There are N number of residents living and working in the economy. Everyone has the equal income I > 0 and shares the same utility function dened over private consumption 0' and housing unit H as follows: U(C, H) = CaHI-, where C? > 0 is an exogenous variable (which we will explore a little further) and 04 e (0, 1) is a parameter. The per-unit price of consumption is equal to 1 (numeraire) and per-unit price for housing is denoted by 7' > 0. 1. . Then, assume C?" = ( Set up a utility maximization problem and solve for the optimal bundle of private consumption and housing unit. . Derive expressions for the own-price elasticity and cross-price elasticity of demand for housing unit (1.e., sameymqmsmp). . What is the economic interpretation of parameter a in this setup? Provide a reasonable number for this parameter and justify. Now, let's assume that there is a income tax of r E (0, 1). This means, the after-tax income for each resident is (1 'r)I. The tax revenue is used to nance government projects (e.g., water quality control, environmental regulations, provision of parks, policing, roads, etc). What is the total revenue? (denote this variable as G) . Does an increase in 1' always lead to an increase in G? If so, provide thoughts on whether or not this feature of the model is compatible with the reality. If not, what are some conditions under which an increase in 7' leads to a decrease in G? . Repeat #1 with the income tax 1'. Express U (C, H ) as a function of rental rate 1', tax rate 1', and income I. G A . . . . ) , where A > 0 and X 2 O. Prov1de some intuitlon about these parameters. . Substituting the expression for 51' into your answer in #6. Under what condition, does utility increase in response to an increase in population (increasing returns to scale)? . Based on the expression you obtained in #8, what is the optimal tax rate? 10. Obviously, there are many simplifying assumptions made throughout the questions here. Identify one assumption that is explicitly or implicitly made in this question and provide your thoughts on the potential implications of relaxing that assumption