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Question: a households utility over consumption c and leisureis u ...

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A households utility over consumption C and leisureis U - U(C,l) Cl. 1. Plot the households indifference curve for U = 80 for values of C and f less than 20 (i.e. find the curve containing all combinations of C andsuch that U(C,e) 80). The household has a time endowment of h 16 hours per day. The wage rate per hour is w=1.25. The households labour income is therefore wN°, where N-h-f=16-f is the time spent working every day. Since this household does not have any other sources of income, its consumption will be C-wN w(h-L)-1.25(16-). 2. How much leisure can this household enjoy at most if it does not buy any consumption goods? How much can it consume at most if it uses all its time endowment to work? 3. Draw the households budget line in the same figure as the indifference curve 4. What is the households optimal consumption bundle (i.e. how many units of C and( will it choose to consume)? Now lets solve the same problem analytically. Remember that the households budget constraint can be written C1.25(16-) 5. Substitute the budget constraint into the utility function to obtain an expression for utility that depends on l only 6. Maximize this utility to obtain the optimal amount of . (Take the derivative of this expression with respect to , set the derivative equal to zero, solve for l.) 7. Find the optimal amount of C by plugging your result for into the budget constraint. 8. How does the optimal consumption bundle you just derived compare to the one you found graphically before? Question 2: Firms Consider a firm that produces output Y from capital K and labour N using the production technology Y -F(K,Nd)-K*(Ndy. The firms capital endowment is given as K-50 Labour is hired to maximize profits. At a wage rate w, the firms labour costs are wN The firms profit (as a function of N) is therefore 1. Find the firms labour demand function by maximizing profits and solving the first- order condition for the wage rate w 2. Plot the labour demand curve ( on the horizontal axis, w on the vertical axis) 3. What wage rate would the firm be willing to pay if it were to hire N-8 units of labour? (Use your labour demand function!) What would the firms profit be in this case? (Use the profit function!) 4.
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