Question 1 Consider the following rickshaw pullers problem m

Question 1: Consider the following rickshaw puller\'s problem: max U(c, l)=(c subject to c+wL Equation (1) is a rickshaw puller\'s utility function in which utility depends on consumption of a numeraire good c and leisure L. The maximization is with respect to consumption and leisure. Equation (2) is a budget constraint: itures of consumption c and a rented rickshaw r cannot exceed aw cash revenue from rickshaw service wl, where w is market wage of ricksh pulling and L is the labor time for rickshaw pulling. Equation (3) defines a time constraint: the puller\'s time endowment L equals his labor time for rickshaw service L and leisure time l. 1. Obtain the optimal consumption level c\" 2. Obtain the optimal leisure time l\' 3. Now suppose the market wage of rickshaw puller w falls down. Find the impacts on the optimal consumption level e and leisure time 1,1 tively espec- 4. Next suppose the rental cost of rickshaw r rises up. Find the impacts on the optimal consumption level c and leisure time l\', respectively 5. Obtain the indirect utility function V (w, 7, L).

Solution

Consider the given problem here the problem of “rickshaw puller’s” is given below.

=> max U = (c*l)^0.5 subject to “c + r < = w*L”, where “l + L = T = L bar”.

So, consider the budget constraint here “c + r < = w*(T – l) = w*T – w*l, => c + w*l < = (w*T – r), the slope of the budget line is “dc/dl = (-1)*w”.

=>Here at the optimum the absolute slope of the budget line must be equal to “MRS”.

So, here “U=c^0.5*l^0.5, => MUl = (0.5)*l^(-0.5)*c^0.5 and MUc = (0.5)*c^(-0.5)*l^0.5.

=> MRS = MUl/MUc = [(0.5)*l^(-0.5)*c^0.5] / [(0.5)*c^(-0.5)*l^0.5] = (c / l).

=> at the optimum MRS = c/l = |dc/dl| = w, => (c / l) = w, => c = w*l………(ii).

Now, substituting the above condition into the budget line we will get the optimum “l”.

=> c + w*l = (w*T – r), => w*l + w*l = (w*T – r), => 2w*l = (w*T – r), => l = (w*T – r) / 2w = T/2 – r/2w.

=> l* = T/2 – r/2w.

Now, substituting “l*” into “ii” we will get “c*”

=> c = w*l, => w*[ T/2 – r/2w] = w*T/2 – r/2, => c* = w*T/2 – r/2.

So, the optimum “consumption” is “c* = w*T/2 – r/2” and the optimum leisure is “l* = T/2 – r/2w”.

3.

We can see that “c” and “l” both of them will depend on “w”. Let’s assume that initially “w=w1”,

=> “c1* = w1*T/2 – r/2” and “l1* = T/2 – r/2w1”.

Now as “w” falls to “w2 < w1”, => now “c2* = w2*T/2 – r/2 < c1*” and “l2* = T/2 – r/2w2 < l1*”, as we can see that “c*” is directly related to “w”, => as “w” will increase => “c*” will increase and vice versa. Similarly is “w” increase, => “r/2w” will decrease, => “T/2 – r/2w” will increase, since “T/2” is fixed here.

=> “c” and “l” both of them are directly related to “w”, => as “w” decreases, => “c” and “l’ will fall down.

4.

Similarly “c” and “l” are negatively related to “r”, => as “r” increases, => both of them “c” and “l” will fall down. So, as the rental cost of rickshaw goes up the optimum value of “c” and “l” will fall.

5).

So, the optimum value of “c” and “l” are “c* = w*T/2 – r/2” and “l* = T/2 – r/2w” respectively and the utility function is “U = c^0.5*l^0.5 = [w*T/2 – r/2]^0.5*[ T/2 – r/2w]^0.5.

So, the indirect utility function is given below.

=> V(w, r, T) = [w*T/2 – r/2]^0.5*[ T/2 – r/2w]^0.5”, where “T = L bar” and “V(w, r, T)” be the indirect utility function”.