Adv. Macro – Ch. 1 Formula Cheatsheet

Variable Glossary

Color groups: blue = output/income · green = capital/factors · amber = growth/time · purple = welfare/utility

$Y$

Real output / GDP

Aggregate production of the economy

$y$

Output per person

$y = Y/N$ — the main welfare proxy

$N$

Population

Total headcount (not employment)

$L$

Employment

Number of workers; $Y/L$ = output per worker

$K$

Capital stock

Physical capital; $K/Y$ = capital–output ratio

$A$

Technology level

Labor-augmenting (Harrod-neutral); $AL$ = effective labor

$r$

Rate of return on capital

$r = \alpha(Y/K)$ on a BGP; splits since 2000 into productive vs. safe rate

$\alpha$

Capital share

$\alpha_K = F_K K / Y \approx 1/3$; labor share $\approx 2/3$

$\sigma$

Elasticity of substitution

Between $K$ and $AL$; Cobb–Douglas has $\sigma=1$; $\sigma>1$ means cheaper $K$ lowers labor share

$\gamma$

Growth rate (generic)

$\gamma_{y,t} = y_t/y_{t-1} - 1$; subscript names the variable

$n$

Population growth rate

Growth rate of $N$; $\gamma_y \approx \gamma_Y - n$

$g$

Growth rate of output per worker

On a BGP, both $K$ and $Y/L$ grow at $g$

$\pi$

Inflation rate

$\pi = (p_t - p_{t-1})/p_{t-1}$; real growth = nominal $-\pi$

$T_2$

Doubling time

Years to double income at rate $\bar{\gamma}$; $\approx 70/(100\bar{\gamma})$

$T$

Catch-up time

Years for country $B$ to equal $A$ given a growth differential

$\lambda$

Consumption-equivalent welfare

Scale factor s.t. a US resident is indifferent between US and country $i$; Jones–Klenow (2016)

$U$

Lifetime utility

Expected discounted utility of a newborn; $U_i = e_i u_i + \text{growth term}$

$u$

Flow utility

$u = \bar{u} + \log C + v(l)$; $\bar{u}$ pins the value of life

$e_i$

Discounted life expectancy

$e_i = \sum_a \beta^a S_i(a)$; with $\beta=1$ this is expected years lived

$S(a)$

Survival function

Probability of surviving from birth to age $a$ in country $i$

$\beta$

Discount factor

$\beta < 1$ down-weights future years; $\beta = 0.99$ in JK calibration

$\sigma^2_i$

Variance of log consumption

Inequality measure; $\mathbb{E}[\log C] = \log \bar{c} - \tfrac{1}{2}\sigma^2$ (lognormal)

$l$

Leisure

$1 -$ hours worked per year; enters $v(l)$ in flow utility

Key Formulas

Growth arithmetic

Eq 10Growth rate definition

$$\gamma_{y,t} = \frac{y_t - y_{t-1}}{y_{t-1}} = \frac{y_t}{y_{t-1}} - 1 \;\in [-1,\infty)$$

Asymmetric: −50% down, +100% back. Use log differences for multi-period work.

Eq 11Log difference (preferred)

$$\Delta \log y_t \equiv \log y_t - \log y_{t-1} = \log(1 + \gamma_{y,t})$$

Additive over time: $\log y_T - \log y_0 = \sum_{t=1}^T \Delta \log y_t$. Differs from $\gamma$ by $\approx \tfrac{1}{2}\gamma^2$.

Eq 14Real GDP per person growth

$$\gamma_y \approx \gamma_{pY} - \pi - n$$

$\gamma_{pY}$ = nominal GDP growth, $\pi$ = inflation, $n$ = population growth. Exact form: $(1+\gamma_Y)/(1+n)$.

Eq 15Average (geometric mean) growth

$$\bar{\gamma} = \left(\frac{y_T}{y_0}\right)^{1/T} - 1, \qquad \log(1+\bar{\gamma}) = \frac{\log y_T - \log y_0}{T}$$

Arithmetic mean of % rates overstates geometric mean (Jensen's inequality). Use geometric mean or OLS log-trend.

