Terrys utility function over leisure ( L ) and other goods ( Y ) is U ( L , Y )

ID: 1094415 • Letter: T

Question

Terrys utility function over leisure (L) and other

goods (Y) is U(L, Y) = Y + LY. The associated marginal

utilities are MUY = 1 + L and MUL = Y. He purchases

other goods at a price of $1, out of the income he

earns from working. Show that, no matter what Terrys

wage rate, the optimal number of hours of leisure that he

consumes is always the same. What is the number of

hours he would like to have for leisure? What must be true about terry's substitution and income effects for leisure when the wage changes

The given utility function is: U(L,Y) = Y+ LY

marginal utility of good Y is: MUY = 1+L

marginal utility of lesiure is: MUL = Y

The price of good Y is : Py= $1

If the wage rate of Teery is w, then the income earned by him is (24 –L)w.

Since Py =1, the number of units of other goods he purchases isY= (24 –L)w

Now, the optimal condition implies that: MRS( L,Y) = PL/PY

Substituting the values in the optimal conditon, we have: Y/(1+L) = w/1

Thus, both the conditions imples that: Y= w(1+L)

Since, Y= (24 –L)w, thus (24 –L)w= w(1+L)

Solving the above equation, we get L = 11.5.

Hence, the above calculation implies that the optimal leisure doesn't depend on wage rate.

Since, the wage rate does not affect the amount of leisure, thus tere will be no substitution and income effect on leisure due to change in wage.

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