad
v = (I - A)^{-1} h = \sum_{k=0}^{\infty} A^{k} hthe series converging when \lambda_{\max}(A) < 1. With A fixed, \Delta v = (I-A)^{-1} \Delta h: a reduction of hours in sector j lowers every value through column j of the Leontief inverse. Values are proportional to direct labor, compounded through the input structure.
4. Solow’s Dynamic Price System
4.1 The dynamic identity
Solow (1959) priced the capital stock. A producer in sector i holds b(ij) units of good j per unit of output. The stock is acquired at p(t), carries interest r, and is worth (1-\delta)p(t+1) one period later. The unit cost of holding capital is therefore (1+r)Bp(t) - (1-\delta)Bp(t+1), and the price identity reads
\quad (2)\qquad p(t) = A\,p(t) + w\,h + (1+r)\,B\,p(t) - (1-\delta)\,B\,p(t+1)
This is a forward difference equation in prices. Its unstable roots make the stationary solution the single bounded path and the one to analyze.
4.2 The stationary reduced form
Setting p(t+1) = p(t) = p in (2) and collecting terms,
\quad (3)\qquad p = A p + w h + (r+\delta) B p
Define the Leontief–Solow matrix
\quad (4)\qquad M \equiv I - A - (r+\delta) B
so that (3) is the linear system Mp = wh with reduced form
\quad
(5)\qquad
p = w\, M^{-1} h = w \sum_{k=0}^{\infty} \big[A + (r+\delta)B\big]^{k} hThe series converges, and M^{-1} is nonnegative element by element, if and only if
\quad
(6)\qquad
\lambda_{\max}\big(A + (r+\delta)B\big) < 1Prices remain proportional to direct labor; the intermediate matrix A and the annualized capital charge (r+\delta)B are compounded together. Labor values (1) are the special case B = 0, r = 0. Since \lambda_{\max}(A + (r+\delta)B) is continuous and increasing in r when B \neq 0, condition (6) defines a maximum rate of return R^{*} by
\quad
(7)\qquad
\lambda_{\max}\big(A + (R^{*}+\delta)B\big) = 1and prices in wage units, p/w, diverge as r \to R^{*}.
5. The Joint AI Perturbation
5.1 An exact finite-change identity
AI adoption in sector j cuts direct hours, \Delta h(j) < 0, and raises the entries of row j of B in the columns of compute goods (processors, servers, data centers), \Delta b(jk) > 0. Write h_{1} = h + \Delta h and B_{1} = B + \Delta B for the new technique, and
\quad
(8)\qquad
N \equiv I - A - (r+\delta) B_{1} = M - (r+\delta)\Delta Bfor the new Leontief–Solow matrix. At the unchanged w and r the new prices satisfy N(p + \Delta p) = w(h + \Delta h). Expanding the left side and using Np = Mp - (r+\delta)\Delta B\,p = wh - (r+\delta)\Delta B\,p,
\quad (9)\qquad N \Delta p = \Delta c, \qquad \Delta c \equiv w\,\Delta h + (r+\delta)\,\Delta B\, p
Equation (9) is exact, not a first-order approximation. The vector \Delta c is the change in unit cost valued at the old prices p; it is propagated through the economy by the new inverse N^{-1}. If the new technique is feasible at the going rate of return, \lambda_{\max}(A + (r+\delta)B_{1}) < 1, then N^{-1} \ge 0 and the sign of each \Delta c(j) travels downstream intact: a cost reduction in j lowers every price that uses j, directly or indirectly, and raises none.
