U = x x x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

Transcription

1 Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities: 00 of the traders own 0 units of good each 50 of the traders own 5 units of good each 50 of the traders own 0 units of good 3 each All the traders have the utility function U = x x 4 x 4 3 What are the equilibrium relative prices of the three goods? traders has members who are best off? Which group of Outline Answer: For each group of traders the Lagrangean may be written log xh + 4 log xh + 4 log xh 3 + υ h [y h p x h p x h p 3 x h 3 where h =,, 3 and y = 0p, y = 5p and y 3 = 0p 3. From the firstorder conditions we find that for a trader of type h: x h = yh p x h = yh 4p x h 3 = yh 4p 3 Excess demand for good and are then: 93

2 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM E = 00x + 50x + 50x E = 00x + 50x + 50x 3 50 Substituting in for x h i and y h and putting E = E = 0 we find p p p 3 p = 0 50 p p p 3 p = 0 that implies p = and p 3 = p p. Using good as numeraire we immediately see that y = y = y 3 = 0. All are equally well off. c Frank Cowell

3 Microeconomics Exercise 7. Consider an exchange economy with two goods and three persons. Alf always demands equal quantities of the two goods. Bill s expenditure on group is always twice his expenditure on good. Charlie never uses good.. Describe the indiff erence maps of the three individuals and suggest utility functions consistent with their behaviour.. If the original endowments are respectively (5, 0), (3, 6) and (0, 4), compute the equilibrium price ratio. What would be the eff ect on equilibrium prices and utility levels if (a) 4 extra units of good were given to Alf; (b) 4 units of good were given to Charlie? Outline Answer:. Let ρ = p p so that values are measured in terms of good. (a) Alf s (binding) budget constraint is ρx a + x a = 5ρ Therefore, given the information in the question, the demand functions are x a = x a = 5ρ ρ +. The utility function consistent with this behaviour is see Figure 7.. (b) Bill s budget constraint is From the question we have U a (x a, x a ) = min {x a, x a } ρx b + x b = 3ρ + 6 Therefore: ρx b = x b x b = + 4 ρ x b = ρ +. The utility function is Cobb-Douglas: see Figure 7.. c Frank Cowell U b = log x b + log x b

4 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM x (5,0) x Figure 7.: Alf s preferences and demand x (3,6) x Figure 7.: Bill s preferences and demand c Frank Cowell

5 Microeconomics (c) Charlie s budget constraint is ρx a + x a = 4 Given the information in the question we have and utility is see Figure 7.3. x c = 4/ρ x c = 0 U c = x c x (4,0) x Figure 7.3: Charlie s preferences and demand. Excess demand for good is : E = 5ρ ρ + + ρ + 0. Putting E = 0 yields ρ = or 4. Hence the equilibrium price ratio is 4. Utility levels are U a = 4, U b = log(54) and U c =. (a) Excess demand is now 9ρ ρ + + ρ + 0 and the equilibrium price ratio is. Utility levels are U a = 6, U b = log(64) and U c =. (b) Excess demand for good is : E = 5ρ ρ + + ρ + 0. Putting E = 0 yields ρ = or 4. Hence the equilibrium price ratio is 4. Utility levels are U a = 4, U b = log(54) and U c = 4 + = 5. c Frank Cowell

6 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM Exercise 7.3 In a two-commodity economy assume a person has the endowment (0, 0).. Find the person s demand function for the two goods if his preferences are represented by each of the types A to D in Exercise 4.. In each case explain what the offer curve must look like.. Assume that there are in fact two equal sized groups of people, each with preferences of type A, where everyone in group has the endowment (0, 0) with α = and everyone in group an endowment (0, 0) with α = 3 4. Use the off er curves to find the competitive equilibrium price and allocation. Outline Answer:. The income of person h is 0. (a) If he has preferences of type A then the Lagrangean is α log x h + [ α log x h + λ [ 0 ρx h x h First order conditions for an interior maximum of (7.) are α λρ = 0 x h α x h λ = 0 0 ρx h x h = 0 Solving these we find λ = 0 and so the demands will be [ 0α x h = ρ 0 [ α (7.) (7.) and the offer curve will simply be a horizontal straight line at x h = 0 [ α. (b) If h has preferences of type B then demand will be x h = x, if ρ > β x h [x, x, if ρ = β x h = (0/ρ, 0), if ρ < β where x := (0, 0), x := (0/β, 0), and their offer curve will consist of the union of the line segment [x, x and the line segment from x to (, 0). (c) If group- persons have preferences of type C then their demands will be x h = x, if ρ > γ x h = x or x, if ρ = γ x h = (0/ρ, 0), if ρ < γ where x := (0, 0), x := (0/ γ, 0), and their offer curve will consist of the union of the point x and the line segment from x to (, 0). c Frank Cowell