Eq 17Rule of 70 (doubling time)

$$T_2 = \frac{\log 2}{\log(1+\bar{\gamma})} \approx \frac{0.693}{\bar{\gamma}} \approx \frac{70}{100\,\bar{\gamma}}$$

At 2%: doubles every 35 years. At 7%: every ~10 years (exact: 10.2). Rule slightly overstates $T_2$.

Key insight: 1 pp difference in growth → 50% level gap over a working life.

Eq 18Catch-up time

$$T = \frac{\log y_{A,0} - \log y_{B,0}}{\log(1+\bar{\gamma}_B) - \log(1+\bar{\gamma}_A)} \approx \frac{\log(y_{A,0}/y_{B,0})}{\bar{\gamma}_B - \bar{\gamma}_A}$$

$B$ starts below $A$ but grows faster. Warning: treats growth rates as constants — they are endogenous (Solow model, Ch. 2).

Eq 16Compounding

$$y_{t+T} = y_t(1+\bar{\gamma})^T = y_t\!\left[\underbrace{1 + T\bar{\gamma}}_{\text{linear}} + \underbrace{\tbinom{T}{2}\bar{\gamma}^2 + \tbinom{T}{3}\bar{\gamma}^3 + \cdots}_{\text{growth on growth}}\right]$$

The binomial expansion separates simple interest from the compounding part. Higher-order terms dominate over long horizons.

Jones–Klenow welfare framework

Eq 2Flow utility

$$u(C, l) = \bar{u} + \log C + v(l), \qquad v'(l) > 0$$

$\bar{u}$ = constant calibrated from the value of a statistical life (~$6 M for a US 40-year-old → $\bar{u} = 5.00$). Log form is essential: makes utility additively separable in $\log C$. $v(l)$ = leisure utility.

A year of life is worth living when $C > e^{-\bar{u}} \approx 0.7\%$ of avg US consumption.

Eq 3Inequality penalty (lognormal)

$$\mathbb{E}[\log C_i] = \log c_i - \tfrac{1}{2}\sigma_i^2$$

$c_i$ = mean consumption in country $i$; $\sigma_i^2$ = variance of $\log C$. Higher inequality lowers expected utility by $\tfrac{1}{2}\sigma^2$. Follows from the lognormal identity $\mathbb{E}[\log X] = \log \mathbb{E}[X] - \tfrac{1}{2}\mathrm{Var}[\log X]$.

Eq 4–5Lifetime utility of a newborn

$$U_i = \underbrace{\sum_{a=1}^{100} \beta^a S_i(a)}_{e_i}\; u_i \;+\; g\sum_{a=1}^{100}\beta^a S_i(a)\, a$$ $$\text{where}\quad e_i \equiv \sum_{a=1}^{100}\beta^a S_i(a), \qquad u_i \equiv \bar{u} + \log c_i - \tfrac{1}{2}\sigma_i^2 + v(l_i)$$

Mortality enters via $S_i(a)$. Later years count less ($\beta < 1$). The growth term rewards countries where incomes rise with age.

Eq 7Welfare measure $\lambda$

$$\log \lambda_i = \frac{U_i(1) - U_{\text{US}}(1)}{e_{\text{US}}}$$

$\lambda_i = 0.5$ means country $i$ has welfare equivalent to half US consumption. Defined by the indifference condition $U_{\text{US}}(\lambda_i) = U_i(1)$: how much would US consumption need to be scaled for a US resident to be indifferent about living in country $i$?

Eq 9Welfare–income gap decomposition ★

$$\log\lambda_i - \log\!\frac{y_i}{y_{\text{US}}} = \underbrace{\frac{e_i - e_{\text{US}}}{e_{\text{US}}}u_i}_{\text{life expectancy}} + \underbrace{\log\frac{c_i}{y_i} - \log\frac{c_{\text{US}}}{y_{\text{US}}}}_{\text{consumption share}} + \underbrace{v(l_i)-v(l_{\text{US}})}_{\text{leisure}} - \underbrace{\tfrac{1}{2}(\sigma_i^2-\sigma_{\text{US}}^2)}_{\text{inequality}}$$

Each term is the welfare gap not captured by income. Additivity follows directly from the $\log C$ form in Eq 2.

