1. Optimal Subsidies. Consider a consumer who consumes two goods, X and Y with baseline prices px and py. Suppose government offers the consumer cash subsidies sx > 0 and sy > 0 so that the prices of the two goods become px = px - sx and Py = py – sy. Denote by I the consumer's income. Let Ê and û denote the consumer's optimal con- sumption under environment (@x, Ôy, I). We wish to show that the government can make the consumer weakly better off without extra costs by sending direct transfers to the consumer. a) Express the total cost of government subsidies S as a function of sx, Sy, Ê, Ê. Consider the environment in which prices are px, py, and the consumer receives income I + S. b) Show the consumer is weakly better off under this new environment than in the original one.