Consider the two period consumption savings problem faced by an individual whoseutility is defined on period consumption. This utility function u(c) has the propertiesthat it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotesthe first derivative while u"(c) represents the second derivative) and satisfies the Inadacondition 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 eitherconsumed or saved. In order to save, the individual must purchase bonds at a price ofq units of the consumption good per bond. Each of these bonds returns a single unit ofthe consumption good in period 2. Total savings through bond purchases is s1 so thattotal expenditures on purchasing bonds is qs1. Let c1 denote the amount of consumptionin period 1 chosen by the individual. In the second period of life, consumption in theamount c2 is financed out of the returns from savings and period 2 income, y2.The problem of the individual is to maximize lifetime utility while respecting thebudget constraints of periods 1 and 2 by choice of (C1, c2, s1). Formally, the individualsolves the problem= 0 (slope of the utility function becomes vertical as consumptionmax {u(ci) + Bи(с2)}C1,C2,81subject to the first period budget constraint,qsi + C1 = yY1along with the second period budget constraint,C2 = Y2 + $1. Using the first-order conditions of this problem with respect to c1, c2 and s1, (i.e.the partial derivatives that have been set equal to zero) construct the optimalintertemporal consumption trade-off condition between c and C2. This trade-off isexecuted by variation in savings.

For the above question let us first derive the lifetime budget constraint : Budget constraint for…

Using the first-order conditions of this problem with respect to C₁, C₂ and s₁, (i.e. the partial derivatives that have been set equal to zero) construct the optimal intertemporal consumption trade-off condition between c₁ and c₂. This trade-off is executed by variation in savings.

Using the first-order conditions of this problem with respect to C₁, C₂ and s₁, (i.e. the partial derivatives that have been set equal to zero) construct the optimal intertemporal consumption trade-off condition between c₁ and c₂. This trade-off is executed by variation in savings.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Similar questions

image attached
Assume you can work as many hours you wish at £12 per hour (net of tax). If you do not work, you have no income. You have no ability to borrow or lend, so your consumption, c, is simply equal to your income. Derive and plot the feasible set, between daily values of consumption c, and “leisure”, l. Label the values at the intercepts (the points where the feasible frontier cuts the two axes).
solve this all these three subparts early i upvote
The utility derived by a consumer from the consumption of two commodities is given by the function U (A, B) = 0.5In (A) + 0.5 In (B) where A are the number of units of the first commodity consumed and B are the number of units of the second commodity consumed each month. A unit of the first commodity costs $8 and a unit of the second commodity costs $ 4 using the Lagrange multiplier method determine the optimal quantity of each of the commodities consumed each month given that consumer has $32 to spend on both commodities each month
Answer please with added diagrams b) Assume that your optimal choice of consumption and leisure is to work 8 hours per day. Illustrate this choice diagrammatically using the feasible set and indifference curves. c) Use indifference curves and the feasible set to show why, given the properties of the optimal choice in part b it is not optimal to work, say, 10, or 6 hours per day.
A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
Anticipated productivity shift An infinitely lived representative household has preferences over a stream of consumption of a single good that are ordered by Where u is a strictly concave, twice continuously differentiable one period utility function, β is a discount factor, and ct is time t consumption. The technology is: Here f(kt)nt is output, where f’ > 0, f” > 0, f t is capital per unit of labor input, and nt is labor input. The household supplies one unit of labor inelastically. The initial capital stock k0 is given and is owned by the representative household. In particular, assume that k0 is at the optimal steady value for k presuming that ψt had been equal to 1 forever. There is no uncertainty. There is no government. a. Formulate the planning problem for this economy in the space of sequences and form the pertinent Lagrangian. Find a formula for the optimal steady state level of capital. How does a permanent increase in ψ affect the steady…
Fred is planning his consumption over two time periods. Fred's preferences for consumption in period and two can be represented by the following utility function: U(c,,c,) = C +(1+p) C" , where pis the subjective discount rate, and c;,c, is consumption in the first and second period. Fred's income in the first period is y, and grows by g % from the first period to the second period. Fred has access to perfect financial markets. The rate of interest is r>0. (a) Derive Fred's demand functions for consumption in the two periods as functions of p,r , y and g. (b) Derive Fred's demand for borrowing/saving as a function of p,r, y and g. (c) Give a condition involving the relationship between r and g for when Fred will borrow and when he will save.
A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
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…
If your utility function is the following form: U(c) = c1 + B c2, where c1 is the consumption in period 1, c2 is the consumption in period 2, the time-preference parameter, B= 1 and period l's endowments el= 500 and period 2's endowments e2 = 400, what is the optimal consumption in each period if (a) the interest rate (r) is zero, and (b) the interest rate (r) is positive? %3D
Find the marginal utility of an additional unit of good z for a consumer with utility function u (a, y) = zi yi when their current consumption level is 2 and y = 16.