Question

Ricardian Equivalence Theorem

For concreteness, suppose that all consumers in the economy earn the same labour income and pay the same amount of taxes in each period, but that one group of consumers ('the poor') enters the economy with a zero level of initial wealth at the beginning of period 1, whereas the remaining consumers ('the rich') start out with a level of initial wealth equal to V₁. Moreover, suppose that disposable labour income in period 1 is so low that the poor would like to borrow during that period, but that the banks are afraid of lending them money because they cannot provide any collateral. In that case the poor will be credit-constrained during period 1, and the consumption of a poor person during that period, C₁^p, will then be given by the budget constraint:

C₁^p = Y₁^L – T₁

A rich consumer does not face any borrowing constraint, and their optimal consumption in period 1, C₁^R is given by the consumption function:

C₁^R = ?{ V₁ + Y₁^L – T₁ + Y₂^L – T₂ / 1 + r }, 0<?,<1

Suppose that the total population size is equal to 1 (we can always normalise population size in this way by appropriate choice of our units of measurement). Suppose further that a fraction ? of the total population is 'poor' in the sense of having no initial wealth.

(a) Derive the economy's aggregate consumption function for period 1, that is, derive

an expression for total consumption C₁ = ?C₁^p + (1 – ?) C₁^R.

(b) Derive an expression for the economy's marginal propensity to consume current disposable income, ?C₁ / ?(Y₁^L – T₁)

(c) Compare this expression with the value of the marginal propensity to consume in an economy with no credit-constrained consumers. Explain the difference.

Answer 1

Solution:

See that the aggregate consumption is the sum of consumption functions of the poor and the rich.

Marginal propensity to consume is the derivative of this consumption function with respect to the current income

If there are no credit constrained consumers such that there mu (u) = 1, we have MPC = 1.