Dalhousie University - Econ-2200
* Note: The most relevant sections of the textbook are 5.1, 5.2 and 3.3 but the material builds on earlier content. 1) Suppose that Fernando has the utility function of U(x,y) = 9x]By25 but the two goods are lasagna units/wk (represented by x) and cake units/wk (represented by y). The marginal utility functions for x and y are respectively: MUx (x, y) = 3 (2) 2/3 [2] and MUy (x, y) = 6(2)1/3 [2] The marginal rate of substitution (MRSxy) (in terms of willingness to give up cake units to get additional lasagna units) was found by using the formula: MRSxy(x, y) = - MUx(x.7) _ 12 . Note: MUy(x.y) 2x We can think of this MRSxy as Fernando's marginal benefit (MB) of lasagna units measured in cake units. Also, recall that the MRSsy (x.y) is the negative of the slope of the indifference curve at (x,y) Recall that the marginal opportunity cost (MOCzy) of lasagna units in terms of cake units is the negative of the slope of the budget line (P./Py) where Px and Py are the respective prices of lasagna and cake units. This is the amount of cake units Fernando is able to trade to get an additional lasagna unit. a) Write down the general budget constraint equation treating Px, P, and M as constants that will act as "placeholders" for exogenous constants to be filled in later. b) Find the general tangency condition equation by setting the MOCxy equal to the MRS.y. Treat Px and Py and M (income) as constants (for which you will fill in exogenous values later). c) Use a) and b) to solve for the Marshallian demand functions for each of x and y. These will have the respective demands as a function of M, Px, and Py. d) Now, we will substitute exogenous values for the constant terms, Px. Py, and M. Suppose that initially, Px1 = $1/lasagna unit and Py = $1/cake unit and Fernando's weekly budget for this food is $400/wk. Find Fernando's budget equation. Plot this on a graph paper and label it as BLI such that your graph is properly drawn to scale.e) Find Fernando's utility maximizing demand for each of x and y (given his budget constraint). Call these XA* and yA*. f) Substitute XA* and ya* back into Fernando's utility function to find his utility and call this UA. g) As in Assignment 1, on the same graph as your budget line, plot the indifference curve corresponding to the utility calculated in f) and label as U1. Label the bundle XA* and yA* on your graph as point A and drop a dashed line to each axis to label xA* and yA" on the appropriate axis. h) Suppose that the price of cheese rises causing the price of lasagna to rise to P,2 = 4$/lasagna unit while the price of cake does not change. Repeat steps d) through g) for this new price, however, label your new budget line (BL2), and your new IC (Uc) and the new utility maximizing bundle as C. i) On your graph, plot a line that has the same slope as the final budget line but is tangent to the initial indifference curve (which you labeled, UA). Label the bundle at the point of tangency as B and drop a line to the x-axis as in the video, slides or textbook (ch. 5). You could label this compensated budget line as "BLE". j) Calculate xs* and ys* at point B (the "decomposition basket"). Hint: Two conditions hold at this point. I) you know the utility value, and, II) The tangency condition holds but remember that the slope of the compensated budget line is the negative of Px/Py (at the final prices after the price change). k) On your graph, label i) the entire change in the demand for lasagna due to the price change and decompose this into the ii) income and ini) substitution components. Calculate the value of the income and substitution effects and the change in demand in its entirety. 1) Plot the Marshallian demand on a graph with price in S/lasagne unit on the y-axis and lasagna units per week on the x-axis. Label the change in the total consumer surplus due to the price change. m) On the same graph as the immediately above question, label two points on the compensated demand equation and join with a smooth curve to provide a rough sketch of the compensated demand curve. Label either the compensating variation or the equivalent variation due to the price change (which-ever is relevant based on the compensation basket found above). n) Is this lasagna a normal or inferior good? Is it a "Giffen good"? Briefly explain in one sentence. o) Are the above solutions "interior" or "corner point solutions? Explain in one sentence
