Consider the application on Gary Becker\'s model of optimal allocation of time.
ID: 1208142 • Letter: C
Question
Consider the application on Gary Becker's model of optimal allocation of time. As with Becker's model, assume your utility is derived from the consumption process rather than from the good itself so that your utility function is U(A_1, A_2) = 1/2 In (A_1) + 1/2 In(A_2), where A_i are the set of activities in which you consume the set of goods G_i. The ith activity requires G_i/n_i units of a good and t_i/h_i hours of time, where n_i and h_i are units of goods and units of time, respectively, per one unit of activity i. Assume that there are only two activities and n_1 = 2, n_2 = 6, h_1 = 2, and h_2 = 2.4. Assume that you are a working parent facing both time and financial constraints. The total time available to you is T = W + t_1 + t_2 = 24, where W represents work, t_1 is the time you spend caring for your children and household, and t_2 is the time you have exclusively for yourself. Your financial constraint reflects the assumption that your total income, wW, is spent on either good 1 or good 2 as given by wW = P_1G_1 + P_2G_2, where P_1 = 3, P_2 = 2, and w = 5. Combining the time and financial constraints into a consolidated constraint that defines the money value of total time and rewriting the constraint in terms of activities rather than goods, we have WT = P_1n_1A_1 + P_2n_2A_2 + wh_1A_1 + wh_2A_2. (a) Set up a Lagrangian function, and determine the values of A_1 and A_2. Use your results to calculate how much time you would spend at work, caring for your household, and pursuing your own leisure activities given your time constraint. (b) Solve for G_1 and G_2 to determine how your total income is spent.Explanation / Answer
ans:
a)
The system for building a Lagrangian capacity is to consolidate the target capacity and all requirements in a way that fulfills two conditions. To begin with, streamlining the Lagrangian capacity must result in the goal capacity's enhancement. Second, all imperatives must be fulfilled. With a specific end goal to fulfill these conditions, utilize the accompanying strides to indicate the Lagrangian capacity.
The application on Gary Becker's Model of ideal designation of time .As Becker's model accept utility got from the utilization prepare somewhat then from the great itself with the goal that uility capacity is
U ( A1, A2) = ½ ln (A1 ) + ½ ln (A2)
Accept that there are just two exercises and
N1= 2 , n2 = 6 , h1 = 2 and h2 = 24
T = W +t1 +t2 = 24
Where W speaks to work , t1 is the time you spend brave for youngsters and family unit and t2 is the time you spend nurturing kids and family and t2 is the time solely for yourself . your money related requirement mirrors the suspicion that your aggregate wage wW, is spent on either good1 or great 2 as given by
wW = P1G1 + P2 G2
This variable is essential in light of the fact that A1 measures the change that happens in the variable being improved given a one-unit change in the limitation. In case you're attempting to minimize the expense of delivering a given amount of yield, wT lets you know the amount of aggregate cost changes in the event that you create one more unit of yield. This empowers you to rapidly evaluate the connections amongst imperatives and the variable being optimized.The Lagrangian capacity is a strategy that consolidates the capacity being improved with capacities portraying the requirement or limitations into a solitary comparison. Explaining the Lagrangian capacity permits you to streamline the variable you pick, subject to the limitations you can't change.
b)
wW = P1G1 + P2 G2
where P1 = 3 ,P2 = 2 and w= 5 joining the time and budgetary requirements into a combined imperatives that characterizes the cash estimation of aggregate time and revamping the limitations regarding action instead of products , we have
wT = P1n1A1 + P2n2A2 + wh1 A1+ wh2A2
Where W speaks to work , t1 is the time you spend brave for youngsters and family and t2 is the time you spend looking after kids and family unit and t2 is the time only for yourself . your money related imperative mirrors the supposition that your aggregate pay wW, is spent on either good1 or great 2 as given by The goal is to amplify the amount delivered subject to the financial backing limitation given the cost of work and capital. This issue can be unraveled utilizing the non-straight strategy for obliged improvement called the Lagrange method.There is a cost capacity that is the whole of the inputs duplicated by the cost of inputs to meet the aggregate expense. There is likewise an iso-cost bend that is a direct blend of the inputs that outcomes in equivalent aggregate cost given the cost of the inputs.
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