consider the following two-period model. The consumer has preferences given by:

Question

consider the following two-period model. The consumer has preferences given by:   u(c,c')=ln c + ß ln c'   where c is current consumption, c' is future consumption, and ß >0 is a discount factor. The consumer receives exogenous income y u tge current period and y' in the future period. The consumer pays proportional tax ts on savyings (i.e. the consumer saves s in the current period but only gets (1-ts) tomorrow). The consumer can save at the interest rate r.

a) Write down the consumer's budget constraint for the current period.

b) Write down the consumer's budget constraint for the future period.

c) Combine the two budget constraints above into one lifetime budget constraint

d) Solve the consumer's optimization problem for c and c'

e) Write down the tax revenue the government receives then imposing the tax on savings.

Explanation / Answer

a) The consumer's budget constraint for the current period:

c + s = y - ts, where ts= taxes in all the constraints given

b)  The consumer's budget constraint for the future period:

c' = (1+r)s +  y' - ts'

c' = (1+r)(1-t) +  y' - ts'

c)  One lifetime budget constraint:

s = c' -  y' + ts'/ (1+r)

d) consumer problem occurs when he has to decide whether to choose present consumption or future consumption. The optimal point occurs where in the graph, the slope of the highest attained indifference curve is equivalent to the slope of the budget constraint.

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