Tax and Retirement Policy: In Chapter 3, we illustrated budgets in which a consumer faced trade-offs between working and leisuring now as well as between consuming now and consuming in the future. We can use a model of this kind to think about tax and retirement policy.
A: Suppose period 1 represents the period over which a worker is productive in the labor force and period 2 represents the period during which the worker expects to be retired. The worker earns a wage w and has L hours of leisure time that could be devoted to work l or leisure consumption „“. Earnings this period can be consumed as current consumption c1 or saved for retirement consumption c2 at an interest rate r. Suppose throughout that consumption in both periods is a normal good - as is leisure this period.(a) Illustrate this worker's budget constraint in a 3-dimensional graph with c1, c2, and „“ on the axes.
(b) For certain types of tastes (as for those used in part B of this question), the optimal labor decision does not vary with the wage or the interest rate in this problem. Suppose this implies that taking „“ˆ— in leisure is always optimal for this worker. Illustrate how this puts the worker's decision on a slice of the 3-dimensional budget you graphed in part (a).
(c) Assume that optimal choices always occur on the 2-dimensional slice you have identified. Illustrate how you could derive a demand curve for c1 -i.e. a curve that shows the relationship between c1 on the horizontal axis and the interest rate r on the vertical. Does this curve slope up or down? What does your answer depend on?
(d) Can you derive a similar economic relationship - except this time with w rather than r on the vertical axis? Can you be certain about whether this relationship is upward sloping (given that consumption in both periods is a normal good)?
(e) Suppose that the government introduces a program that raises taxes on wages and uses the revenues to subsidize savings. Indicate first how each part of this policy - the tax on wages and the subsidy for savings (which raises the effective interest rate) - impacts current and retirement consumption.
(f) Suppose the tax revenue is exactly enough to pay for the subsidy. Without drawing any further graphs, what do you think will happen to current and retirement consumption?
(g) There are two ways that programs such as this can be structured: Method 1 puts the tax revenues collected from the individual into a personal savings account that is used to finance the savings subsidy when the worker retires; Method 2 uses current tax revenues to support current retirees-and then uses tax revenues from future workers to subsidize current workers when they retire. (The latter is often referred to as "pay-as-you-go" financing.) By simply knowing what happens to current and retirement consumption of workers under such programs, can you speculate what will happen to overall savings under Method 1 and Method 2 (given that tax revenues become savings under Method 1 but not under Method 2)?
B: Suppose the worker€™s tastes can be summarized by the function

(a) Set up the budget equation that takes into account the tradeoffs this worker faces between consuming and leisuring now as well as between consuming now and consuming in the future.
(b) Set up this worker€™s optimization problem€”and solve for the optimal consumption levels in each period as well as the optimal leisure consumption this period.
(c) In part Awe assumed that the worker would choose the same amount of work effort regardless of the wage and interest rate. Is this true for the tastes used in this part of the exercise?
(d) How does consumption before retirement change with w and r? Can you make sense of this in light of your graphical answers in part A?
(e) In A(e) we described a policy that imposes a tax t on wages and a subsidy s on savings. Suppose that the tax lowers the wage retained by the worker to (1ˆ’t)w and the subsidy raises the effective interest rate for the worker to (r +s). Without necessarily redoing the optimization problem, how will the equations for the optimal levels of c1, c2 and „“ change under such a policy?
(f) Are the effects of t and s individually as you concluded in A(e)?
(g) For a given t, how much tax revenue does the government raise? For a given s, how much of a cost does the government incur? What do your answers imply about the relationship between s and t if the revenues raised now are exactly offset by the expenditures incurred next period (taking into account that the revenues can earn interest until they need to be spent)?
(h) Can you now verify your conclusion from A (f)?
(i) What happens to the size of personal savings that the individual worker puts away under this policy? If we consider the tax revenue the government collects on behalf of the worker (which will be returned in the form of the savings subsidy when the worker retires), what happens to the worker€™s overall savings€”his personal savings plus the forced savings from the tax?
(j) How would your answer about the increase in actual overall savings change if the government, instead of actually saving the tax revenue on behalf of the worker, were to simply spend current tax revenues on current retirees. (This is sometimes referred to as a pay-as-you-go policy.)

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(1-6) C2 ul(1.2.0) = (40-a) ß (1-B) u(съ,с2,0)