7 Microeconomics 0 A x' B indifference curve 0[ α x β x'' x x' x x C D γ δ x'' x x Figure 7.4: Offer Curves for Four Cases (d) If group- persons have preferences of type D then their demands will be x h = [ 0 ρ+δ 0δ ρ+δ and their offer curve is just the straight line x h illustrated in Figure 7.4. = δx h. These are. If a type-a person had an income of 0 units of commodity then, by analogy with part, demand would be [ x 0α = 0ρ [ α (7.3) and the offer curve will simply be a vertical straight line at x h = 0α. From (7.) and (7.3) we have x = 0 = 5, x = 0 [ 4 3 = 5. Given that there are 0 units per person of commodity and 0 units per person of commodity the materials balance condition then means that c Frank Cowell

8 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM the equilibrium allocation must be x = x = [ 5 5 [ 5 5 Solving for ρ from (7.) and (7.3) we find that the equilibrium price ratio must be 3. c Frank Cowell

9 Microeconomics Exercise 7.4 The agents in a two-commodity exchange economy have utility functions U a (x a ) = log(x a ) + log(x a ) U b (x b ) = log(x b ) + log(x b ) where x h i is the consumption by agent h of good i, h = a, b; i =,. The property distribution is given by the endowments R a = (9, 3) and R b = (, 6).. Obtain the excess demand function for each good and verify that Walras Law is true.. Find the equilibrium price ratio. 3. What is the equilibrium allocation? 4. Given that total resources available remain fixed at R := R a + R b = (, 9), derive the contract curve. Outline Answer:. To get the demand functions for each person we need to find the utilitymaximising solution. The Lagrangean for person a is L a (x a, ν a ) := log (x a ) + log (x a ) + ν a [9p + 3p p x a p x a First-order conditions are x x ν a p = 0 ν a p = 0 9p + 3p p x a p x a = 0 Define ρ := p /p and normalise p arbitrarily at. Then, rearranging the FOC we get ν = ρx a ν = x a (7.4) 9ρ + 3 = ρx a + x a Subtracting the first two equations from the third in (7.4) we can see that ν a = +3ρ. Substituting back for the Lagrange multiplier ν a into the first two parts of (7.4) we see that the first-order conditions imply: [ [ x a 3 + x a = ρ (7.5) 6ρ + Using exactly the same method for person b we would find [ [ x b x b = ρ 4ρ + (7.6) Using the definition we can then find the excess demand functions by evaluating: E i := x a i + x b i Ri a Ri b (7.7) c Frank Cowell 006 0

10 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM Doing this we get [ E = E [ 5 ρ 0 0ρ 5 Now construct the weighted sum of excess demands. It is obvious that (7.8) ρe + E = 0 (7.9) thus confirming Walras Law. In equilibrium the materials balance condition must hold and so excess demand for each good must be zero, unless the corresponding equilibrium price is zero (markets clear).. Solving for E = 0 in (7.8) we find ρ = for the (normalised) equilibrium prices. [ [ x b 6 3. The allocation is x b = 4 4. The contract curve is traced out by the MRS condition and the materials balance condition MRS a = MRS b (7.0) E = 0 (7.) From (7.) we have Applying (7.0) we then get [ x b x b [ x a = 9 x a (7.) x a x a = xa [9 x a (7.3) which implies that the equation of the contract curve is: x a = xa x a (7.4) + 7. c Frank Cowell 006 0

11 Microeconomics Exercise 7.5 Which of the following sets of functions are legitimate excess demand functions? E (p) = p + 0 p E (p) = p (7.5) E 3 (p) = 0 p 3 E (p) = p+p3 p E (p) = p+p3 p E 3 (p) = p+p p 3 E (p) = p3 p E (p) = p3 p E 3 (p) = (7.6) (7.7) Outline Answer: The first system is not homogeneous of degree zero in prices. The second violates Walras Law. The third one is both homogeneous and satisfies Walras Law. c Frank Cowell

12 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM Exercise 7.6 In a two-commodity economy let ρ be the price of commodity in terms of commodity. Suppose the excess demand function for commodity is given by 4ρ + 5ρ ρ 3. How many equilibria are there? Are they stable or unstable? How might your answer be affected if there were an increase in the stock of commodity in the economy? Outline Answer: The excess demand for commodity at relative price ρ can be written So that E(ρ) := 4ρ + 5ρ ρ 3 = [ ρ [ ρ. de(ρ)/dρ = 4 + 0ρ 6ρ see Figure 7.5. From this we see that there are two equilibria as follows:. ρ = 0.5. Here de(ρ)/dρ < 0 and so it is clear that the equilibrium is locally stable ρ =. Here de(ρ)/dρ = 0. But the graph of the function reveals that it is locally stable from above (where ρ > ) and unstable from below (where ρ < ). If there were an increase in the stock of commodity the excess demand function would be shifted to the left in Figure 7.5 then there is only one, stable equilibrium..5 ρ 0.5 E Figure 7.5: Excess demand c Frank Cowell

