"LUCAS has fixed money income, I which spent two goods X and Y. The prices of X and Y are fixed. Lucas,s Utility is based on following utility function. U(x,y)= min(4X,16Y). His income share for X is SX where Sx = PxX/I
and his income share for Y is Sy, where Sy = PyY/I
a: derive his demand function for X and Y.
b; using your answers from a, derive the own-price elasticity of demand, cross-price elasticity of demand and the income elasticity of demand for X and Y.
Thanks your help is appreciated."
This looks like a problem from a textbook. My preference is to not provide answers to those but only some general guidelines to help you think it through. Here I will content myself with part a of the question question. Part b asks you to do some grinding based your answer to part a.
The question is asking about choice for a particular class of preferences called "perfect complements" or fixed proportion preferences or Leontief preferences, after the economist Wassily Leontief. It turns out that the demands generated by these preferences have no substitution effect. The have only an income effect.
This first graph gives an idea of what the indifference curves look like when the proportions are 1:1.
The indifference curve has a right angle at the 45 degree line. Above and to the left of the 45 degree line Good 2 is redundant and Good 1 is scarce. Then utility is determined by the amount of Good 1. Below and to the right of the line Good 1 is redundant and Good 2 is scarce. Then utility is determined by the amount of Good 2. In the problem posed the proportions are not. 1:1. It looks like they are either 4:1 or 1:4. Figuring out which is something you'll need to determine.
The second graph should give you and idea about how to solve for the demands.
Since it is always optimal to consume the goods such that neither is redundant, the choice will always end up on the dashed line, ergo the fixed proportions. The choice will also be on the budget line. That gives two linear equations that must be solved to get the demands.