5.2 The payback bound
For an adopting sector j define the substitution ratio
\quad
(10)\qquad
\varphi(j) \equiv \frac{(\Delta B\,p)(j)}{-\,w\,\Delta h(j)} \;\ge 0dollars of capital added per unit output per dollar of annual payroll removed, both valued at prevailing prices. Then
\begin{aligned}
&(11)\quad
\Delta c(j) = w\,\Delta h(j) + (r+\delta)(\Delta B\,p)(j)
\\[8pt]
&\phantom{(11)\quad \Delta c(j)}
= -\,w\,|\Delta h(j)|\,\big[\,1 - (r+\delta)\,\varphi(j)\,\big]
\end{aligned}so that
\quad
(12)\qquad
\Delta c(j) < 0
\quad\Longleftrightarrow\quad
\varphi(j) < \frac{1}{r+\delta}The right side is a payback period: the inequality says that the annual capital charge per dollar invested, r+\delta, must fall short of the annual payroll that dollar saves, or equivalently that the outlay must repay itself from saved wages within 1/(r+\delta) years. Compute equipment depreciates fast; with \delta near 0.25 and r near 0.10 the bound is about three years of saved payroll per dollar invested. Adopters below the bound lower their own prices in wage units and the prices downstream of them. AI producers whose capital outlays per dollar of payroll saved exceed the bound raise their prices unless their own layoffs are deep enough to compensate. Any given price moves by the inner product of the corresponding row of N^{-1} with the vector \Delta c, a weighted sum of these opposing cost changes.
5.3 Okishio’s theorem
Inequality (12) is the criterion of Okishio (1961): the technique is adopted because it reduces cost at prevailing prices and wage. The theorem adds that such a technique raises the uniform rate of return at a given real wage. The proof is short in the present notation. Let the real wage be fixed by a consumption basket d, so that w = d'p. At the old rate r the new prices are p_{1} = p + N^{-1} \Delta c \le p, with strict inequality for at least one good, hence d'p_{1} < w: at rate r the money wage would buy more than the basket. Since prices in wage units rise with r (Section 6.1), restoring d'p_{1} = w requires a higher rate r_{1} > r. The realized rate of return rises; what falls is its ceiling, as the next section shows.
6. Consequences
6.1 Interest sensitivity
Differentiating Mp = wh with respect to r at fixed w,
\quad
(13)\qquad
\frac{\partial p}{\partial r} = M^{-1} B\, p \;\ge 0Every price rises with the rate of return, and the derivative is increasing in B at given prices and given M^{-1}; with B_{1} \ge B element by element, N^{-1} \ge M^{-1} as well. Price formation migrates from payroll to the capital charge: prices of AI-intensive output become interest-rate prices.
6.2 The wage share
Multiplying (3) on the left by x' and rearranging, aggregate value added splits into wages and the capital charge:
\quad (14)\qquad x'(I-A)p = w\,h'x + (r+\delta)\,x'Bp
Define the price-form organic composition q \equiv x'Bp / (w\,h'x). The wage share of value added is
\quad
(15)\qquad
\omega = \frac{1}{1 + (r+\delta)\,q}The AI perturbation lowers h'x and raises B. Unless capital-goods prices fall by more than the proportional rise in B, q rises and \omega falls: the wage share declines on both counts, fewer hours in the numerator and a larger capital charge in the denominator.
6.3 The ceiling descends
By the monotonicity of the Perron–Frobenius root in the elements of a nonnegative matrix, \lambda_{\max}(A + (r+\delta)B_{1}) \ge \lambda_{\max}(A + (r+\delta)B) for every r, hence from (7)
\quad
(16)\qquad
R^{*}(B_{1}) \le R^{*}(B)Capital deepening lowers the maximum rate of return consistent with positive prices. Combined with Okishio’s result, the realized rate rises toward a ceiling that falls: the interval [\,r, R^{*}\,] compresses from both sides. This is the rigorous content of the statement that capital deepening tightens the system; it is not a prediction about the realized rate, which Okishio settles in the opposite direction.
