Dynamic and Intertemporal Budget Constraints Economics 203

Budget Constraints from Micro  Budget constraints in standard microeconomics have expenditures on one side of the equation and “income” on the other p 1 x 1 +p 2 x 2 + … + p N x N = I  they are linear in the choice variables (amounts consumed of each of the N goods x 1,..., x N ), with prices as the only coefficients on choice variables  Goods are the only choice variables  There is only one budget constraint to be analyzed in a standard microeconomics problem  Budget constraints in standard microeconomics have expenditures on one side of the equation and “income” on the other p 1 x 1 +p 2 x 2 + … + p N x N = I  they are linear in the choice variables (amounts consumed of each of the N goods x 1,..., x N ), with prices as the only coefficients on choice variables  Goods are the only choice variables  There is only one budget constraint to be analyzed in a standard microeconomics problem

Intertemporal Budget Constraint  A budget constraint in macroeconomics is not, on first inspection, so simple  There are many of them – one for every period  Important trick in macroeconomics is to transform these “common sense” constraints (dynamic budget constraints) into single budget constraint, which is called the intertemporal budget constraint  A budget constraint in macroeconomics is not, on first inspection, so simple  There are many of them – one for every period  Important trick in macroeconomics is to transform these “common sense” constraints (dynamic budget constraints) into single budget constraint, which is called the intertemporal budget constraint

 Dynamic budget constraints can be transformed into intertemporal budget constraint in four steps: 1. Write down dynamic budget constraints ("sources of funds" = expenditures) 2. Express each dynamic budget constraint in common units. This involves multiplying each constraint by an interest rate factor 3. Sum the dynamic budget constraints 4. Impose terminal condition(s)  Dynamic budget constraints can be transformed into intertemporal budget constraint in four steps: 1. Write down dynamic budget constraints ("sources of funds" = expenditures) 2. Express each dynamic budget constraint in common units. This involves multiplying each constraint by an interest rate factor 3. Sum the dynamic budget constraints 4. Impose terminal condition(s) Intertemporal Budget Constraint

Three Souces of Funds  Household enjoys three sources of funds in each period (measured in period t dollars): P t y t – Labor income P t y t b t-1 – Principal from loans to other households b t-1 R t-1 b t-1 – Interest from loans to other households R t-1 b t-1  Household enjoys three sources of funds in each period (measured in period t dollars): P t y t – Labor income P t y t b t-1 – Principal from loans to other households b t-1 R t-1 b t-1 – Interest from loans to other households R t-1 b t-1

 Funds are spent on consumption P t c t and the accumulation of bonds b t  Notice that labor income and consumption are measured in “real units” (units of the consumption good) while bonds are measured in dollars  Price level P t is used to convert dollars to real units (P t dollars are required to buy one unit of the consumption good at date t)  Funds are spent on consumption P t c t and the accumulation of bonds b t  Notice that labor income and consumption are measured in “real units” (units of the consumption good) while bonds are measured in dollars  Price level P t is used to convert dollars to real units (P t dollars are required to buy one unit of the consumption good at date t) Two Uses of Funds

Two Dynamic Budget Constraints  Notice the convention chosen for timing in the model  $1 bond purchased at date t-1 pays $1 in principal and $R t-1 in interest at date t  Quantity of such bonds held is denoted b t-1 and R is called the nominal interest rate  With two periods, we have total of two dynamic budget constraints equating the sources and uses of funds: P 1 y 1 + (1+R 0 )b 0 = P 1 c 1 + b 1 P 2 y 2 + (1+R 1 )b 1 = P 2 c 2 + b 2  Notice the convention chosen for timing in the model  $1 bond purchased at date t-1 pays $1 in principal and $R t-1 in interest at date t  Quantity of such bonds held is denoted b t-1 and R is called the nominal interest rate  With two periods, we have total of two dynamic budget constraints equating the sources and uses of funds: P 1 y 1 + (1+R 0 )b 0 = P 1 c 1 + b 1 P 2 y 2 + (1+R 1 )b 1 = P 2 c 2 + b 2

