goods market

gdp

• consumption: $C$, goods/services purchased by consumers
• investment: $I$, fixed investment, sum of non-residential and residential investments
• government spending: $G$, purchases of goods/services by national, regional, local governments
  • does not include government transfers
• imports: $IM$, purchases of foreign goods and services by domestic
• exports: $X$, purchases of domestic goods and services by foreigners
• gdp: $Z\equiv C+I+G+X-IM$

consumption

• consumption is based on disposable income, $Y_D$
• $C=c_0+c_1(Y_D)$, $c_1$ is ‘marginal propensity to consume’
  • effect an additional euro has on consumption
  • $0<c_1<1$
• $Y_D\equiv Y-T$, income minus taxes ($C=c_0+c_1(Y-T)$)

investment

• endogenous → derived, exogenous → given
• $I=\bar{I}$ (given)
• or $I=S+T-G$

government spending

• $G$ and $T$ also taken as exogenous, given
• no reliable rule can be written for governments bc of irregularity

gdp substitution

• $Z=C+I+G$, simplified form
• $Z=c_0+c_1(Y-T)+\bar{I}+G$, substituting

equilibrium

• assuming firms do not hold inventory, production ($Y$) must equal demand ($Z$), $Y=Z$
• replacing $Z$, get $Y=c_0+c_1(Y-T)+\bar{I}+G$
• $Y$ is GDP from both production and consumer side

solving

• $Y=c_0+c_1(Y-T)+\bar{I}+G$ → $Y=\frac{1}{1-c_1}[c_0+\bar{I}+G-c_{1}T]$
• $c_0+\bar{I}+G-c_{1}T$ is autonomous spending, demand that does not depend on output
• $\frac{1}{1-c_1}$ is multiplier, increases as $c_1$ approaches $1$, also $\frac{\Delta I}{\Delta G}$

following

• first-round increase in demand moves $A$ up to $B$, $\text{€1 B}$ distance
• first-round increase in demand leads to equal increase in production, also shown by $A$ up to $B$ with same distance
• first-round increase in production leads to equal increase in income, $B$ right to $C$ by $\text{€1 B}$
• second-round increase in demand moves $C$ up to $D$, distance is $c_1\times \text{€1 B}$
• second-round income in demand leads to equal increase in production, by $C$ up to $D$, and then increase in income, right from $D$ to $E$
• third-round increase in demand is $c_1\times c_1\times \text{€1 B}$
• geometric series as $1+c_1+c_1^2+\cdots +c_1^n$, approaches $\text{€1}/(1-c_1)\text{ B}$ sum

saving

• private saving ($S$), saving by consumers, disposable income minus consumption, $S=Y_D-C$
• in terms of income and taxes, $S=Y-T-C$
  • public savings are $T-G$, positive is surplus, negative is deficit
• $Y=C+\bar{I}+G$ → $Y-T-C=\bar{I}+G-T$ → $S=\bar{I}+G-T$
• equilibrium is $\text{production = demand}$ or $\text{investment = saving}$

solving

• $S=Y-T-C=Y-T-c_0-c_1(Y-T)$