610: ECON Microeconomic Theory II
Prof. Blume
Problem Set 1
1
Problem Set 1
This is a solution key for problem set 1. Some intermediate steps are left to the reader. If you still have questions regarding any of the problems, please visit Prof. Blume or one of the TAs during office hours. Thanks, Liyan and Max
Problem 1: In the coconut economy, Rob can produce at most al coconuts with l hours of work. His utility function for coconuts c and leisure time t is u(c, t) = b ln(c) + t/12. He has 24 hours in a day to allocate between work and leisure. Suppose a = 6 and b = 1. (a) Find the Pareto optimal allocation. (b) Write this problem as a Kuhn-Tucker problem (if you did not do so in part (a) and find the multiplier for the production constraint. (c) How does the optimal allocation change with respect to the parameters a and b? (d) What happens when b > 4? Solution: (a) The Pareto problem is to choose an allocation of leisure, work and coconuts to maximise Rob's utility given the time and production constraints, as well as nonnegativity constraints on each of these variables. That is:
(c,l,t)0
max u(c, t) = b ln c + t/12 s.t. c al t + l 24
You can show that the solutions to this problem are fully characterised by the following KKT conditions (where is the multiplier on the production constraint): b -=0 c 1 a - 0 12 2 a(24 - t) t 1 a - 12 2 a(24 - t) =0 (1) (2) (3) (4)
a(24 - t) - c = 0
ECON 610: Microeconomic Theory II
Prof. Blume
Problem Set 1
2
The solution to this set of equations is: 24 - 6b if b 4 0 if b > 4 6ab if b 4 c = 24a if b > 4 b if b 4 6a = b if b > 4 24a t =
So, for (a, b) = (6, 1) the Pareto optimal allocation is (c, t, l) = (6, 18, 6). (b) For (a, b) = (6, 1) the multiplier on the production constraint is 1/6. (c) For b 4 the optimal allocation of time between leisure and labour depends only on the utility parameter b, and is linearly decreasing in this parameter. The optimal allocation of coconuts depends in a positive way on both a and b, but is concave in both. (d) When b > 4 all time is allocated to production and the maximum number of coconuts, given the production and time constraints, is consumed. No leisure is consumed when b > 4.
Problem 2: Find the competitive equilibrium in the coconut economy, taking coconuts to be the numeraire good. Solution: The PMP is: max
l
al - wl
Solving the FOC for this problem we find that a ld = and 4w2 The UMP is: max u(c, t)
(c,t)0
=
a 4w
s.t. c + wt 24w + 0 t 24
ECON 610: Microeconomic Theory II
Prof. Blume
Problem Set 1
3
It should be clear that c = 0 cannot be part of a solution and that t = 0 is possible at a solution to this problem, but how do we rule out t = 24? Solutions to the UMP are: l = =
s
12b - 24 12b - 24
w
if 24 + if 24 +
w w
12b < 12b
a 4w2 a 4w2
a 4w2
if 24 + if 24 +
12b < 12b
So, at an interior solution the equilibrium wage rate is obtained from the following market clearing condition: ls = ld 12b - a a = w = 2 2 4w 4w a 24b
Given this wage rate, labour demand is interior if 24 + a 12b b 4 4w2
Likewise, we determine that the equilibrium wage at a corner solution is w = (1/4)(a/6)0.5 and that this occurs for b > 4. Hence, the equilibrium wage rate is w (a, b) =
1 2 1 2 a 6b a 6
if b 4 if b > 4
For (a, b) = (6, 1) the equilibrium allocation is (c, t, l) = (6, 18, 6).
Problem 3: Let's think about taxes. (a) Suppose firms profits are taxed at rate . Suppose the tax revenues are handed back to Rob as a lump sum. How does the equilibrium vary with for all 0 < 1. (b) Suppose that a sales tax of t per unit is imposed on the sale of coconuts. The tax revenues are handed back to Robinson as a lump sum. How does the equilibrium vary with t the in neighborhood of t = 0? (c) In each case, what happens to social welfare, which in this case is Rob's utility? Solution: In both cases the PMP of the firm is unaltered, so we can concentrate attention on how the taxes affect Rob's behaviour as a consumer.