Dynamic Budget Constraint in Common Units  Above, period 1 constraint is expressed in period 1 dollars and period 2 constraint in period 2 dollars  Because the possibility of borrowing and lending, one dollar at period 1 is equivalent to $(1+R 1 ) at period 2  So dividing the period 2 constraint by (1+R 1 ) expresses it in period one dollars:  Above, period 1 constraint is expressed in period 1 dollars and period 2 constraint in period 2 dollars  Because the possibility of borrowing and lending, one dollar at period 1 is equivalent to $(1+R 1 ) at period 2  So dividing the period 2 constraint by (1+R 1 ) expresses it in period one dollars: P2y2P2y2P2y2P2y2 + b 1 = P2c2P2c2P2c2P2c2 + b2b2b2b2(*) 1 +R 1

Interest Rate Arithmetic  Define the real interest rate r t earned between periods t and t+1 as in the “Interest Rate Arithmetic” chapter: (*)  Using the definition of r 1, (*) becomes:  Define the real interest rate r t earned between periods t and t+1 as in the “Interest Rate Arithmetic” chapter: (*)  Using the definition of r 1, (*) becomes: 1 + r t = PtPtPtPt (1 + R t ) (1 + R t ) P t+1 P1y2P1y2P1y2P1y2 + b 1 = P1c2P1c2P1c2P1c2+ b2b2b2b2(**) 1 + r 1 1+R 1

Summing Dynamic Budget Constraints (**)  Summing (**) and the dynamic budget constraint for period 1: P 1 y 1 + b 0 (1 + R 0 ) + P1y2P1y2P1y2P1y2 + b 1 = P 1 c 1 + b 1 + P1c2P1c2P1c2P1c2 + b2b2b2b2 1 + r R 1

Impose Terminal Condition  Require b 2 = 0  b 2 is amount lent in period 2 to be paid back in period 3  There is no period 3 – no one would make such a loan  Using b 2 = 0, canceling b 1 from both sides of the equation and dividing by P 1 :  Require b 2 = 0  b 2 is amount lent in period 2 to be paid back in period 3  There is no period 3 – no one would make such a loan  Using b 2 = 0, canceling b 1 from both sides of the equation and dividing by P 1 : y 1 + b 0 (1 + R 0 )/P 1 + y2y2y2y2 = c 1 + c2c2c2c2 1 + r 1

Resemblance with Prototypical Microeconomic Budget Constraint  just one constraint  Choice variables enter linearly (prices 1 and 1/(1+r 1 ))  Each of choice variables are “goods” and rest of variables (namely, the LHS of the equation) are beyond control of consumer  In microeconomics, LHS is “income” but here it is more accurately described as “the present value of income” because incomes in both periods are counted, with future income discounted  just one constraint  Choice variables enter linearly (prices 1 and 1/(1+r 1 ))  Each of choice variables are “goods” and rest of variables (namely, the LHS of the equation) are beyond control of consumer  In microeconomics, LHS is “income” but here it is more accurately described as “the present value of income” because incomes in both periods are counted, with future income discounted y 1 + b 0 (1 + R 0 )/P 1 + y2y2y2y2 = c 1 + c2c2c2c2 1 + r 1

slope = -(1+r)

Interest Rates have Wealth and Substitution Effects ? ?

 high r always discourages current consumption via a (intertemporal) substitution effect  high r may encourage or discourage current consumption via a wealth effect, depending on asset position  net borrowers experience an adverse wealth effect, which reinforces the substitution effect  net lenders experience a favorable wealth effect, which offsets the substitution effect  typical person is a net lender  s = y – c  r has offsetting wealth and substitution effects on saving  consumption growth c t+1 /c t has just a substitution effect, because the wealth effects cancel  high r always discourages current consumption via a (intertemporal) substitution effect  high r may encourage or discourage current consumption via a wealth effect, depending on asset position  net borrowers experience an adverse wealth effect, which reinforces the substitution effect  net lenders experience a favorable wealth effect, which offsets the substitution effect  typical person is a net lender  s = y – c  r has offsetting wealth and substitution effects on saving  consumption growth c t+1 /c t has just a substitution effect, because the wealth effects cancel