Question

* Note: The most relevant sections of the textbook are 3.1 and 3.2 but the material builds on earlier content 1) Suppose that Nadeem has the same utility function as Lisa did in assignment 1 of U(x,y) - x""ybut the two goods are chickpea curry units/wk (represented by x) and rice units/wk (represented by y). As before, his marginal utility functions for x and y are respectively: MU_(x,y)==0)** [2] and MU,(x, y) =*)*** [2] In assignment 1, the marginal rate of substitution (MRSxy) (in terms of willingness to give up rice to get additional curries) was found by using the formula: MRSxy(x, y) = MUy(x,y) 3x Note: We can think of this MRSxy as Nadeem's marginal benefit (MB) of curry units measured in rice units. Also, recall that the MRSxy (x,y) is the negative of the slope of the indifference curve at (x,y) Recall that the marginal opportunity cost (MOCxy) of currie units in terms of rice units is the negative of the slope of the budget line (P/Py) where P, and Py are the respective prices of currie and rice units. This is the amount of rice Nadeem is able to trade to get an addition unit of curry. a) Find the general tangency condition equation by setting the MOC y equal to the MRSxy. Treat Ps and Py as constants (for which you will fill in exogenous values later). b) Write down the general budget constraint equation treating Px. Py and M as constants. c) Given that you have two unknowns variables [the demand for currie (x) and the demand for rice (y)] and two linear equations from a) and b), solve for the Marshallian demand function for each of x and y. d) Now we will substitute exogenous values for the constant terms, Px. Py, and M. Suppose that initially, Px1 = $1/curry unit and Py = $1/curry unit and Nadeem's weekly budget for this food is $200/wk. Find Nadeem's budget equation. Plot this on a graph paper and label it as BL¡ such that your graph is properly drawn to scale. e) Find Nadeem's utility maximizing demand for each of x and y (given his budget constraint). Call these xi* and y*. f) Substitute xl* and y* back into Nadeem's utility function to find his utility and call this U. g) As in Assignment 1, on the same graph as your budget line, plot the indifference curve corresponding to the utility calculated in f) and label as Ui. Label the bundle xi* and y* on your graph as point A and drop a dashed line to each axis to label xi* and y on the appropriate axis. h) Now suppose that due to a locust swarm, that damages many chick pea crops, the price of curry rises to P2 = 3$/curry unit while the price of rice does not change. Repeat steps d) through g) for this new price, however, label your new budget line (BL2), and your new IC (U2) 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 old indifference curve (which you labeled, U1). Label the point of tangency as B and drop a line to the x-axis as in class. You could label this BLB. i) Calculate x and y at point B. The book calls this the "decomposition basket" (see pg. 170 - 171). Hint: Two conditions hold at this point. I) you know what utility is and you know the utility equation, and, II) The tangency condition holds but remember that the slope of the compensated budget line is the negative of P/P, (at the new prices) k) On your graph, label the entire change in the demand for curry due to the price change and decompose this into the income and substitution components. Calculate the values of the change in demand in its entirety) and that which is due to each of the income and substitutione 1) Is curry normal or inferior? Is it a "Giffen good”? Briefly explain in one sentence. m) Are the above solutions "interior" or "corner solutions? Explain in one sentence. Give an example of a utility function that generates comer solutions. n) Make a graphical sketch of the Marshellian Demand. You only need to include 2 points. Does it slope downwards or upwards? Plot the compensated demand on the same graph (2 points only)

Answer 1

a) MOCxy = - Px Py (The bodas (The budget line is xp. + ypy - M). MRSxy - 1 The general targency condition giver > 20 3 ye px 6) The budget constraint equation is xPx + ypy = M. or (1ypy olx + y Py = M given by (from part (1)) part (a)) УР, 2 м c) The Marshalllan demand for rice is y = 3 M Chrom past part ) (from 0 4 y la - Also x = 1 x 2 m PL d) Nadeem's budget constraint is given by x + y = 200

Rice write/wk 200 150 100 M 50 BLL 200 Curry writs /wk U (20 e Given the budget constraint, xty - - 200 x* = 4 (200) >>(* = 50 4 1

Rice write/wk 200 150 100 M 50 BLL 200 Curry writs /wk U (20 e Given the budget constraint, xty - - 200 x* = 4 (200) >>(* = 50 4 1