Term 2 Lecture 7 Flashcards
(10 cards)
Summary: allocation of spending over time can be seen as a further example of
The theory of demand with endowments.
- intertemporal choice can be seen as an extension of consumer theory with endowments, where people have income at different points in time and must decide when to consume or save
- preferences often exhibit discounting, meaning people typically prefer present consumption over future consumption
Intertemporal Consumption budget constraint:
C0 + c1/(1+r) = y0 + y1/(1+r)
- any resources not consumed in period 0 can be saved and earn interest r
- any additional consumption in period 0 must be borrowed and repaid in period 1 with interest
Demand for current consumption
- how do we derive it?
C0 = f0(y0 + y1/1+r, r)
- current consumption depends on the present value of lifetime income
- the interest rate, r
Derive this by rearranging the budget constraint for c1, then subbing into the lifetime utility function and then maximising that for c0
Effect of interest rate changes on individual, given the current consumption demand function
Borrower: consumes more than their initial income in period 0, meaning they must borrow and repay in period 1
- higher interest rate, means higher repayment costs, so since borrowing funds current consumption, c0 falls
- lower interest rate, reduces borrowing costs, makes borrowing cheaper, increasing c0
Saver: consumes less than their income and invests excess income at rate r, receiving 1+r times their savings in period 1
- higher interest rate means greater future returns, so saves more and consumes less today, c0 falls
- lower interest rate, reduces the benefit of saving, reducing the incentive to save, leading to a higher c0
Therefore saver remains saves and borrower remains borrower.
Intertemporal utility function:
U(c0,c1) = v(c0) + (1/1+k).v(c1)
- v is the within period utility function
- k is the subjective discount rate, how much the individual values future utility relative to present utility
Convexity and consumption smoothing for the intertemporal utility function
To ensure a desire to smooth consumption over time, preferences must be convex, requires the within period utility function to be concave, meaning v’’(c) < 0, the 2nd derivative of the utility functions with respect to consumption c
- concavity implies DMU of consumption
- means individuals prefer a balanced consumption path, rather than everything in one and none in the other
Deriving the Euler Equation
Max v(c0) + 1/(1+k).v(y1 + (y0 - c0)(1 + r)) s.t. The intertemporal budget constraint
- FOC with respect to c0, to get:
(V’(c0))/(v’(c1)) = (1+r)/(1+k) —> the Euler Equation
- LHS is the MRS between current and future consumption
- RHS is the tradeoff imposed by the market interest rate
Implications of the Euler Equation
- r = k
- r > k
- r < k
- role of concavity
- r = k, then c0 = c1
- r > k, then individual saves more, so c1 > c0
- r < k, individual consumes more today, so c0 > c1
- c0 and c1 are all increasing functions of life time resources y0 + y1/1+r
- more concave v(.) means more sensitivity to interest rate changes
Intertemporal elasticity of substitution, IES
How sensitive the ratio of future to current consumption, c1/c0 is to changes in the interest rate r
- o = Dln(c1/c0)/Dln(1+r) = Dln(c1/c0)/Dln(v’(c1)/v’(c0))
- high o means people are more willing to adjust their consumption between today and the future when interest rates change
- IES is inversely related to the concavity of v(c), if its very concave, MU falls sharply as consumption increases, means people are less willing to change consumption across periods -> a low o
Optimisation problem to max lifetime utility, with both consumption c0 and investment in the family enterprise X
Max v(c0) + (1/1+k).v(y1 + F(x) + (y0 - c0 - X)(1+r))
- F(X) is the future return form investing in the family business
- in period 0, individual can either consume, invest in bonds, or invest in the business
- in period 1, receive returns from both investments and any unspent money from period 0
FOC: condition for consumption:
- v’(c0) = 1+r/1+k
FOC: condition for X:
- F’(X) = 1+r, so the individual will invest in the family enterprise until the marginal return on investment equals the returns from bonds, so if f’(x) is less than 1+r, then its better to invest in bonds and vice versa.
