Transcription of Solving for the Walrasian Equilibrium: Two examples Example 1
1 Econ 401A October 9, 2016 1 Solving for the Walrasian Equilibrium: Two examples Example 1: Every consumer has the same utility function function 1112()Uxxx . There is an initial endowment of 30 units of commodity 1. Commodity 1 is both consumed and used as an input in the production of commodity 2. The production function is 4qz . The utility function is homothetic since 1( )( )UxUx . Then ( ) ( )UxUy implies that ( ) ( )UxUy . (a) Solve for the production plan that maximizes the utility of the representative consumer. (b) What must be the WE price ratio? Example 2: Every consumer has the same utility function function 412()Uxxx . There is an initial endowment of 81 units of commodity 1. Commodity 2 is a produced using commodity 1 as an input. The production function is 1/26qz . (a) Show that the utility function is homothetic. (b) Solve for the production plan that maximizes the utility of the representative consumer. (c) What must be the WE price ratio?
2 (d) What is the equilibrium profit of the firm? Econ 401A October 9, 2016 2 Example 1 Step 1: Feasible outcomes No production: 1 21 2( , ) ( , ) (30,0)xxx . Use 1 unit of commodity 1 in production. 4qz . So the maximum output of commodity 2 is 4. Use z units of commodity 1. The maximum output is 4qz . Then 130xz and 24xz . The maximum output of commodity 2 is 4*30 = 120. The set of feasible alternatives is therefore as depicted. Preferences Instead of maximizing U we maximize 12ln 4ln lnuUxx Note that 1111144uMUxxx and 122221uMUxxx . These are both positive so utility is strictly increasing. Note also that 2121221( , )( )MUxMRSxxMUx 2x Econ 401A October 9, 2016 3 Consider moving from A to B on the level set depicted below. The ratio 21xx increases. Therefore the 12( , )MRSxx increases. Thus the steepness of the level set is greater at B. Thus without worrying about the exact shape, we can draw level sets as shown below.
3 Superimposing the level sets on the first figure, the optimum for the representative consumer is the point on the boundary of the feasible set tangential to the indifference curve. Econ 401A October 9, 2016 4 Step 2: Solve for the optimum For any zthe output must satisfy 4qz . Since utility is increasing this must be an equality for a maximum . All the output of commodity 2 will be consumed therefore 24xqz . Commodity 1 is used both as an input in the production of commodity 2 and as consumption. Therefore if z units are used in production, 1130xzz . Substitute these into the utility function. 111114(30 ) (4 )(30 )Uzzzz . Look on the margin 22142211(30 )(30 ) 4dUzzdzzz This must be zero if the maximizer, *0z . Then 22(30 ) 4zz . Take the square root. 302zz Therefore *10z and so **12( , ) (20,40)xx Econ 401A October 9, 2016 5 Step 3: Solve for prices that support the optimal production plan.
4 In the model, firms are price takers. Consider any pair of prices 12( , )ppp and a production plan ( , )zq. That is, the firm purchases z units of commodity 1 and sells q units of commodity 2. The profit is 21( , )zqpqpz . For any z the firm will produce the maximum possible output to maximize profit. In this Example the set of feasible plans is the set {( , ) | 4 }Szqqz so the firm will choose 4qz . Then profit is 214pzpz The marginal profit to increasing the input is therefore 214dppdz . Supporting prices The price vector p is said to support the optimum if **( , )zq is profit-maximizing. Then marginal profit must be zero. Then the price vector is supporting if 124pp . If the relative price of the input is higher then marginal profit is always negative so *10z is not supported. This is depicted in the figure. The level set of zero profit, 210pqpz , passes through ( , ) (0,0)zq . Profit is higher in the direction of the arrows (more output and less input).
5 The firm therefore maximizes profit by producing nothing. Econ 401A October 9, 2016 6 If 12ppis below 4 it is always strictly profitable to increase output so again the prices are not supporting. Therefore the only possible WE price ratio is 4. In this case the zero profit line is the green line (the boundary of the feasible set.) Step 4: Explain why consumer demand is equal to supply at these prices Since the endowment of commodity 2 is zero, 2xq . Since the endowment of commodity 1 is 130 , 11xz . Therefore 2 21 121 1()pxpxpqpz 211 1pqpzp 11( , )zqp In Step 3, we showed that for profit maximization, equilibrium profit must be zero. Therefore the budget constraint of the representative consumer is 2 21 11 1pxpxp Econ 401A October 9, 2016 7 In the figure below this is the green line through the endowment point Therefore *(20,40)x is utility maximizing, the choice of the representative consumer.
6 Econ 401A October 9, 2016 8 Example 2 Step 1: Feasible outcomes No production: 1 21 2( , ) ( , ) (81,0)xxx . With an input of zunits of commodity 1, the maximum output is 1/26qz . Then 181xz and 1/226xqz . The maximum output of commodity 2 is 1/26*(81) 54 . The set of feasible alternatives is depicted below Preferences It is simpler to maximize 12ln 4ln lnuUxx . Note that 1111144uMUxxx and 122221uMUxxx . These are both positive so utility is strictly increasing. Note also that 121221( , )4 MUxMRSxxMUx Econ 401A October 9, 2016 9 Arguing as in Example 1, moving from A to B on the level set depicted below, the 12( , )MRSxx increases. Thus the steepness of the level set is greater at B. Thus without worrying about the exact shape, we can draw level sets as shown below. Superimposing the level sets on the first figure, the optimum for the representative consumer is the point on the boundary of the feasible set tangential to the indifference curve.
7 Econ 401A October 9, 2016 10 Step 2: Solve for the optimum For any zthe output must satisfy 1/26qz . Since utility is increasing this must be an equality for a maximum . All the output of commodity 2 will be consumed therefore 1/226xqz . Commodity 1 is both consumed and used as an input in the production of commodity 2. Therefore if z units are used in production, 1181xzz . Substitute these into the utility function. 1/2124ln(81 ) ln64ln(81 ) ln6 lnuzzzz . Look on the margin 41812dudzzz This must be zero if the maximizer, *0z . Then 818zz . Therefore *9z and so **12( , ) (72,18)xx Econ 401A October 9, 2016 11 Step 3: Solve for prices that support the optimal production plan. In the model, firms are price takers. Consider any pair of prices 12( , )ppp and a production plan ( , )zq. That is, the firm purchases z units of commodity 1 and sells q units of commodity 2.
8 The profit is 21( , )zqpqpz . For any z the firm will produce the maximum possible output to maximize profit. In this Example the set of feasible plans is the set 1/2{( , ) | 6 }Szqqz so the firm will choose 1/26qz . Then profit is 1/2216pzpz The marginal profit to increasing the input is therefore 1/222111/233pdpzppdzz . Supporting prices The price vector p is said to support the optimum if **( , )zq is profit-maximizing. Then marginal profit must be zero at*9z . Then the price vector is supporting if * 1/212()13pzp . The zero profit level set is the green line through the endowment point. The maximum profit level set is the green line, 21pqpz , tangential to the boundary of the production set at **( , )zq So this must have a slope of 1. Econ 401A October 9, 2016 12 Step 4: Explain why consumer demand is equal to supply at these prices Since the endowment of commodity 2 is zero, 2xq.
9 Since the endowment of commodity 1 is 181 , 11xz . The level set for maximized profit is 21pqpz But 2qx and 11zx . Therefore 2 21 11()pxpx Rearranging this equation, 2 21 11 1pxpxp . Thus the maximum profit line is also the budget line for the representative consumer. Therefore *(72,18)x is utility maximizing, the choice of the representative consumer.
