ECON 1116 Vocab

solow model production function

Y = AK^alpha power * L^(1 - alpha power)

𝑌=𝐴𝐾𝛼𝐿1−𝛼

capital accumulation equation

Kt+1=(1−delta)Kt +It

𝐾𝑡+1=(1−𝛿)𝐾𝑡+𝐼𝑡

what does the capital accumulation equation say

next period’s capital is equal to today’s capital minus the amount lost to depreciation, plus new investment

constant savings rate assumption equation

It = sYt

what does the constant saving rate assumption say

says country saves constant fraction s of output

expenditure equation for GDP

Yt=Ct+It+G+Nx

explain the expenditure equation for gdp

arises because we are ignoring government purchases and net exports for simplicity

what does the expenditure equation for gdp imply

Ct = (1-s)Yt: output not saved is consumed

where is the steady state on the solow diagram

where the investment curve crosses the depreciation line (Kt = K*)

what does the phase diagram show

there is a unique steady state and we always converge to it

continuously compounded growth rate formula

yx = (lnx - lnx-n)/n

per-worker production function

y=Akalpha

growth accounting formula for per-worker GDP

γy​=γA​+αγk​, Yy = YA + alphaYk

formula for constant returns to scale

Y = A*K^(1/3)*L^(2/3)

formula for annual growth rate from 1 year to the next

g = (xt + 1) - 1 = (xt + 1 - xt)/xt

formula for gdp of a country after multiple years of growth

x(t+n) = (1 + g)^(n) * xt

growth rate for multiple years of growth

g = ((x(t+n)) / (xt))^(1/n)-1

continuously compounded growth rate from one year to the next

Y = lnx(t+1) - lnx(t)

continually compounded multi-year average growth (per year)

Y = (lnx[t] - lnx[t-n])/n

how do we compare real gdp per capita across countries?

use the same prices in each country to avoid low-price countries looking artificially poor

How much has the U.S. real GDP per capita grown per year

about 1.7% per year since about 1870.

Pre‑1700 vs modern frontier growth:

• Before 1700, even frontier countries had growth of ≤ 0.1% per year.

• After about 1820, frontier growth accelerated to about 2% per year.

Divergence across countries:

• Before 1600, real GDP per capita was much more similar across countries.

• Today, it differs by a factor of about 100 between the poorest and richest countries.

Growth miracles and disasters:

• Miracles like China have caught up substantially to the frontier, especially since 1980.

• Disasters like parts of sub‑Saharan Africa have seen long periods of stagnation or decline

Convergence:

• There has been convergence of GDP per capita among OECD (rich) countries,

• but not convergence for the world as a whole.

Broadly shared growth recently:

Growth has been broadly shared across much of the world in the last 50 years.

marginal product of capital

𝑀PK = 𝛼𝐴𝐾^(𝛼−1) 𝐿^(1−𝛼) = 𝛼𝐴(𝐾/𝐿) ^(𝛼−1) = 𝛼𝑌/𝐾

marginal product of labor

𝑀P𝐿 = (1 − 𝛼)𝐴𝐾^𝛼𝐿^−𝛼 = (1 − 𝛼)𝐴(𝐾/𝐿)^𝛼 = (1 − 𝛼)𝑌/𝐿

cobb-douglas production function

𝑌 = 𝐴𝐾^𝛼*𝐿^(1−𝛼)

Cobb–Douglas with competition

Y = rK + wL

consumption equation (combining investment equation w/national income identity)

C[t] = Y[t]*(1-s)

equation for change in capital

ΔK[t]​=K[t+1] ​− K[t] ​= sAK[t]^(α)*​L^(1−α)−δKt

transitional dynamics

period when a permanent change in a parameter (like the saving rate s) makes the economy grow temporarily as it moves to a new steady state

what are limitations of the solow model

• But differences in saving rates alone are too small to explain the huge gaps in GDP per capita across countries.

• The model also cannot explain sustained long‑run growth in rich countries like the U.S.: if only capital accumulates, growth must eventually stop.

  • Persistent growth suggests something else (TFP AA) must keep improving.

what is the golden-rule saving rate for cobb-douglas

s[golden] = alpha

romer model

growth of TFP is related to the amount of research and development that a country does; a country can affect the growth rate of TFP by incentivizing research (patents, subsidies)