Question

4. Consider the consumption-leisure choice model we discussed in class. Suppose

individual utility is represented by the function U(c, L) = min {c, 10L}, where c is

consumption and L is leisure. Individuals have a total h = 16 hours that could be

divided into work and leisure. Market wage rate is w = 10.

(a) Sketch the individual’s indifference curve.

(b) Find the optimal consumption and leisure choice.

(c) Now suppose wage increases to w = 12. Find the new optimal consumption

and leisure choice. What can you say about the income and substitution effect

of wage on labor supply?

Answer 1

a).

Consider the given problem here the utility function is given by, “U = min(C, 10L)”, => at the optimum “C=10*L”, => C/L = 10”. So, here “Consumption” and “Leisure” are consumes with a constant ratio. The following fig shows the IC.

b).

Here the budget line is given by.

=> C = W*(h-L), => C + W*L = W*h, where “W=10” and “h=16”. Now, the utility maximization problem is given by.

=> Maximize “U = min(C, 10*L)” subject to “C+WL = Wh”.

Now, at the optimum “C=10*L”, => C+WL = W*h, => 10*L+WL = W*h, => (10+W)L = W*h, => L = W*h/(10+W), be the optimum leisure.   

Now, C = 10*L = 10*[W*h/(10+W)], C= 10*W*h/(10+W)], be the optimum consumption.

So, the optimum consumption and leisure are given by, “L = W*h/(10+W) = 10*16/(10+10) =8” and “C = 10*W*h/(10+W)] = 10*10*16/(10+10)] = 80”.

c).

Now, let’s assume wage increases to “W=12”, the optimum choice of “consumption” and “leisure” are given by.

=> L = W*h/(10+W) = 12*16/(10+10) = 9.6 hours.

=> C = 10*L = 96.

So, consumption and leisure both increases as a result of increase in wage rate. Here goods are perfect complement, => both goods are consumed at the constant ratio, => there is not “substitution effect”, => the total effect of price change is “income effect”.