Suppose that Erin spends her income on two goods, food (X) and clothing (Y), and that her utility function is given by U(X,Y)=.20X2Y2 (the marginal utility for X is .40XY2 and that for Y is .40X2Y). The prices of food and clothing are $10 and $20 for each unit, respectively. Her income is $500 per week.
1. Find the level of her utility when she purchases 10 units of food and 20 units of clothing every week.
2. In (a), find her marginal rate of substitution.
3. Find her choice for X and Y when she maximizes her utility given her budget constraint.
Erin spends money on two products: food (X) and clothing (Y). The price of one unit of food is $10 and the price of one unit of clothing is $20.
Her utility function is given by U(X, Y) = 0.2*X^2*Y^2
When she purchases 10 units of food and 20 units of clothing, her level of utility is equal to 0.2*10^2*20^2 = 8000.
The utility curve U(X, Y) = 0.2*X^2*Y^2. The marginal utility of X is given by MUX = 0.4*X*Y^2 and the marginal utility of Y is MUY = 0.4*X^2*Y. Using the two, the marginal rate of substitution is equal to MRS = Y/X
As one unit of food costs $10 while one unit of clothing costs $20, to maximize utility let her buy x and y units of the two.
10x + 20y = 500, this gives:
x = (500 - 20y)/10 = 50 - 2y
U = 0.4*X^2*Y^2
Substituting x = 50 - 2y and y gives:
U = 0.4*(50-2y)^2*y^2 = (8*y^4-400*y^3+5000*y^2)/5
To maximize this, take the derivative with respect to y, equate that to 0 and solve for y.
(32*y^3-1200*y^2+10000*y)/5 = 0
2*y^2-75*y+625 = 0
y = 25 and y = 25/2
The second derivative of U with respect to y is negative for y = 12.5. This is the point where she maximizes utility.
Erin should buy 12.5 units of clothing and 25 units of food.