Question

4. Suppose that the consumer’s preferences are given by

   U(c,l)=2c ^(1/2) +2l ^(1/2)

   where c is the level of consumption and l is leisure. The consumer has to allocate 50 hours between leisure and labour. The real wage rate is 10 per hour and the real non-wage income is 160. Assume that there is no

   government. Note that (∂c ^(1/2)) / (∂c) = (1/2) c^(-1/2)

   1. (a) Write the budget constraint of the household. (b) Solve for the tangency condition using the marginal rate of substitution MRS l,c, and the wage

   2. (c) Solve for optimal consumption and leisure.

*Would highly appreciate graphs*

Answer 1

Solution:

a) Whatever wage is earned, plus the non-labor income is allocated between leisure and consumption. The price for 1 hour of leisure is the amount of wage given up had he worked for that 1 hour, thus price of leisure is the wage rate. As we are not given any price for consumption good, we normalize it to 1. Lastly, with total available hours of 50, labour hours + leisure hours = 50

So, labour hours = 50 - leisure hours, and wage income earned is thus, wage rate*labour hours = wage rate*(50 - leisure hours). So, combining all this, we can write the budget constraint as:

Pc*c + w*l <= M + w*(50 - l) ;

where Pc is price of consumption good (which we have normalized to 1), c is amount of consumption good, w is wage rate, l is leisure hours, M is non-wage income

So, 1*c + 10*l <= 160 + 10*(50 - l) is the required budget constraint

On simplifying it further, we get: c + 10*l <= 160 + 500 - 10*l

c + 20*l <= 660

b) We are given the utility function for consumption and leisure as: U(c, l) = 2*c1/2 + 2*l1/2

So, we can find the marginal rate of substitution (that is the slope of this indifference curve) as marginal utility of leisure/marginal utility of consumption good

MUl = dU(c, l)/dl = 2*(1/2)*l-1/2 = l-1/2

MUc = dU(c, l)/dc = 2*(1/2)*c-1/2 = c-1/2

Then, MRS = l-1/2/c-1/2 = (c/l)1/2 (notice the change in sign of powers, thus reversal of the denominator and numerator)

Tangency condition says that slope of indifference curve must equal the slope of budget line (or budget line must be tangent to the indifference curve) at the optimal allocation point. Slope of the budget line = price of leisure/price of consumption good = w/Pc

So, with MRS = w/Pc

(c/l)1/2 = w/1

or (c/l)1/2 = w

c) Then, continuing the above optimality/tangency condition, we can solve further as follows:

c1/2 = 10*l1/2

On squaring on both sides, we get: c = 100*l

Then, substituting this condition in the budget line: c + 20*l = 660

100*l + 20*l = 660

l = 660/(100 + 20) = 5.5

And consumption, c = 100*5.5 = 550

So, optimal level of consumption is 550 units, and optimal hours of leisure is 5.5 hours (and so, correspondingly, optimal hours of labor = 50 - 5.5 = 44.5 hours).