13 Microeconomics Exercise 7.7 Consider the following four types of preferences: Type A : α log x + [ α log x Type B : βx + x Type C : γ [x + [x Type D : min {δx, x } where x, x denote respectively consumption of goods and and α, β, γ, δ are strictly positive parameters with α <.. Draw the indiff erence curves for each type.. Assume that a person has an endowment of 0 units of commodity and zero of commodity. Show that, if his preferences are of type A, then his demand for the two commodities can be represented as [ [ x 0α x := = 0ρ [ α x where ρ is the price of good in terms of good. What is the person s offer curve in this case? 3. Assume now that a person has an endowment of 0 units of commodity (and zero units of commodity ) find the person s demand for the two goods if his preferences are represented by each of the types A to D. In each case explain what the offer curve must look like. 4. In a two-commodity economy there are two equal-sized groups of people. People in group own all of commodity (0 units per person) and people in group own all of commodity (0 units per person). If Group has preferences of type A with α = find the competitive equilibrium prices and allocations in each of the following cases: (a) Group have preferences of type A with α = 3 4 (b) Group have preferences of type B with β = 3. (c) Group have preferences of type D with δ =. 5. What problem might arise if group had preferences of type C? Compare this case with case 4b Outline Answer:. Indifference curves have the shape shown in the figure The income of a group- person is 0ρ. If group- persons have preferences of type A then the Lagrangean is α log x + [ α log x + λ [ 0ρ ρx x c Frank Cowell

14 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM z z () () z z z z (3) (4) z z Figure 7.6: Indifference Curves First order conditions for an interior maximum are α λρ = 0 x α x λ = 0 0ρ ρx x = 0 Solving these we find λ = 0ρ and so the demands will be [ x 0α = 0ρ [ α and the offer curve will simply be a vertical straight line at x = 0α. 3. The income of a group- person is 0. So, if group- persons have preferences of type A, then their demands will be [ 0α x = ρ 0 [ α and their offer curve will simply be a horizontal straight line at x = 0 [ α. If group- persons have preferences of type B then their demands will be x = x, if ρ > β x [x, x, if ρ = β c Frank Cowell x = (0/ρ, 0), if ρ < β

15 Microeconomics where x := (0, 0), x := (0/β, 0), and their offer curve will consist of the union of the line segment [x, x and the line segment from x to (, 0). If group- persons have preferences of type C then their demands will be x = x, if ρ > γ x = x or x, if ρ = γ x = (0/ρ, 0), if ρ < γ where x := (0, 0), x := (0/ γ, 0), and their offer curve will consist of the union of the point x and the line segment from x to (, 0). If group- persons have preferences of type D then their demands will be x = [ 0 ρ+δ 0δ ρ+δ and their offer curve is just the straight line x = δx. 4. In each case below we could work out the excess demand function, set excess demand equal to zero, find the equilibrium price and then the equilibrium allocation. However, we can get to the result more quickly by using an equivalent approach. Given that an equilibrium allocation must lie at the intersection of the offer curves of the two parties the answer in each case is immediate. (a) From the above computations we have x = 0 = 5, x = 0 [ 4 3 = 5. Given that there are 0 units per person of commodity and 0 units per person of commodity the materials balance condition then means that the equilibrium allocation must be [ x 5 = 5 [ x 5 = 5 Solving for ρ we find that the equilibrium price ratio must be 3. (b) We have x = 5, and so x = 5. Using the fact that the equilibrium must lie on the group- offer curve we see that the solution must lie on the straight line from (0, 0) to (0/3, 0) we find that x = 5 and, from the materials balance condition x = 0 5 = 5 (as in the previous case). By the same reasoning as in the previous case the equilibrium price must be ρ = 3. (c) Once again we have x = 5, and so x = 5. Given that the group- offer curve in this case is such that the person always consume s equal quantities of the two goods we must have x = 5 and so again x = 0 5 = 5 (as in the previous cases). As before the equilibrium price must be Note that the demand function and the offer curve for the group- people is discontinuous. So, if there are relatively small numbers in each group c Frank Cowell

16 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM there may be no equilibrium (the two offer curves do not intersect). In the large numbers case we could appeal to a continuity argument and have an equilibrium with proportion of group at point x and the rest at x. The equilibrium would then look very much like case 4b. c Frank Cowell