France example: income 67.2% of US, welfare $\lambda = 91.8\%$. Gap $= +0.155$ (life) $-0.152$ (C/Y) $+0.083$ (leisure) $+0.102$ (ineq.) $+0.124$ (hrs ineq.)

Across countries: $\mathrm{corr}(\log\lambda,\, \log y) = 0.98$ — income is still a good proxy, but Western Europe systematically looks closer to the US in welfare than in income.

Kaldor Facts — Status Check

Fact	Statement	US value	Since 2000
K1	$Y/L$ grows at a roughly constant rate	~2% / yr trend 1870–2000	changed TFP growth fell below 0.5% after 2005
K2	$K/L$ grows at a similar rate to $Y/L$	Both ~1.5–2% / yr	holds
K3	Real return $r$ roughly constant	~9% net; ~12–14% gross	changed Return on capital stable/rising; safe rate fell to ≈ 0%. Spread widened 4 pp.
K4	$K/Y$ roughly constant	~2–3×	changed Holds for productive capital; fails for wealth (Piketty–Zucman: 2–3× → 4–6×, mainly housing revaluation)
K5	Labor share $\approx 2/3$, capital share $\approx 1/3$	Labor share ~65–70%	broken Global labor share fell ~5 pp since 1975. Cause disputed: cheaper capital ($\sigma > 1$), superstar firms, or accounting change.
K6	Growth rates differ substantially across countries	Std dev ~2–2.7 pp/decade	holds

Why K1–K5 together define a balanced growth path (BGP): If $Y/L$ and $K/L$ grow at rate $g$, then $K/Y$ is constant (K4). With constant $K/Y$ and capital share $\alpha = rK/Y$, the return $r = \alpha(Y/K)$ is constant (K3). With $Y$ and $K$ growing at $g+n$, investment $I$ and consumption grow at the same rate → $I/Y$ constant. The BGP therefore forces technology to be purely labor-augmenting (Uzawa's theorem) — which is why every model in Chs. 2–7 writes $Y = F(K, AL)$.

PWT 11.0 Variable Quick Reference

Variable	Use it for	Don't use it for
`rgdpe / pop`	Level comparisons of living standards across countries	Growth rates over time (prices move)
`rgdpo / emp`	Productivity comparisons (output-side, per worker)	Welfare/consumption comparisons
`rgdpna / pop`	Growth rates over time (constant national prices)	Cross-country level comparisons
`cgdpe / cgdpo`	Cross-country comparison in a given year (current PPPs)	Long time-series growth

Chapter 1 · Advanced Macroeconomics · University of Ljubljana · Oct 2026

Fact	Statement	US value	Since 2000
K1	\(Y/L\) grows at a roughly constant rate	~2% / yr trend 1870–2000	changed TFP growth fell below 0.5% after 2005
K2	\(K/L\) grows at a similar rate to \(Y/L\)	Both ~1.5–2% / yr	holds
K3	Real return \(r\) roughly constant	~9% net; ~12–14% gross	changed Return on capital stable/rising; safe rate fell to ≈ 0%. Spread widened 4 pp.
K4	\(K/Y\) roughly constant	~2–3×	changed Holds for productive capital; fails for wealth (Piketty–Zucman: 2–3× → 4–6×, mainly housing revaluation)
K5	Labor share \(\approx 2/3\), capital share \(\approx 1/3\)	Labor share ~65–70%	broken Global labor share fell ~5 pp since 1975. Cause disputed: cheaper capital (\(\sigma > 1\)), superstar firms, or accounting change.
K6	Growth rates differ substantially across countries	Std dev ~2–2.7 pp/decade	holds

Advanced Macroeconomics — Chapter 1 Cheatsheet