Question

Consider a scenario with n consumers, each possessing positive wealth levels Wi (i = 1, ..., n). All consumers share identical utility functions over two goods, x and y, where x represents rice and y is another commodity. The utility function is given by U(x, y) = u(x) + y, with u(x) = (8000 - (80 - x)²)/2. They maximize utility given prices p for good x and 1 for good y.

Derive the expression for the aggregate demand for rice.

Answer 1

The aggregate demand for rice can be derived using the marginal utility per price ratio. For the utility function U(x, y) = u(x) + y, with u(x) = (8000 - (80 - x)²)/2, the aggregate demand for rice is a straight line with slope -n and y-intercept 80n, derived from the sum of the individual demands 80 - p for each consumer.

Step-by-step explanation:

To derive the aggregate demand for rice given the utility function U(x, y) = u(x) + y, and the marginal utility of rice u'(x), consumers will equate the marginal utility per price of rice with that of the other commodity y. Consumer's utility-maximizing condition will be:

u'(x)/p = 1

Since u(x) = (8000 - (80 - x)²)/2, then u'(x) = 80 - x. Plugging into the utility-maximizing condition gives:

80 - x = p

Now, solving for x (the quantity of rice) in terms of price p:

x = 80 - p

If we have 'n' consumers each with different wealth levels Wi, the aggregate demand for rice is the sum of individual demands. Denoting Xi as the quantity of rice demanded by the ith consumer:

Aggregate Demand for Rice = ∑ Xi for i = 1 to n

Since Xi = 80 - p, we have:

Aggregate Demand for Rice = ∑ (80 - p) for i = 1 to n

Which simplifies to:

Aggregate Demand for Rice = n(80 - p)

Therefore, the aggregate demand is a straight line with intercept at 80n and slope of -n with respect to price p.