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05 Question (5 points) See page 100 One of the most common functional forms used in economics is Cobb-Douglas. A Cobb-Douglas utility function takes the formu(I, y) = zºy , where 0 <a < 1. Note that we have imposed the condition that the exponents sum to 1. A monotonic transformation of the function could change that characteristic and still represent the same preferences. However, you will see below that having the exponents sum to 1 leads to a clear interpretation of the demand functions, so it is worth transforming the utility so that the exponents sum to 1. This problem will take you through the steps to finding the general-form demand functions for a Cobb-Douglas utility. These functions have some special properties and a very clear interpretation 1st attempt Part 1 (1 point) See Hint Given u(2,y) = "y-a, find the marginal rate of substitution. MRS = Choose one: A. az-ly-a B. - ay (1-2) ochy O D. (1-a)z"ya (1-axy Part 2 (1 point) See Hint To find demand, we use the condition that the MRS will be equal to the price ratio at an interior optimal bundle. Set the MRS from Part 1 equal to and then solve the resulting equation for y. y= Choose one: apy (1-a)p3 B. (1-0), PE (1-2) (1-ap- Part 3 (2 points) See Hint Now substitute the y equation from Part 2 into the budget constraint to find the demand, 3(Pa, Py, m). Choose one: A. (1-2) (1-a)m Once you have found (PxPy, m), substitute it into the y equation to find demand for y(P. Py, m). Choose one: A. (1-a)m B (1-2) Part 4 (1 point) See Hint Based on the demand functions for I and y, what is the economic interpretation of a (and, by extension, (1 - a)? Choose one: A. a is the price elasticity of demand for I, and (1 - a) is the price elasticity of demand for y. B. a is the income elasticity of demand for I, and (1 - a) is the income elasticity of demand for y. O C. a is the share of income the consumer will spend on I, and (1 - a) is the share of income the consumer will spend on y. O D.a is the share of utility the consumer will receive from I, and (1 - a) is the share of utility the consumer will receive from y

Answer 1

462, 42= xay l-x 1) 20 Ох ao = xich-lyld a mun δυ (1-x) xx y 1-X-7 = muy oy MRS = more muy xxx-lyl-x (1-x) xx y1d-1 = - x x get (12) x yt yl (B) is correct option. - (27 MRS = Pre a (17 is correct option.

- (3) Prezet Pyy = m i Pekt Bo | 1-8) P2 x = m. a ZTP & PR 2 (1-x) Pre-2 = m PRK [&t 127 - xm. Р.ҡ = am xCP2, py, my = am P2 (D) is correct option. YIPs, Py, m) = (1-x) m (A) is correct option.

(4) Share of income spent on a = Prein m = Beam PK m share of income spent ony = Pyy m. = Py (2-x) m m (1-2).m ma = 11-0) (22 is correct option.