Question

Given that U=f(q1, q2) and p1=2$, p2= 5$ per kg. If the budget constraint of a consumer is 100$, find the maximum quantity of goods q1 and q2 which the consumer can consume to maximise utility

Answer 1

The budget constraint of the consumer can be written as follows:

$$B=p_1q_1+p_2q_2,$$ $$100=2q_1+5q_2.$$

Let's find the Marginal Rate of Substitution (MRS):

$$MRS=\dfrac{p_1}{p_2}=\dfrac{2}{5}=0.4$$

Let's consider a special case of the "Cobb-Douglas" utility function, which has the form:

$$U(q_1,q_2)=q_1^aq_2^b,$$

where $a$ and $b$ are two constants. In this case the marginal rate of substitution (MRS) for the Cobb-Douglas utility function is:

$$MRS=(\dfrac{a}{b})(\dfrac{q_2}{q_1})$$

regardless of the values of $a$ and $b$.

Therefore, we can write:

$$MRS=\dfrac{p_1}{p_2}=0.4=\dfrac{q_2}{q_1},$$ $$q_2=0.4q_1.$$

Therefore, the budget constraint becomes:

$$100=2q_1+5\cdot0.4q_1=4q_1,$$ $$q_1=\dfrac{100}{4}=25.$$

Finally, we can find $q_2$:

$$q_2=0.4q_1=0.4\cdot25=10.$$

Answer:

$q_1=25, q_2=10.$