Question

3. Consider two consumers, A and B, who both maximize the same logarithmic utility function, U = In 1 + In C2, and face the same real interest rate, r=.04: Consumer Y1-TI Y2-T2 A 20,000 10,400 10000 20, 800 What are the levels of C1, C2 and saving in period 1 for each? Suppose both face a borrowing constraint that limits borrowing to no more than 1000. Now describe the consumption choices for each. Explain. With the constraint in place, describe how each responds to a tax cut of 1000 in period 1.

Answer 1

J= .04 O= In G + In C2 Consumer Yi-Ti A 20,000 B 10,000 - T2 10,400 20,800 (a) Optimal choice MRS = MRE Joldal - (ltr) Jolder 25 - 17o.ch It 0.04 -> 1.04 - ca- 100uc, For A 1 - Budget constraint in Terms of Future value Cz + ci(+2) = (1, -T)+(12-T2 ) (172) - 1004 C1 + 1.046 = 20,000 + 10400 (1 +0.04) = 2.08 = 20,000 + 10816 (1) => a 308 16 = 148 15:38 2 C2 = 1.04 (14815. 38) = 15407.99 = 15408.

For B (6) ( ) 2.08 c = (0,000 + 2oºoo (1.0) C = 15207069 C2 = 1004 cl - C2 = 1581599 = C2 = 15816 Savings For A - a = 14815. 38 YI-T = 20,000 - S=5184.62 Saving of B. O a = 15207.69 Yi-Ti = 10,000 =) SI = 10,000 - 15207.69 From (ii) - - 5207.69 2.08 C1 = 11,000+ - 20800 (1.4) 1 Borrowing onstraint Ci= 15688. Since in Period-1 only Consumer one is consuming more than his Avcillable income, So it it can Borrow only, 1000 Then Gi= Hoog 15688

and Consumption of consumer A will remain unchanged at C = 14815538 A Tax cut of a $1000 in Period - I For Consumer - A Now Y,- Ti = 21,000 from (i) 1. So, " 2.0801 - 2.08c, = 21000 + 10816 - 15296.1538 Now G = 15296. 1538 and si = (y1 - TJ-CI = 21,000 - 15296-1538 = 5703.8462 For consumer-B Now Y,- T = 10,000 + 1,000+ 1,000 (Browwing, From - cii, 2.08 C1 = 12,000 t 20800 (1.04) = 12,000 + 21632 C = 16,169 Si= 11,000 - 16 169 = -5169