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A consumer views a unit of consumption in period 1 as a perfect substitute (one for one) for a unit of consumption in period 2. If the real interest rate is positive and if p1=p2, the consumer will

consume only in period 1.

consume more in period 1 than in period 2 if income elasticity exceeds 1, otherwise consume more in period 2 than in period 1.

consume only in period 2.

consume equal amounts in each period.

equalize expenditures but not consumption in the two periods.

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QUESTION 33

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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.

Instructions: Please answer the following questions related to current (Period 1) and future (Period 2) demands for groundwater within the context of our two-period model.

Suppose that Period 1 demand for groundwater is given by the linear equation QUOTE PToday=$10-0.4*Q Today. Furthermore, suppose climate change causes Period 2: demand for groundwater to be different linear equations, QUOTE PFuture=$12-0.3*Q future. The current reserves of groundwater to be allocated across both periods is Q=Q1+Q2=16, and future demand is discounted at a rate of so that the present value of future demand is given by 11.08*P2 and the present value of future net benefits is 11.08*NB2. The marginal cost of extraction is constant and equal in both periods: MC1=MC2=$3.

a) Determine the optimal quantity of groundwater to extract in Periods 1 and 2 when users in both periods are considered in one problem. Is more extracted today or in the future, and why? What does the marginal user cost?

b) Determine the number of units extracted in Periods 1 and 2 if current Period 1 user lack secure property rights over the aquifer. That is, solve the problem as if Period 1 users considered only their consumption in their problem.

c) Explain in words/in graphs/in equations (as you prefer) why the quantities extracted in b) represent a loss of overall welfare/net benefits when compared to the extraction in a).

d) What would the discount rate need to be to optimally solve the two-period model with equal quantities extracted in both periods? That is, what r sets present value of demand or net benefits equal in both periods while maintaining equilibrium extraction quantities at Q1*=Q2*=8?

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