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Analyzing Equilibrium Prices and Incomes in a General Equilibrium Model

This document explores a General Equilibrium model to derive equilibrium prices and incomes as a function of endowment. It aims to illustrate the limitations of redistribution within the model, employing different social welfare functions (SWFs). Various methods, including budget constraints and optimization strategies based on Cobb-Douglas preferences, are utilized to calculate the demands for two types of agents. The analysis further investigates income possibility sets and welfare optimum points, emphasizing the relationship between resource allocation and economic equality.

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Analyzing Equilibrium Prices and Incomes in a General Equilibrium Model

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  1. Exercise 9.6 MICROECONOMICS Principles and Analysis Frank Cowell February 2007

  2. Ex 9.6(1): Question • purpose: to derive equilibrium prices and incomes as a function of endowment. To show the limits to redistribution within the GE model for a alternative SWFs • method: find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for r

  3. Ex 9.6(1): budget constraints • Use commodity 2 as numéraire • price of good 1 is r • price of good 2 is 1 • Evaluate incomes for the two types, given their resources: • type a has endowment (30, k) • therefore ya= 30r + k • type b has endowment (60, 210 k) • therefore yb= 60r + [210 k] • Budget constraints for the two types are therefore: • rx1a +x2a≤ 30r + k • rx1b +x2b ≤ 60r + [210 k]

  4. Ex 9.6(1): optimisation • We could jump straight to a solution • utility functions are simple… • …so we can draw on known results • Cobb-Douglas preferences imply • indifference curves do not touch the origin… • …so we need consider only interior solutions • also demand functions for the two commodities exhibit constant expenditure shares • In this case (for type a) • coefficients of Cobb-Douglas are 2 and 1 • so expenditure shares are ⅔ and ⅓ • (and for b they will be ⅓ and ⅔ ) • gives the optimal demands immediately… Jump to “equilibrium price”

  5. Ex 9.6(1): optimisation, type a • The Lagrangean is: • 2log x1a + log x2a + na[ya rx1a x2a ] • where nais the Lagrange multiplier • and ya is 30r + k • FOC for an interior solution • 2/x1a nar = 0 • 1/x2a na = 0 • ya rx1a x2a= 0 • Eliminating na from these three equations, demands are: • x1a= ⅔ ya / r • x2a = ⅓ ya

  6. Ex 9.6(1): optimisation, type b • The Lagrangean is: • log x1b + 2log x2b + nb[yb rx1b x2b ] • where nbis the Lagrange multiplier • and yb is 60r + 210 k • FOC for an interior solution • 1/x1b nbr = 0 • 2/x2b nb = 0 • yb rx1b x2b= 0 • Eliminating nb from these three equations, demands are: • x1b= ⅓ yb / r • x2b = ⅔yb

  7. Ex 9.6(1): equilibrium price • Take demand equations for the two types • substitute in the values for income • type-a demand becomes • type-b demand becomes • Excess demand for commodity 2: • [10r+⅓k]+[40r +140− ⅔k]−210 • which simplifies to 50r− ⅓k−70 • Set excess demand to 0 for equilibrium: • equilibrium price must be: • r= [210 + k] / 150

  8. Ex 9.6(2): Question and solution • Incomes for the two types are resources: • ya= 30r + k • yb= 60r + [210 k] • The equilibrium price is: • r= [210 + k] / 150 • So we can solve for incomes as: • ya= [210 + 6k] / 5 • yb= [1470 3k] / 5 • Equivalently we can write yaand ybin terms of ras • ya= 180r 210 • yb= 420  90r

  9. Ex 9.6(3): Question • purpose: to use the outcome of the GE model to plot the “income-possibility” set • method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps.

  10. Income possibility set yb • incomes for k = 0 • incomes for k = 210 • incomes for intermediate values of k • attainable set if income can be thrown away 300 • (42, 294) • yb = 315  ½ya 200 • (294, 168) 100 ya 300 0 100 200

  11. Ex 9.6(4): Question • purpose: find a welfare optimum subject to the “income-possibility” set • method: plot contours for the function W on the previous diagram.

  12. Welfare optimum: first case yb • income possibility set • Contours of W = log ya + log yb • Maximisation of W over income-possibility set 300 • W is maximised at corner • incomes are (294, 168) • here k = 210 • so optimum is where all of resource 2 is allocated to type a 200 • 100 ya 300 0 100 200

  13. Ex 9.6(5): Question • purpose: as in part 4 • method: as in part 4

  14. Welfare optimum: second case yb • income possibility set • Contours of W = ya + yb • Maximisation of W over income-possibility set 300 • again W is maximised at corner • …where k = 210 • so optimum is where all of resource 2 is allocated to type a 200 • 100 ya 300 0 100 200

  15. Ex 9.6: Points to note • Applying GE methods gives the feasible set • Limits to redistribution • natural bounds on k • asymmetric attainable set • Must take account of corners • Get the same W-maximising solution • where society is averse to inequality • where society is indifferent to inequality

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