I have attached 2 screenshots, these are related with each other. One of them is giving info, another one contains questions. Could you please explain a and b with clear step by step solution? Please, do not write only answer.Could you please explain and provide step by step solution i) and ii) ?

(0.) Consider the following optimization problem: mam U = U (1:, y) subject to pain +pyy = B where pmpy and B are constants. (For your reference, the above model is very common in both Micro and Macro. This is called a standard utility maximization problem. U is the consumer's payoff or utility which is a function of the quantities of two goods 3: and y, B represents the consumers's available income (or, budget constraint) and, Pa; and Py denote the prices of good :3 and :9, respectively.) (i) (5 marks) Write the Lagrangian and write the first order conditions (FOC) for the Lagrangian. (ii) (6 marks) Recall that Marginal rate of substitution (MRS) is defined as the ratio of marginal utilities (i.e., U. ). Show that the FOC in (i) implies that MRS should be the same as the price ratio of two goods (i.e., Px/Py)