7. Sector-Specific Rates of Return
A uniform r is the premise of competitive equalization. Oligopoly in the compute sector suggests replacing the scalar by a diagonal matrix R = \operatorname{diag}(r(1), \ldots, r(n)). The stationary identity becomes
\begin{aligned}
&(17)\quad
p = Ap + wh + (R + \delta I)Bp
\\[8pt]
&\quad
M(R) \equiv I - A - (R+\delta I)B
\\[8pt]
&\quad
p = w\,M(R)^{-1} h
\end{aligned}and the finite-change algebra goes through line by line with N(R) = M(R) - (R+\delta I)\Delta B:
\begin{aligned}
&(18)\quad
N(R)\,\Delta p = \Delta c
\\[8pt]
&\quad
\Delta c(j) = w\,\Delta h(j) + \big(r(j)+\delta\big)(\Delta B\,p)(j)
\\[8pt]
&\quad
\Delta c(j) < 0 \iff \varphi(j) < \frac{1}{r(j)+\delta}
\end{aligned}The rate earned by the compute producers, r(k), enters the adopter’s test through the price p(k) at which \varphi(j) is valued. Differentiating (17),
\quad
(19)\qquad
\frac{\partial p}{\partial r(k)} = M(R)^{-1} e(k)\,e(k)'B p = (Bp)(k)\;\big[M(R)^{-1}\big]_{\cdot k} \;\ge 0column k of the inverse scaled by sector k’s own capital charge base. Hence \partial \varphi(j)/\partial r(k) \ge 0 for every adopter j that buys from k. The consequence is a partition of adopters by the payback test. Those with \varphi(j) below 1/(r(j)+\delta) at competitive compute prices but above it at oligopoly prices switch from the deflationary branch to the inflationary one; the set of sectors whose prices fall shrinks as r(k) rises. The surplus those adopters would have transmitted downstream as lower prices is instead retained upstream in sector k. The U.S. configuration since 2023 fits this partition: the large technology firms carry the capital expenditure, \Delta b(jk), while the dominant processor supplier carries the rate r(k), visible as an operating margin well above its customers’. Equalization is the restriction R = rI, and the dispersion of the r(j) is an empirical question rather than an axiom.
8. The Firm-Level Reduced Form
Multiply the unit price identity of sector j by its output x(j). With Y = p(j)x(j) (revenue), X = [(Ap)(j) + w h(j)]\,x(j) (operating cost before depreciation), K = (Bp)(j)\,x(j) (net property, plant and equipment) and \delta K depreciation, the accounting identity is
\quad (20)\qquad Y \equiv X + \delta K + S
where S is operating profit after depreciation. The structural alternatives, each with a stochastic error \varepsilon, are
\begin{aligned}
&(21)\quad
\text{Solow:}
&&
S = r\,K + \varepsilon
\\[8pt]
&(22)\quad
\text{Marx:}
&&
S = r\,(X + K) + \varepsilon
\\[8pt]
&(23)\quad
\text{Kalecki:}
&&
S = \kappa\,(X + \delta K) + \varepsilon
\end{aligned}the first charging a return on the capital stock alone, the second on circulating and fixed capital at a uniform rate, the third a markup \kappa on cost price. All three are nested in one reduced form amenable to least squares,
\quad (24)\qquad Y(j,t) = \alpha + \beta\, X(j,t) + \gamma\, K(j,t) + \varepsilon(j,t)
with the restrictions
| Model | β | γ |
|---|---|---|
| Solow | \beta = 1 | \gamma = r + \delta |
| Marx | \beta = 1 + r | \gamma = \beta - 1 + \delta |
| Kalecki | \beta = 1 + \kappa | \gamma = \beta\,\delta |
Since \delta is observable as depreciation over lagged net PPE, each restriction is testable. The sectoral identity is linear homogeneous and therefore imposes \alpha = 0; a nonzero intercept in firm data is the “intercept problem” and reads as a unit fixed cost \alpha / x. The diagonal matrix R of Section 7 makes \beta a firm-specific coefficient, \beta(j) = \bar\beta + u(j), estimable by Swamy’s (1970) random coefficient method; its between-firm variance measures the distance from profit-rate equalization. Equation (24) is the expanded form of the markup regression in Silva (2026).
A practical remark closes the note. The matrix A is available from the BEA use tables. The matrix B requires a capital-flow table by industry and asset type, which BEA last published for 1997 to my knowledge. The firm-level reduced form (24) sidesteps that gap: Compustat reports Y, X, K and depreciation for every public company, and the price-system restrictions become hypotheses on \beta and \gamma.
References
Marx, K. (1894). Capital, Volume III. Ed. F. Engels. Hamburg: Otto Meissner.
Okishio, N. (1961). Technical change and the rate of profit. Kobe University Economic Review 7: 85–99.
Silva, E. A. (2026). U.S. Corporate Profits, 1950–2024: Operating Profit Markups in Leading U.S. Firms. SpringerBriefs in Economics. Cham: Springer.
Solow, R. M. (1959). Competitive valuation in a dynamic input-output system. Econometrica 27(1): 30–53.
Swamy, P. A. V. B. (1970). Efficient inference in a random coefficient regression model. Econometrica 38(2): 311–323.