Consider the following utility function of 3 goods, x, y and z: U(x,y,z)=ax+by+cz; x,y,z?0 and a, b, c are constants. The prices of good x and y is denoted by pX and pY respectively. The income is denoted by m. Good z is provided by the government free of cost but the quantity of good z provided by the government depends on the consumption of good x and y chosen by the consumer. For example, if in equilibrium, the consumer chooses to consume 10 units of good x and 20 units of good y, then government will provide max{10,20}=20 units of good z

So z*=max{x*, y*}, where z*, x*, y* denote the consumption of each good that maximizes utility of the consumer. a) Find the consumption bundle that maximizes utility for the consumer. [6 points] b) Suppose now, the government changed it’s rule for provision of free good z, now z*=min{x*, y*}. Will that change your answer to the previous part? If yes, then find out consumption bundle that maximizes utility for the consumer. If no, just write “consumption in equilibrium remains the same”.