Lecture 6: Multivariable Functions, Limits, and Continuity

Learning objectives

Analyze domains and level sets; define multivariate limits and continuity; distinguish pathwise behavior from genuine limits; interpret surfaces and economic production functions.

1. Functions of several variables

A scalar function $f:D\subseteq\mathbb R^n\to\mathbb R$ maps a vector input to a scalar output. Its domain, range, level sets, and contour surfaces describe its geometry. In economics, examples include utility, production, cost, and welfare functions.

2. Multivariate limits

The limit $\lim_{x\to a}f(x)=L$ requires that every sequence approaching $a$ within the domain produces function values approaching $L$. A limit must be independent of the path. Showing two paths with different limiting values disproves existence; checking a finite number of paths cannot generally prove existence.

3. Continuity

A function is continuous at $a$ if its multivariate limit equals $f(a)$. Polynomials are continuous everywhere; rational functions are continuous wherever denominators are nonzero. Compositions preserve continuity under compatible domains.

4. Open and closed sets

A ball around a point defines a neighborhood. Open sets contain a neighborhood around every point; closed sets contain their limit points. Compact subsets of Euclidean space are closed and bounded. Continuity on a compact set implies boundedness and attainment of maximum and minimum.

5. Level sets and economic interpretation

A level set $\{x:f(x)=c\}$ defines an indifference curve or isoquant in common applications. The gradient, developed in the next lecture, is normal to a regular level set. Curvature and feasibility depend on the domain and constraints.

Worked example

Consider $f(x,y)=xy/(x^2+y^2)$ away from the origin. Along $y=x$, the value is $1/2$; along $y=-x$, it is $-1/2$. Therefore the limit at the origin does not exist.

Problem set

Determine the domain of $\log(x-y)$.
Test a multivariate limit along two paths.
Explain why checking coordinate axes alone is insufficient.
Describe level sets of a Cobb–Douglas production function.
State the extreme value theorem on compact subsets of Euclidean space.

Computational lab

Plot surfaces and contour maps for quadratic, Cobb–Douglas, and non-convergent examples. Numerically explore path dependence near a candidate limit.

Coding exercises

These exercises apply the calculus of this lecture to real economic data in Python, mixing symbolic (sympy), numerical (scipy) and statistical (statsmodels) tools. Install once with pip install sympy scipy numpy pandas statsmodels wooldridge. Try the starter code first, then open the solution. Every printed result below comes from running the solution code exactly as shown.

Exercise 6.1 — The CES production function and its Cobb–Douglas limit

Data: We calibrate to the U.S. labor share of about 2/3 (capital share $\alpha=1/3$) and evaluate at the capital–output ratio implied by macrodata investment data.

The CES function is $Y=\big(\alpha K^{\rho}+(1-\alpha)L^{\rho}\big)^{1/\rho}$, with elasticity of substitution $\sigma=1/(1-\rho)$.

Show with sympy that $\lim_{\rho\to0}Y=K^{\alpha}L^{1-\alpha}$ (take logs and use L'Hôpital's rule).
Evaluate the CES function at $K=3$, $L=1$ for $\rho$ approaching 0 from both sides, and show convergence to the Cobb–Douglas value.
Compute the capital share $\frac{\partial Y}{\partial K}\frac{K}{Y}$ as $\rho$ varies. Why does the Cobb–Douglas case imply a constant factor share, and what does the U.S. evidence on a falling labor share suggest about $\sigma$?

import sympy as sp
K, L, a, r = sp.symbols("K L alpha rho", positive=True)
logY = sp.log(a * K**r + (1 - a) * L**r) / r
# TODO: sp.limit(logY, r, 0)

Solution, output and discussion

import numpy as np, sympy as sp
K, L, a = sp.symbols("K L alpha", positive=True); r = sp.symbols("rho", real=True)
logY = sp.log(a * K**r + (1 - a) * L**r) / r
lim = sp.simplify(sp.limit(logY, r, 0))
print("lim_{rho->0} log Y =", lim, " -> Y = K^alpha L^(1-alpha)")
ces = lambda rho, k=3.0, l=1.0, al=1/3: (al * k**rho + (1 - al) * l**rho)**(1 / rho)
print(f"Cobb-Douglas value at K=3, L=1: {3**(1/3):.8f}")
for rho in (-0.5, -0.1, -0.01, -0.001, 0.001, 0.01, 0.1, 0.5):
    Y = ces(rho); share = (1/3) * 3**rho / ((1/3) * 3**rho + 2/3)
    print(f"  rho = {rho:+.3f} (sigma = {1/(1-rho):.3f}): Y = {Y:.8f}, capital share = {share:.4f}")

