Production functions
2026-04-27 — 2026-10-08
Wherein Production Functions Are Scrutinized for Latent Assumptions, Isoquants Are Plotted as Contours, and the Constant Elasticity of Substitution Is Tested Against Cobb-Douglas, Leontief, and Translog Models.
I’m annoyed that people use dumb production functions when speculating about AI.
I’m not going to name any names here—you know who you are. But, before you next write about neural scaling, make sure that relations other than \(\text{output} = \text{input 1}\times \text{input 2}\) can exist between inputs and outputs, or I will be not so much angry as disappointed.
A production function \(F\) maps inputs to output: \(Y = F(K, L, \dots)\). In practice, we pick one for its analytical convenience, which is dangerous because it smuggles in assumptions. It’s worth checking those before going all-in on something. That said, “choosing not to choose” a production function and defaulting to \(Y=KL\) is still making a choice; ceteris paribus, a stupid one.
An isoquant is the set of input combinations that produce the same output, \(\{(K, L) : F(K, L) = \bar Y\}\), i.e. a contour line of \(F\) when we plot \(L\) against \(K\).
Returns to scale describes what happens to output when we scale every input by the same factor \(t\): \(F(tK, tL) = t^k F(K, L)\), with constant returns at \(k = 1\) (double every input, double the output), increasing returns at \(k > 1\), and decreasing returns at \(k < 1\).
The cost function gives the cheapest way to produce output \(Y\) when the inputs \(x = (x_1, \dots, x_n)\) have prices \(w = (w_1, \dots, w_n)\):
\[C(w, Y) = \min_{x}\,\{\, w \cdot x \;:\; F(x) \ge Y \,\}.\]
If \(F\) is well-behaved (more of any input never hurts, and isoquants are convex), we can recover \(F\) from \(C\). The two carry the same information, so we may work with whichever is easier; this is duality. The cost function also determines the input demands: by Shephard’s lemma, \(\partial C/\partial w_i = x_i^*(w, Y)\) is the cost-minimizing quantity of input \(i\). That makes \(C\) the usual choice for direct estimation. A firm takes its input prices as given but chooses its input quantities, so a regression on prices suffers less from the endogeneity that hampers a regression on quantities.
1 Let us care about elasticity of substitution
We usually want a production function that allows us to consider the impact of the elasticity of substitution \(\sigma = \dfrac{d\log(K/L)}{d\log(\text{MRTS})}\).
Here MRTS is the marginal rate of technical substitution, \(\text{MRTS} = \dfrac{\partial F/\partial L}{\partial F/\partial K}\): how many units of capital one more unit of labour can replace at fixed output, which is the slope of the isoquant. A cost-minimising firm sets the MRTS equal to the wage–rental ratio \(w/r\), so \(\sigma\) is the percentage change in the input mix \(K/L\) for a 1% change in relative factor prices.
Capital and labour are a textbook pair because of labour economics. However, we can generalize to any kind of substitution between inputs. With inputs \(Y = F(x_1, \dots, x_n)\) there is an MRTS for every pair, \(\text{MRTS}_{ij} = \dfrac{\partial F/\partial x_i}{\partial F/\partial x_j}\), holding output and all the other inputs fixed. The MRTS generalization is still well defined there. \(\sigma\) ceases to be unique, as we must decide what the other inputs do while \(i\) and \(j\) trade off. Various choices give varying specialty elasticities, which are out of scope for the moment.
2 The Constant Elasticity of Substitution Family
A.k.a. the CES family. Includes many classics.
\[Y = A\left[\alpha K^{\rho} + (1-\alpha)L^{\rho}\right]^{1/\rho}, \qquad \sigma = \frac{1}{1-\rho}\]
Three famous special cases fall out as \(\rho \to 1, 0, -\infty\):
2.1 Linear / perfect substitutes
(\(\rho = 1\), \(\sigma = \infty\)): \(Y = aK + bL\). One input can replace the other at a fixed rate, so a cost-minimizing buyer checks which input is cheaper per unit of output and uses only that one. That fits electricity, where a kilowatt-hour from a wind farm does the same job as one from a gas plant, but most interesting production functions are not perfectly substitutable.
