Question

7. Marvin is deciding between consuming his income today (period 1) and a year from now (period 2). His utility function is u(c1, c2) = 10√c1 + c2. His income today, m1, is 100. His income in a year, m2, is also 100. The interest rate is 10% and there is no inflation. How much will he consume today?

Answer 1

Answer: Marvin's budget constraint will be

=> c1 + c2/(1+r) = m1 + m2/(1+r)

Where r is the interest rate. This equation is interpreted as consumption today plus consumption next period (discounted in the present) must be equal to total income today and next period (discounted in current period).

Since m1 and m2 = 100, and rate of interest = 10% or 0.1 the equation becomes:

=> c1 + c2/(1+0.1) = 100 + 100/(1+0.1)

=> c1 + c2/(1.1) = 100 + 100/(1.1)

Now utility function is given as: u(c1, c2) = 10*c1^1/2 + c2

In equilibrium, the ratio of marginal utilities must be equal to the ratio of price. The 'price' in this case is p1 = 1 in period 1 and p2 = 1/(1+r) in period 2. Thus ratio of price =

=> p1/p2 = 1 / (1/(1+r))

=> p1/p2 = (1 + r)

=> p1/p2 = 1.1 as r = 0.1.

Marginal utility is the additional utility a consumer receives by consumption one more unit of a good.To calculate marginal utility of a good the utility function is differentiated with respect to that good. And other variable present in the equation is treated as a constant. Hence Marginal utility of c1 = MU1 =

MU1 = d(u(c1, c2))/dc1

=> MU1 = d(10*c1^1/2 + c2) / dc1

=>MU1 = 10*1/2*c1^1-1/2

=>MU1 = 5/c1^1/2 This is the marginal utility form c1.

Similarly, marginal utlity of c2 = MU2 =

=> MU2 = d(u(c1, c2))/dc2

=>MU2 = d(10*c1^1/2 + c2)/dc2

=>MU2 = 1    (as d(10*c1^1/2 )/dc2 = 0 as c1 here will be a constant. Hence only d(c2)/dc2 will remain which equals 1)

Thus ratio of marginal utilities = price =

=> MU1/MU2 = p1/p2

=>5/c1^1/2 = 1.1

=> c1^1/2 = 5/1.1

=>squaring both sides:

=>c1 = (5/1.1)² = 25/1.21

=> c1 = 20.66 approx

Hence Marvin will consume 20.66 units today. The income will have no effect on his consumption. This is a consequence of having quasi-linear function, where one one term is expressed in functional form (in this case c1 in the utility function) while the other term is expressed in linear form (in this case c2 in the utility function).

Hence c1 = 20.66 is the answer.