Question

Consider a consumer who derives utility from two goods: consumption (Good C) and leisure (Good H, in hours). The consumer has a total of L hours available. The consumer's income comes from time spent at work, which pays a wage of w per hour. Assume the three activities are mutually exclusive: While at work, the consumer cannot spend time on leisure or consumption. (a) What is the consumer's budget constraint? (b) Assuming the consumer's utility function is U(c,h)=a*ln(c)+(1-a)ln(h), derive the demand functions for consumption and for leisure. Assuming the price of consumption is not zero, how does a change in the wage affect consumption.

Answer 1

Two goods: Consumption (Good C) and Leisure (Good H, in hours)

Total hours available: 'L' hours

Wage = W/hour

Utility function: U(c,h)=a*ln(c)+(1-a)ln(h)

Let the price of good C be 'P' and the quantity consumed be 'c'. and let 'h' be the hours spent on leisure.

Then, we can write budget constraint as:

P*c = w*(L-h) .....(i)

Now subject to the above budget constraint, we have to maximize utility.

dU/dc = a/c ; dU/dh = (1-a)/h

so, by equating (dU/dc)/P = (dU/dh)/W

we will get c = hwa/(P-Pa)..........(ii)

Now after substituting the value of c from (ii) into (i), we will get the demand function for leisure:

h = L(1-a) --> demand function for leisure.

Now substitute the above value of h into (i) to get the demand function for consumption:

c = WLa/P -----> demand function for consumption

From the above equation we can write dc/dW = change in consumption due to change in wage = La/P.