1 (19 marks): Sarah has a monthly income of $1,800 to spend on goods X and Y, where good Y is the composite good (therefore priced at $1 per "unit"). Good X is a type of food. The price of X is initially $5 per unit, then this price increases to $7 per unit. Sarah's MRS, measured at any point along her indifference curve, is given by the formula: Y/3X. In one diagram (for parts 1(a)-1(d)), show: 1(a) The budget constraints for Sarah, both before and after the price change for good X. Note: Remember to label the axes, and show on diagram, the numerical values where the budget constraint meets the vertical and the horizontal axes, and the slope value of the budget constraint, and show in answer how have derived them. 1(b) The optimal consumption bundles for Sarah, both before and after the price change for good X (include the indifference curves). 1(c) According to diagram in part 1(b), show the income and substitution effects arising, as a result of the change in price of good X. 1(d) (i) Based on your analysis and referring to diagram, note whether find that good X is a normal or an inferior good for Sarah. (ii) Explain clearly how can determine whether a good is a normal or an inferior good (e.g. when the price of good X increases), and justify answer in part 1(e)(i) referring to your analysis and diagram. 1(e) For the two price levels for X given in this question ($5 and $7), and based on analysis in parts 1(a) to 1(b), draw a new diagram of the demand curve for Sarah for good X, and derive the equation of the demand curve. Remember to label the axes (Note: In deriving the equation of the demand curve's slope (and for deriving the P axis intercept value) can use fractions, or decimal points up to 3 digits and round result for the Y axis intercept)). 1 (f) (i) Following from question 1(b), show the formula, and derive the value of the MRS of Y for X for Sarah at the optimum consumption bundle point, when the per unit price of X is $5 (at optimum point A) and when the price of X increases to $7 (at optimum Point B). (ii) Do agree with the statement that for normal goods: 'As the price of X increases, the new optimum consumption bundle's numerical value of the MRS of Y for X is expected to increase in absolute terms.'? Is answer YES or No?! Explain Why Note: In order to obtain the marks for (1(a)), 1(b), 1(c), 1(e) and 1(f) will need to provide your workings for all the relevant points that have shown in diagram; and the diagram can be stylised (it does not need to be plotted to scale)