17 Microeconomics Exercise 7.8 In a two-commodity exchange economy there are two equal-sized groups of people. Those of type a have the utility function U a (x a ) = [xa [xa and a resource endowment of (R, 0); those of type b have the utility function U b ( x b) = x b x b and a resource endowment of (0, R ).. How many equilibria does this system have?. Find the equilibrium price ratio if R = 5, R = 6. Outline Answer: For consumers of type a the relevant Lagrangean is [xa [xa + λ [pr px a x a where p is the price of good in terms of good. The FOC for a maximum are [x a 3 pλ = 0 [x a 3 λ = 0 Rearranging and using the budget constraint we get So px a + x a = x a = p /3 λ /3 x a = λ /3 [ p /3 + λ /3 = pr x a = λ /3 = pr p /3 + For consumers of type b the Lagrangean is log x b + log x b + µ [ R px b x b The FOC for a maximum are [ x b pµ = 0 [ x b µ = 0 Rearranging and using the budget constraint we get x b = pµ x b = µ c Frank Cowell

18 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM So px b + x b = µ = R x b = µ = R The excess-demand function for good is therefore. Excess demand is 0 where pr p /3 + R pθ p /3 = 0 where θ := R /R. This is equivalent to requiring p /3 = pθ The expression p /3 is an increasing concave function through the origin. It is clear that the straight line given by pθ can cut this just once. There is one equilibrium.. If (R, R ) = (0, 3) then θ := 0/3 = 5/8 and p = 8. c Frank Cowell 006 0

19 Microeconomics Exercise 7.9 In a two-person, private-ownership economy persons a and b each have utility functions of the form V h ( p, y h) ( ) = log y h p β h p β h log (p p ) where h = a, b and β h, β h are parameters. Find the equilibrium price ratio as a function of the property distribution [R. Outline Answer: Using Roy s identity we have, for each h and each i : Now we have x h i = V i h V h y Vi h = β h i [y h p β h p β h /p i, Vy h = [y h p β h p β h. Combining the two results we find for each h: x h i = β h i + [ [ p [R h β h + p R h β h p i Defining β i = β a i + β b i and R i = Ri a + Rb i we obtain the excess demand for good : E = β R + [p [R β p + p [R β Hence putting E = 0 we get the equilibrium price ratio thus: p p = R β R β.. c Frank Cowell 006

20 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM Exercise 7.0 In an economy there are large equal-sized groups of capitalists and workers. Production is organised as in the model of Exercises.4 and 6.4. Capitalists income consists solely of the profits from the production process; workers income comes solely from the sale of labour. Capitalists and workers have the utility functions x c x c and x w [R 3 x w 3 respectively, where x h i denotes the consumption of good i by a person of type h and R 3 is the stock of commodity 3.. If capitalists and workers act as price takers find the optimal demands for the consumption goods by each group, and the optimal supply of labour R 3 x w 3.. Show that the excess demand functions for goods, can be written as Π + p [p A p Π A p p where Π is the expression for profits found in Exercise 6.4. Show that in equilibrium p /p = 3 and hence show that the equilibrium price of good (in terms of good 3) is given by p = [ /3 3 A 3. What is the ratio of the money incomes of workers and capitalists in equilibrium? Outline Answer:. Given that the capitalist utility function is x c x c it is immediate that in the optimum the capitalists spend an equal share of their income on the two consumption goods and so Worker utility is and the budget constraint is x c i = Π p i. x w [R 3 x w 3 (7.8) p x w R 3 x w 3 (7.9) Maximising (7.8) subject to (7.9) is equivalent to maximising c Frank Cowell 006 p [R 3 x w 3 [R 3 x w 3. (7.0)

21 Microeconomics The FOC is which gives optimal labour supply as: p [R 3 x w 3 = 0 (7.) R 3 x w 3 = p (7.) and, from (7.9), the workers optimal consumption of good is x w = [p. (7.3). The economy has no stock of good or good ; workers do not consume good ; so excess demand for the two goods is, respectively: x c + x w y = Π p + [p A p (7.4) x c y = Π p A p (7.5) To find the equilibrium set each of (7.4) and (7.5) equal to zero. gives Substituting in for profits in (7.7) we have and so Substituting for Π and p we get This Πp + = A [p 3 (7.6) Π = A [p (7.7) [p + [p = [p 4 p p = 3. (7.8) p A [p + 3 [p + = A [p 3 4 and, on rearranging, this gives [ /3 3 p = (7.9) A 3. Profits in equilibrium are A [p = A 3 [p = A 3 [ /3 3 = A [ /3 A /3. 3 Given that the price of good 3 is normalised to, using (7.)and (7.9) total labour income in equilibrium is = [ /3 [ /3 3 A = /3 p A 3 c Frank Cowell 006 3

22 Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM So, workers and capitalists get the same money income in equilibrium! Note that this is unaffected by the value of A; increases in A could be interpreted as technical progress and so the income distribution remains unchanged by such progress. c Frank Cowell 006 4