Savings and Investment

Investment

• cost benefit decision to additional unit of $K$ = MPK
• investment becomes K with lag $⟹MP{K}^f$

User Cost of Capital: interest cost + deprecation

$$uc=r{p}_k+\delta p_k$$

$MP{K}^f>uc:$ Increase K

$MP{K}^f<uc:$ Decrease K

Desired Capital Stock ($MP{K}^f=uc$)

$MP{K}^f$ downward sloping; $uc$ is constant

Factors that affect $MP{K}^f$

• Technological changes

Factors that affect $uc$

• changes in the $r,\delta$
• Taxes

Desired Investment

$$I_t=(K^{\ast }-K_t)+\delta K_t$$

Desired Investment = net increase in K + investment needed to replace depreciated capital

With Corporate Taxes

$MP{K}^f=\frac{uc}{(1-\tau )}=\frac{(r+d)P_k}{1-\tau }$

$\tau \uparrow :K\downarrow$

Savings and Consumption

Saving allows for consumption smoothing

Budget Constraint

$(1+r)c_1+c_2=(1+r)Y_1+Y_2$

• lifetime wealth: $(1+r)Y_1+Y_2$
• price of current consumption determined by $r$

Consumption smoothing: $c_1=c_2=c$

Marginal Propensity to Consume:

• fraction of additional income a person consumes in current period

$MPC=\frac{\Delta c_1}{\Delta Y_1}$ estimated between 0 and 1

Savings in Period 1:

$$s_1=Y_1-c=\frac{Y_1-Y_2}{2+r}$$

Factors that affect consumption and savings

$r:$ opposite IE for savers and borrowers, SE + IE make it ambiguous

• empirically $r\downarrow :c_1\uparrow$ slightly

Lump-Sum Tax

BC: $(1+r)c_1+c_2=(1+r)Y_1+Y_2-(1+r){\tau }_1-{\tau }_2$

lump sum tax does not affect individuals BC

Ricardian Equivalence

${\tau }_1\downarrow$ has to be balanced by future tax increases $⟹$ lump-sum tax cuts do not affect $c_1$

National Savings

Private Savings = $(Y-T)-C$

Gov Savings = $T-G$

National Savings: $S_{\text{pvt}}+S_{\text{gov}}$

$Y-C-G$