ECON 610: Microeconomic Theory II (a) The UMP with a profit tax is: max u(c, t)
(c,t)0
Prof. Blume
Problem Set 1
4
s.t. c + wt 24w + (1 - ) + T 0t You can see that in the UMP (1 - ) + T is a constant (i.e. contains no choice variables). Hence, simply replace the in the solutions to the previous problem with this constant, to get the following solutions: ls = 12b - 24
(1- )+T w
if 24 + if 24 +
(1- )+T w (1- )+T w
12b < 12b
The fiscal condition that T = can then be used to show that the joint solution to the UMP and fiscal constraint is: l = =
s
12b - 24 12b - 24
w
if 24 + if 24 +
w w
12b < 12b
a 4w2 a 4w2
a 4w2
if 24 + if 24 +
12b < 12b
Hence, equilibrium is not affected by the profit tax. (b) The UMP with a sales tax is: max u(c, t)
(c,t)0
s.t. (1 + )c + wt 24w + + T 0t Solutions to this UMP are: ls = =
12b 1+
- -
w
24
12b 1+ a 4w2
if 24 + if 24 +
w w
<
a 4w2 a 4w2
12b 1+ 12b 1+
24
if 24 + if 24 +
<
12b 1+ 12b 1+
Hence, the market clearing condition for the labour market (ls = ld ) implies an equilibrium wage rate of 1 (1 + )a w ( ) = 2 6b at an interior solution to the UMP, and w ( ) = 1 2
ECON 610: Microeconomic Theory II
Prof. Blume
Problem Set 1
5
at a corner solution to the UMP. Given these wage rates, the solution to the UMP is interior if b 4(1 + ), and at a corner otherwise. Hence, for (a, b) = (6, 1), the equilibrium allocations of coconuts, leisure and labour are: c = 6 1+ 6 t = (4 + 3) 1+ 6 l = 1+
That is, the equilibrium consumption of coconuts is decreasing in , while the consumption of leisure is increasing in . (c) Social welfare without a tax, or with the profit tax is u(6, 18) = ln 6 + (1/12)18 = ln 6 + 2 - With the sales tax 0, social welfare is u 6 6 , 24 - 1+ 1+ = ln 6 - ln(1 + ) + 2 - ln 6 + 2 - 1 2 1 2(1 + ) 1 2
Problem 4: Another coconut economy. The production function is f (x) = (3 - 2x)x for 0 x < 3/4, and 9/8 for x > 3/4. Suppose Rob is endowed with 1 unit of leisure. (a) Utility is of the form u(c, l) = c + al for a > 0. As a varies between 0 and 1, identify the Pareto optimal allocations. Which allocations cannot be supported as competitive equilibria? (b) Utility is of the form u(c, l) = min c, al. As a > 0 varies, identify the Pareto optimal allocations. Which allocations cannot be supported as competitive equilibria? Solution: Drawing a diagram is helpful when facing linear or Leontief utility functions. Shift indifference curves upward subject to feasibility conditions to obtain Pareto optimal allocations. If a Pareto optimal allocation along with a price vector solves UMP and PMP, and satisfies market clear condition, then it is supported by a competitive equilibrium (CE).
ECON 610: Microeconomic Theory II
Prof. Blume
Problem Set 1
6
(a) If a > 3 = f (0). x = 0, l = 1, c = 0 is the Pareto optimal allocation. If 0 < a 3, Pareto optimal allocation is obtained by solving MRS=MRT, a = f (x) = 3 - 4x x = 3-a , l = 1+a , c = f 3-a . Letting wage rate w equal to a, we could find all the 4 4 4 Pareto optimal allocations could be supported by a CE.
9/8 (b) If a > 1-3/4 = 9/2, x 3/4, 1 - 9/8 , c = 9/8, l = 1 - x are the Pareto optimal a allocations. These allocations could be supported by a CE with w = 0. If 0 < a <
9/2, x = (solving c = al = a (1 - x) = f (x) = 3x - 2x2 ), c = f (x), 2 l = 1 - x is the Pareto optimal allocation, which could also be supported by a CE with w = f (x).
(3+a)-
(3+a)2 -8a