4. Consider the following utility function of a consumer: u(x₁, x₂) = a ln x₁ + (1 - a) ln x₂ where, x₁ and x₂ are units of Good 1 and Good 2 consumer consumes. In the above utility function, substitute a = number of letters in your first name For example, a == for SONALI. Also, assume that the price of good 1 is p₁ and price of good 2 is p₂; and income of the consumer is m. (a) Write an expression for marginal utility for good 1, marginal utility of good 2, and the marginal rate of substitution. (b) Are the preferences of this consumer monotonic? Check and explain. (c) Solve for the consumer's optimal consumption bundle (your answers should be given in terms of P1₁, P2 and m). Explain all your steps providing all the details relevant for your answers. (d) What fraction of this consumer's income is spent on good 1? How does this fraction change when P₁ increases? (e) Draw the Engel curve for good 1. Label the slope.

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