In the two-period model, which equation represents the consumer's lifetime budget constraint? +c' 1+r * +y' – – t' 1+r 1+r c' y' t' c + 1+r = y +- 1+r :-t - 1+r c(1+r) + c' = y(1+ r) + y' – t(1+r) t' | c + c' = y + y' – t – t'
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- Suppose that a consumer/investor has an initial endowment only for the current period, which is Eo =450. She may consume today or in the next period only (two-period model). The interest rate for borrowing and lending in the capital market is 5% (a)Depict the budget constraint for the investor in an inter-temporal consumption diagram! What is the maximum amount the consumer is able to consume in the next period? (b)The consumption preferences of the consumer/investor are best described by a square root function, defined over current and future consumption. What is his optimal consumption plan? Show your calculations! Depict the results in appropriate diagram. Which amount is invested in the capital market?There are some simplifying assumptions in order to generate simple expressions. One of these assumptions is that r (interest rate) = ρ (rate at which household discounts future). Suppose we relaxed this assumption (i.e. allowed r to differ from ρ). Two results of the model are: i) The household keeps the expected value of consumption constant over time. ii) The household responds differently to permanent versus temporary income changes. Discuss the implications of allowing r to differ from ρ on each of these resultsQ1. Consider the following two-period model of consumption and saving: Utility = C1^0.5 + B*C2^0.5 C1 + C2/(1+r) = Y1 + Y2/(1+r) where Y1 = 4, Y2 = 1, r = 0.17 and B = 0.5. Find a numerical solution for period 1 consumption, C1. (State your answer to 2 decimal places.)
- Consider the two-period household-maximization model discussed inclass. The model is modified in order to look at applications including credit constraints,interest-rate markups, and taxation. A representative household lives for two periods andmaximizes utility of consumption in period 1 and in period 2. The utility is represented bylog(c) where c denotes consumption. Assuming no discounting between period 1 and period 2. The maximization problem for the representative household can be written asmax{log c1 + log c2}c1 + a1 = y1 − τ1 + (1 + r)a0c2 = y2 − τ2 + (1 + r)a1where y1 and y2 denote income levels in period 1 and period 2, τ1 and τ2 are taxes in the twoperiods, and a0 and a1 denote the assets of the households in each period. a0 is exogenouslygiven. Assume the interest rate r = 0, and the government can borrow or save at the sameinterest rate so that its present-value budget constraint is given byg1 + g2 = τ1 + τ2where g1 and g2 are exogenous government expenditures in the two…What is meant by “excess sensitivity” of consumption? Is this view of consumption consistent with the permanent-income hypothesis? Explain. How does the stock market affect consumption according to the permanent-income hypothesis? Is this prediction in line with the empirical evidence? Explain.Assume a set of well-behaved (i.e. strictly monotone and strictly convex) intertemporal indifference curvesbetween period 1 and 2. Then suppose that the nominal interest rate r decreases. Explain what happens to thenew interior solution if current and future consumption are normal and inferior goods, respectively.
- Consider the two-period household-maximization model discussed in class. The model is modified in order to look at applications including credit constraints, interest-rate markups, and taxation. A representative household lives for two periods and maximizes utility of consumption in period 1 and in period 2. The utility is represented by log(c) where c denotes consumption. Assuming no discounting between period 1 and period 2. The maximization problem for the representative household can be written as below (see image): Question: Show consumption c1 and c2 (you can use algebraic or graphical methods). In theanswer, you should discuss whether a1 ≥ 0 or a1 < 0 and provide an economic interpretation.What determine(s) the sign of a1 and why?We consider a two-periodendowmenteconomy. Suppose you have the following utility function U （c , c ′）=lnc +βlnc ′ where c and c ′ refer to consumption in the first and second periods, respectively, and β =0.5. The household income in the first and second periods is denoted by y and y ′. Assumegovernment collects lump-sum tax in both periods (T, T ′)to finance the public spending inboth periods (G , G ′)and issue bonds B in the first period. A real net interest rate r is 1. 1. Write down the household’s problem. Does the household perfectly smooth their con-sumption over lifetime? Show it using the optimality condition. 2. Solve for optimal c , c ′ and s (saving) when y −T > y ′−T ′. 3. Given the conditions in part (2), there is an increase in future income y ′ such that y −T =y ′−T ′. Find new optimal c , c ′ and s . Explain how c , c ′ and s change compared to part(2). 4. Focusing on part (3),(a) Suppose that the government increases T by a small amount and decreases T ′ bythe…Consider the two-period model. The consumerís preferences over current and future consumption (c and c 0 ) are: REFER TO IMAGE FOR NEXT STEP (a) Find lifetime wealth, we. (b) Set up the Lagrangian and FINDd the optimal levels of current consumption (c), future consumption (c') and saving (y-t-c) c) Confirm that the allocation you found in part (b) is in fact optimal, by completing the following table. refer to second image
- In a two-period model, an individual earns and consumes C1 in period 1 and only consumes C2 in period 2. Suppose the saving interest rate is 3.3% and the income in period 1 is $4,500. Assuming consumption smoothing, the consumption (C1 or C2) for period 1 and period 2 should be $ A . Compute A.In a two-period model, an individual earns and consumes C1 in period 1 and only consumes C2 in period 2. Suppose the saving interest rate is 3.3% and the income in period 1 is $4,500. Assuming consumption smoothing, the consumption (C1 or C2) for period 1 and period 2 should be $ A . Compute A.Consider an individual who lives for two periods and has utility of lifetime consumption U = log(C1) + 1/1+δ log(C2), where C1 and C2 are the consumption levels in the first and second period respectively, and δ, 0 1 > 0 in the first period and no income in the second period, so Y2 = 0. He can transfer some income to the second period at a before-tax rate of return of r, so saving $S in the first period gives $[1 + r]S in the second period. The government levies a capital tax at rate τ on capital income received in the second period. The tax proceeds are paid as a lump-sum transfer to the following generation. The present generation does not care about the next one. a. What is the lifetime consumption profile of this individual? What is his lifetime indirect utility function expressed as a function of Y1 and b. Evaluate the change in initial income Y1 that is required to compensate the individual for the welfare loss due to the capital income tax τ. c. What is…Assume a consumer has current-period income y = 200, future-period income y′ = 150, current and future taxes t = 40 and t′ = 50, respectively, and faces a market real interest rate of r = 0.05, or 5% per period. The consumer would like to consume according to the following utility function: U (c, c′ ) = ln(c) + ln(c′ ). Show mathematically the lifetime budget constraint for this consumer. Find the optimal consumption in the current and future periods and optimal saving. Suppose that instead of r = 0.05 the interest rate is r = 0.1. Repeat parts (a) and (b). Does the substitution effect or the income effect dominate?
