1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.
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1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.
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- Consider the two period consumption savings problem faced by an individual whose utility is defined on period consumption. This utility function u(c) has the properties that it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotes the first derivative while u"(c) represents the second derivative) and satisfies the Inada condition lim.-→0 u'(c) approaches zero). The individual's lifetime utility is give by u(cı) + Bu(c2). In the first period of life, the individual has y1 units of income that can be either consumed or saved. In order to save, the individual must purchase bonds at a price of q units of the consumption good per bond. Each of these bonds returns a single unit of the consumption good in period 2. Total savings through bond purchases is s1 so that total expenditures on purchasing bonds is qs1. Let c1 denote the amount of consumption in period 1 chosen by the individual. In the second period of life, consumption in the amount c2 is financed out of the…Clare is contemplating her possible consumption patter for this year and next. She know that she will have income of $50,000 this year and $55,000 next yea. Her plan is to consume $40,000 this year (t=0). She is also going to invest 30,000. This investment has a positive NPV of $450. She decides to take the investment; in addition, the return on the investment is 9.62%. What consumption she can expect at t=1? (show a detailed procedure)2. Ann has started working and is saving to buy a house, which requires a down-payment of d. She has a monthly income of y and faces goods prices pa and p3. Ann's utility function is given by u = 3 In a. + (1- 3) In a. %3D Her minimum standard of living renders a utility of uo. Solve for the minimum number of months it takes for Ann to save up to the down-payment, as a function of (d, y, uo, Pa, Ps).
- Rodrigo is taking a year between high school and college to work and save up. His utility from consumption each year is U(c) = discounts future utility by B. Rodrigo is going to make $I his year of working, and whatever he doesn't consume from that income will a savings account which will earn return r before he consumes next year. He has to pay for school expenses E in year two, before he consumes (but after return has been realized). 1-o and he go intoA consumer's consumption-utility function for a two period horizon is 0.5 U(Cg,G) =C,G" he consumer's earned income stream is given by mo, m1 and the market rate of interest is r. a) Write the intertemporal budget constraint in present value terms. If the consumer does not consume anything in peripd 0, what is the most she can consume in period 1? b) Draw a graph that shows optimal consumption in each period co* and c1*. What is the slope of her budget line? c) Solve the problem for optimal consumption in each period co* and c;*. d) Suppose mọ is S50 and mị is S110 and r = 0.1. Is the consumer a borrower or a lender? Show this outcome by drawing co*. C1*, mo, mį, and bond-holdings on your graph.no chagpt answer urgent. The marginal rate of substitution of current consumption for future consumption is A) the slope of the indifference curve. B) minus the slope of the difference curve. C) the downward slope of the budget constraint. D) the endowment point. E) the slope of the lifetime budget constraint.
- The consumer choice is not restricted to the choice of consumptiongoods. In fact, it can apply to all our decisions that involve trade-offs. Suppose Mary has awage per hour of 10 euros. With her earned income she consumes. That isC=wH per day.She also decides how many hours to work of take leisure time each day.H=24-N, whereHis work and N is leisure. Her utility is given by (picture) Solve for the optimal decision of labor/leisure. Plot the budget constraint and the indif-ferent curve. What is the labor supply function?2. Mr. A has the following utility function and budget constraints: Max 0.1Ln(C1) + 0.7Ln(C2) Subject to S1 + C1 = 100 C2 + S2 = (1 + r)S1 where C1 and C2 are consumption level at young and that at old respectively. Likewise, S1 and S2 are saving at young and saving at old respectively. a) Find out Mr. A’s optimal consumption levels (i.e. C1*, C2*) and optimal savings (i.e. S1*, S2*) in terms of interest rate r. b) Show clearly the results in part a) in a suitable diagram (with C1 as x-axis and C2 as y-axis). c) Is Mr. A a saver ? or a borrower ? d) If r is equal to 0 (i.e. saving gives no returns), will Mr. A still choose to save when he is young (i.e. is S1 still bigger than 0) ? Why ? e) Suppose that Mr. A is not allowed to save (i.e. S1 = 0). What are his optimal consumption levels ? Show his optimal consumption levels in the same diagram you prepare for part a) (with a suitable indifference curve). f) If r increases,…On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according tothe functionU(c, b) = 20c− c²+ 18b − 3b²a. How many cigars and glasses of brandy does he consume during an evening? (Cost isno object to J. P.)b. Lately, however, J. P. has been advised by his doctors that he should limit the sum ofglasses of brandy and cigars consumed to 5. How many glasses of brandy and cigarswill he consume under these circumstances?
- During any year, I can consume any amount that doesnot exceed my current wealth. If I consume c dollars duringa year, I earn ca units of happiness. By the beginning of thenext year, the previous year’s ending wealth grows by afactor k.a Formulate a recursion that can be used to maximizetotal utility earned during the next T years. Assume Ioriginally have w0 dollars.b Let ft(w) be the maximum utility earned during years t, t 1, . . . , T, given that I have w dollars at the be-ginning of year t; and ct(w) be the amount that should be consumed during year t to attain ft(w). By workingbackward, show that for appropriately chosen constantsat and bt,ft(w) btwa and ct(w) atwInterpret these results.Anna has endowment 1500 now and 500 later. Internet rate is 2.0%. She prefers smooth consumption to time (i.e., u0=u1=u). a. Assume utility function, u(c)= log c. What are the optimal consumption c0and c1if Anna's beta=1, and she wants to maximize her utility? b. Now assume that the utility function, u(c)=c0.5. If everything else remains the same as Problem 1(a), what are the optimal consumption c0and c1if Anna wants to maximize her utility?What is the logarithmic transform of the utility function U= 52, given the budget constraint - M. Select one: O a. In U = a ln a +Bin 2 +y In os %3D O b. In U = In a + In + In e In U = a ln a -Bln a -yln es Od. InU=n 2? + In e? + In a