ECON0016: Macroeconomic Theory and Policy
Term 1, Problem set 1
1. The permanent income hypothesis and credit constraints
How would households respond to an announcement at time t of a decrease in income at time t+1.
(a) If households were not credit constrained
· For this example assume initially the individual has no assets a = 0 so, if the individual expected income always to be y1, then initial consumption will be equal to y1 as in the diagram (the red lines are income, the heavy black line consumption)
· At time t, when the announcement happens, individuals reduce their estimate of lifetime wealth and so the optimal level of consumption falls.
· So at time t consumption is reduced and individuals save the difference between consumption and income from t to t+1
· Afterwards consumption is above income since individuals earn interest on the assets they accumulated between t and t+1
(b) In the presence of credit constraints
· “credit” constraints are constraints on borrowing. No borrowing is needed in this example so consumption would be exactly as in (a)
(c) If saving were not possible
· Consumption would track income
(d) What is the relation between consumption and current income for the two cases in (b) and (c)? What are the implications of this for the marginal propensity to consume and the multiplier?
o Under the PIH consumption does not change when an expected change to current income occurs so the MPC is zero and the multiplier is one.
o If households are unable to borrow or save, consumption tracks current income so the MPC is 1 and the multiplier is undefined. Remember the PIH is derived in terms of after-tax income so the multiplier is 1/t
(e) This is what we discussed in class. To smooth consumption, you don’t need saving but borrowing. Therefore, credit constraint is important here.
2. The permanent income hypothesis example
Assuming the real interest rate is 1%, calculate how consumption and borrowing would change in each of these two cases:
(a) A stock market crash permanently reduces the value of an individual’s assets by 1000.
(b) Households are told that in one year they will receive a one-off bonus of 1000. Then in one year’s time it is not paid.
(a) If assets are reduced by 1000, the impact on lifetime wealth is (1+r)1000. The impact on consumption is r/(1+r) (1+r) 1000 = r1000 = 10. There is no change in borrowing/saving.
(b)
· The impact on lifetime wealth is the discounted value of the bonus 1/(1+r) *1000.
· So, the impact on consumption = 1+r/r 1+r/1, i.e. the impact of the bonus is smoothed over the whole of the lifetime.
· To consume extra this period, households borrow this amount.
· If the bonus does not arrive, household assets will be permanently lower by the amount borrowed i.e. 9.8 i.e. lifetime wealth will be lower by (1+r) x 9.8
· So, consumption after one year (when the bonus is not paid) will be reduced from the initial level by the interest on this amount i.e. 1+r/r x (1+r) x 9.8 = 0.098.
3. The algebra of the IS curve
(a) Write down an expression for the government spending multiplier when both consumption and investment depend on current income (hint: follow the steps on slide 8 and slide 9 of lecture 2)
Consumption and investment are
c = c0 + cyy + crr
i = i0 + iyy + irr
Aggregate demand is:
ad = c + i + g = (c0 + i0) + (cy + iy)y + (cr + ir)r + g
Imposing market clearing:
y = ad
Then collecting terms:
So the government spending multiplier is
(b) Is this multiplier larger or smaller than the one derived in the lectures? Explain why.
Larger if iy > 0. The multiplier effect occurs because an initial increase in income causes consumption to rise which increases output and hence income further. If investment is sensitive to output too, there is another channel by which this happens so the multiplier is bigger.
(c) Derive an expression for the IS curve (i.e. a relation between y and r) when both consumption and investment depend on current income
Rearranging the expression above:
y = (c0 + i0) + (cy + iy)y + (cr + ir)r + g
We have
Note: the intercept is positive as marginal propensity to consume+ invest, cy + iy, is less than 1. The slop is negative because cr < 0 and ir < 0.