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

Part 1: Example of time allocation with labor, leisure choice with heme production 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 dal,I {T=15]. They get utility from leisure and consumption of market goods (C) and home goods (G) with these marginal utilities: MUL = IQDI'J- 2L MU: = 5011}- 10C MUG = 4m} - G The; purchase market goods {C} at a price p=1, using their earnings from paid work lw'W}. They,r produce home goods {G} using a production method that makes h units of G per hour of time H lso G = h'H]. The wage rate w=5 and the household productivity rate is h=5. 3. 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)? 4. What is the marginal utility of paid work for the first hour of paid work (W=0)? What is the opportunity cost of paid work for the first 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? 5. What is the optimal time allocation for this agent, in terms of hours of work (W), leisure (L) and home production (H)?Step 1: solve for leisure/home production . This part didn't change, same solution . Start by setting W = 0 . Time constraint is now: T = 24 = H + L . Find values of H and L that maximize utility > set MU, = MUH . Find intersection of lines: MU,=110-L and MUH = 100 . That happens at L = 10 . Going back to time constraint, we have that H = 14 . With no work, we have L = 10, H = 14 . This means MU =110-10 = 100 = MUH = 100Step 2: compare MU of work to non-work . MU of time outside of paid work is 100 for last hour . MU of work? Go back to the equation . MUw= 150 - 4W . Set W = 0 . Get MUw= 150 . Is MUw= 150 > MUL= MUH = 100? . If yes, the person will work W > 0. . If no, the person will be at a corner with W =0.Step 3: find the interior solution - We know that w > O - Now we set our 3 MU equations equal to one another - MUL=110L - MU\": 100 - MUW=150 4W - Using first two equations, we have L = 10 - Using second two equations, we have W = 50/4 = 12.5 - Using time constraint, we have 10+12.5+H=24 -) H=1.5 Part 1: Example of time allocation with labor, leisure choice with home production 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 = l 2L MU: = 500 10C MUG = 400 - G They purchase market goods {C} at a price p=1, using their earnings from paid work lw'W). They produce home goods {G} using a production method that makes h units of G per hour of time H [so 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 lT=15) between leisure [L] and home production [H]? What is the marginal utility of paid work for the rst hour of paid work [Will]? 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}