ECON 214: Intermediate Macro (2024/25, T2)

Assignment 2

due on February 5th

Question 1: Households

A household’s utility over consumption C and leisure  is U = U(C , ) = C.

1.   Plot the household’s indifference curve for  U = 80  for values of C and  less than 20 (i.e. find the curve containing all combinations of C and  such that  U(C , ) = 80 ).

The household has a time endowment of h=16 hours per day. The wage rate per hour is w=1.25. The household’s labour income is therefore wNs, where Ns=h-  =16-  is the  time spent working every day. Since this household does not have any other sources of income, its consumption will be C=wNs=w(h- )=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 household’s budget line in the same figure as the indifference curve.

4.   What is the household’s optimal consumption bundle (i.e. how many units of C and  will it choose to consume)?

Now let’s solve the same problem analytically. Remember that the household’s budget constraint can be written C= 1.25(16- ).

5.   Substitute the budget constraint into the utility function to obtain an expression for utility that depends on  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 .)

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 Nd using the production technology  The firm’s capital endowment is given as K=50.  Labour is hired to maximize profits. At a wage rate w, the firm’s labour costs are wNd.

The firm’s profit (as a function of Nd) is therefore

1.   Find the firm’s labour demand function by maximizing profits and solving the first- order condition for the wage rate w.

2.   Plot the labour demand curve (Nd  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 Nd=8 units of labour? (Use your labour demand function!)

4.   What would the firm’s profit be in this case? (Use the profit function!)