2.2 Cobb-Douglas
(\(\rho \to 0\), \(\sigma = 1\)): \(Y = A K^{\alpha} L^{\beta}\). The default workhorse. It’s everywhere because it is realistic if you squint right and also still pretty tractable:
- Log-linear, so estimation is OLS on \(\log Y = \log A + \alpha \log K + \beta \log L\).
- “Constant income shares”: if each input (let us say, labour and capital again) is paid according to its marginal product, owners of capital always receive the fraction \(\alpha\) of output and workers the fraction \(\beta\). For much of the twentieth century labour’s share of national income did hold roughly steady, which bolstered the credibility of Cobb-Douglas.
- Self-dual — the cost function is also Cobb-Douglas, so demand systems come out in closed form.
- Returns to scale is just \(\alpha + \beta\); set \(=1\) and we get constant returns for free.
- Nice interior solutions, everything differentiable, no inputs ever go to zero (each input is essential: \(Y = 0\) if any input is 0).
That said, setting \(\sigma = 1\) is dangerous. If we can estimate the other parameters from data, we should estimate this one too. It’s clearly important. Whether income shares rise or fall as capital deepens depends on whether \(\sigma \gtrless 1\). Piketty has famously noted that labour share has been declining.
2.3 Leontief
(\(\rho \to -\infty\), \(\sigma = 0\)): \(Y = \min\left(\frac{K}{a}, \frac{L}{b}\right)\). Fixed proportions, zero substitution. Think “one pilot per plane”, “two wheels per bicycle”.
- It’s the basis of input–output analysis (what Leontief’s renown stems from) and of activity-analysis / linear programming models of production.
- It’s the right short-run model under putty-clay: substitutable ex ante when designing the plant, rigid ex post once the concrete is poured.
- Isoquants have kinks and it’s non-differentiable, so calculus-based comparative statics fail — but LP duality replaces them.
2.4 General CES
(\(\sigma\) free): what we use when \(\sigma\) is a variable of interest. Famously a battleground in environmental economics — whether manufactured capital can substitute for natural capital (\(\sigma > 1\) vs \(\sigma < 1\)) is deep in the Solow-vs-Daly sustainability argument.
3 Non-constant elasticity of substitution
3.1 Nested CES / KLEM
\(Y = F\big(E, G(K, L)\big)\) with different \(\sigma\) at each level. This represents a tree hierarchy of inputs funneling into a single output of interest. Standard in CGE and climate-economy models, because the claim there is usually “capital and skilled labour are complements but both substitute for energy” — which no single-level CES can express.
3.2 Translog (Christensen–Jorgenson–Lau)
\[\log Y = a_0 + \sum_i a_i \log x_i + \tfrac{1}{2}\sum_i\sum_j b_{ij}\log x_i \log x_j\] This is a quadratic in the log inputs, i.e., a second-order Taylor expansion in the log domain. Around any chosen input mix, it can match how output responds to each input and how easily every pair of inputs substitutes for each other; so, unlike CES, it lets \(\sigma\) differ between pairs and change as the mix changes. Setting every \(b_{ij} = 0\) recovers Cobb-Douglas, so we can check whether Cobb-Douglas is sufficient by testing whether those terms are zero, insofar as estimation is actually feasible, although some introspection will reveal that here be dragons. For one, the parameter count grows with the square of the number of inputs, and, moreover, we need to fit it in a tiny, localized range, which is hard. In practice, most translog work fits the cost function \(C\) in this form rather than \(F\), for the identification reason above.
3.3 Others I have not heard of
Surely many
4 Practical advice
Rule of thumb from a non-economist:
If I find myself using something dumber than Cobb-Douglas, I should switch to Cobb-Douglas, which is typically the minimum viable complexity that still captures substitution between inputs.
Then, if the problem proves more complex, I can get fancier.