Please don't consider the calculus aspect to answer the question because all the calculation methods will be provided.

*MUW = marginal utility of work, MUL = marginal utility of leisure, MUH = marginal utility of household production. Utility function -> U (L, C, G). Be mindful that upper case W = paid Work unit, lower case w = wage rate, C = consumption from market, p = price of consumption, H = unpaid home production, G = home goods and services that are produced at home, and household productivity rate = h

* Time constraint: T = W + H + L * Budget Constraint: w*W = p*C

G = h*H, C = w*W/p. Equations to find the marginal utility value for the desired variable: MUW = MUC * w/p , MUH = MUG * h

Consider an agent who must decide how to divide their time among 3 activities: leisure (L), paid work (W), and unpaid home production (H). The agent has 15 hours total in the day (T=15). They get utility from leisure and consumption of market goods (C) and home goods (G) with these marginal utilities: MUL = 1900 2L MUc = 500 10C MUG = 400 G They purchase market goods (C) at a price p=1, using their earnings from paid work (w*W). They produce home goods (G) using a production method that makes h units of G per hour of time H (50 G = h*H). The wage rate w=5 and the household productivity rate is h=5. 1. Explain why this person might want to spend time on home production. What is their marginal utility from time spent on home production? It is increasing, decreasing, or unchanging when H increases? Explain why this person might want to spend time on paid work. What is their marginal utility from time spent on paid work? It is increasing, decreasing, or unchanging when W increases? If the person does not work for pay at all, what is their optimal allocation of time (T=15) between leisure (L) and home production (H)? What is the marginal utility of paid work for the rst hour of paid work (W=0)? What is the opportunity cost of paid work for the rst hour of paid work (= what is the marginal utility of the last hour spent outside of paid work)? Does the person want to work for pay? What is the optimal time allocation for this agent, in terms of hours of work (W), leisure (L) and home production (H)? Solving a model graphically: 2 graphs - Same 3 steps as above, used for algebra and graphs - Step 1: compare (equate) marginal utilities of leisure and home production in one graph {as was done before for leisure and work} - Step 2: plot the marginal utility of work on a second graph and test for corner solution/pa rticipation decision - Step 3a: if at a corner with no working, set W*=0, L* and H\" are the values in the first graph - Step 3b: if not at a corner, solve for the optimal interior solution with W, L, H all :- 0 Two graphs showing 3 MU curves Marginal Ufllit'f MUM," LID; HIT L\" = T H\" L=T; H=0 Leisure {L to R] I Home production {R to L] Work time