Output:

lim_{rho->0} log Y = log(K**alpha*L**(1 - alpha))  -> Y = K^alpha L^(1-alpha)
Cobb-Douglas value at K=3, L=1: 1.44224957
  rho = -0.500 (sigma = 0.667): Y = 1.35486374, capital share = 0.2240
  rho = -0.100 (sigma = 0.909): Y = 1.42327646, capital share = 0.3094
  rho = -0.010 (sigma = 0.990): Y = 1.44031910, capital share = 0.3309
  rho = -0.001 (sigma = 0.999): Y = 1.44205619, capital share = 0.3331
  rho = +0.001 (sigma = 1.001): Y = 1.44244302, capital share = 0.3336
  rho = +0.010 (sigma = 1.010): Y = 1.44418736, capital share = 0.3358
  rho = +0.100 (sigma = 1.111): Y = 1.46195368, capital share = 0.3582
  rho = +0.500 (sigma = 2.000): Y = 1.54757814, capital share = 0.4641

sympy evaluates the 0/0 limit of $\log Y$ and returns the Cobb–Douglas form, and the numerical values converge to $3^{1/3}$ from both sides. The capital share is constant at $\alpha=1/3$ only in the Cobb–Douglas case: with $\sigma<1$ ($\rho<0$), a high capital–labor ratio lowers capital's share, and with $\sigma>1$ it raises it. A falling U.S. labor share alongside capital deepening is therefore consistent with $\sigma>1$ — one explanation debated in the literature (Karabarbounis and Neiman, 2014); others stress markups and automation.

Exercise 6.2 — Level sets of an estimated two-input investment function

Data: grunfeld (statsmodels) — investment, firm market value and capital stock of 11 large U.S. firms, 1935–1954.

Estimate the log-linear investment function $\log I=\beta_0+\beta_1\log V+\beta_2\log K$ (pooled OLS).
The fitted function $I(V,K)=e^{\beta_0}V^{\beta_1}K^{\beta_2}$ is a function of two variables. Find the level curve on which predicted investment equals the sample median, solving for $K$ as a function of $V$.
Compute the slope $dK/dV$ of that level curve at the median firm-year (the marginal rate of substitution between market value and capital in generating investment).

import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
g = sm.datasets.grunfeld.load_pandas().data.query("invest > 0 and capital > 0")
r = smf.ols("np.log(invest) ~ np.log(value) + np.log(capital)", data=g).fit()
# TODO: K(V) on the level set: K = (I0 / (e^b0 V^b1))^(1/b2); slope -(b1 K)/(b2 V)

Solution, output and discussion

import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
g = sm.datasets.grunfeld.load_pandas().data.query("invest > 0 and capital > 0")
r = smf.ols("np.log(invest) ~ np.log(value) + np.log(capital)", data=g).fit()
b0, b1, b2 = r.params
print(f"log I = {b0:.3f} + {b1:.3f} log V + {b2:.3f} log K   (R2 {r.rsquared:.3f})")
I0, V0, K0 = g["invest"].median(), g["value"].median(), g["capital"].median()
Kcurve = lambda V: (I0 / (np.exp(b0) * V**b1))**(1 / b2)
for V in (0.5 * V0, V0, 2 * V0):
    print(f"  on the level set I = {I0:.1f}: V = {V:8.1f} -> K = {Kcurve(V):8.1f}")
print(f"slope dK/dV of the level curve at V = {V0:.1f}: {-(b1 * Kcurve(V0)) / (b2 * V0):.4f}")

Output:

log I = -2.777 + 0.803 log V + 0.375 log K   (R2 0.880)
  on the level set I = 52.4: V =    202.3 -> K =    720.7
  on the level set I = 52.4: V =    404.6 -> K =    163.6
  on the level set I = 52.4: V =    809.3 -> K =     37.1
slope dK/dV of the level curve at V = 404.6: -0.8650

The fitted function explains 88% of the variation in log investment. On the level curve through the median investment, doubling market value lets predicted investment stay constant with much less capital: the curve is steep and convex, as for any Cobb–Douglas level set. The slope at the median value says that $1 more market value substitutes for about $0.87 less capital in generating the same predicted investment. This is a descriptive relation across firms, not a production technology.

Reading

Apostol, Calculus, Vol. 2; Simon and Blume, Mathematics for Economists; Chiang and Wainwright, Fundamental Methods of Mathematical Economics.