Question

1. The utility function for an individual is given by the equation: 𝑈 = 𝑋1 0.25𝑋2 0.75 and their budget constraint is 𝑃1𝑋1 + 𝑃2𝑋2 = 𝑀 (i) Using the Lagrange, and showing all work, Derive the demand functions for good 𝑋1 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑋2 (ii) Given that the price for good 1, p1= k2.5 and the price for good 2, p2=k5, and the consumer’s income is k100. Calculate and graphically present the optimal choice for good 1 and good 2. (iii) Calculate the optimal level of utility for the consumer derived from the above optimal bundle. (iv) Show (by calculating) that at a consumer’s optimal point, the slope of the indifference curve equals the slope of the budget line. (v) Suppose government needs to raise revenue through imposing a quantity tax of 0.5 kwacha per unit of good

Answer 1

$$U=x_1^{0.25}x_2^{0.75}$$

$$MU_1=0.25(\frac{x_2}{x_1})^{0.75}$$

$$MU_2=0.75(\frac{x_1}{x_2})^{0.25}$$

$$\frac{MU_1}{p_1}=\frac{MU_2}{p_2}$$

$$x_2=1.5x_1$$

$$2.5x_1+5x_2=100$$

$$x_1=10$$

$$x_2=15$$