Question

The numbers in this question may end up not making sense; just treat this as an exercise. There are n consumers, with positive wealth levels W₁, i = 1,..., n. All of them have identical utility functions over 2 goods, x and y, where x is rice, and y is the other good, given by

U(x, y) = u(x)+y
where u(x) = (8000 - (80-x)²)/2. They maximize utility given prices p for good x and 1 for good y.
(a) Obtain the expression for aggregate demand for rice. There are m farmers. Each of them has 1 hectare of land under rice cultivation and an identical production function:
f(K, L) = AKLẞ
where a and ẞ are positive and a + B < 1.

Answer 1

The aggregate demand for rice is derived from the utility maximization of consumers, considering their budget constraints, and the aggregate supply is determined by profit-maximizing farmers with a given production function. The equilibrium is reached when aggregate demand equals aggregate supply, providing insights into the market dynamics for rice.

To obtain the aggregate demand for rice, we need to consider the optimization problem of the consumers. The utility maximization problem for a consumer i can be stated as:

`max_{x_(i),y_(i) } u(x_(i))+ y_(i)` subject to
`px_(i) + y_(i) = W_(i)`

where
`W_(i)` is the wealth of consumer i and p is the price of good x. The utility function
`u(x)` is given by
`(1)/(2)(8000-(80-x)^(2) )`

Solving this consumer optimization problem, we find the demand function for good
`x(x_(i))` for each consumer. The aggregate demand for rice (X) is the sum of individual demands:

X=∑n i=1 xi

​Now, let's consider the farmers with the production function

`f(K,L) = AKL^(\beta )`

where

A and β are positive parameters, and K is the capital and L is the labor. The farmers aim to maximize their profit, which is the difference between revenue and cost. Given the one-hectare land constraint, the profit-maximization problem is:

`max_(KLP). f(K,L) - wL - rK`

where w is the wage rate and r is the rental rate of capital.

The resulting optimal inputs
`K^(*)` and
`L^(*)` give us the aggregate supply of rice.