In: Economics
1. Anqi pays 45 dollars per unit of x and 30 dollars per unit of y. Her preferences are u(x, y) = x1/2y1/2. Anqi's income is 1800 dollars. How many units of good x should she purchase?
2 .A consumer has preferences u(x, y) = x1/2 + y1/2. She usually consumes 4 units of x and 4 units of y but the store has sold out of good y. How many units of good x must she consume to keep her utility unchanged?
1) let me denote the price of x as Px and price of y as Py and Ani's income an M.
Problem of Ani is Maximize U(x,y)= X1/2y1/2
Subject to xPx. + y.Py = M
The lagrange expression of the above maximization problem is
L= x1/2y1/2 + (M - x.Px - y.Py) where is the lagrange multiplier
The first order conditions are as follows:
= 1/2 x-1/2y1/2 - Px = 0 1/2 x-1/2y1/2 = Px.......(1)
= 1/2 x1/2y-1/2 - Py = 0 1/2 x1/2y-1/2 = Py........(2)
= M - xPx - yPy= 0............(3)
Dividing (1) by (2) we get , y/x= Px/Py y = xPx/Py....(4) putting this in eqn (3) we get,
M - xPx - xPxPy/Py = 0 2xPx= M x = M/2Px Putting the value of M and Px , x = 1800/(2 x 45) = 20
Ani should purchase 20 units of x.
2) The utility function of the consumer is U = x1/2 + y1/2. She usually consumes 4 units of x and 4 units of y. So from this combination the utility derived by the consumer is U= 41/2 + 41/2 = 2+2 = 4 utils.
When the store has sold out of y, then utility function will be U = x1/2. To derive 4 utils of utility units of x to consume is 4=x1/2 x= (4)2 = 16.
the consumer must consume 16 units of x to keep the utility unchanged