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Mathematical Postscripts

Mathematical Postscripts. Chapter 5 Chapter 6 Chapter 8 Chapter 21. The Factor-Proportions Model. Factor Prices and Costs The unit isoquant in Figure 5P-1 shows combinations of capital and labor that can be used to produce one unit of the good.

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Mathematical Postscripts

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  1. Mathematical Postscripts Chapter 5 Chapter 6 Chapter 8 Chapter 21

  2. The Factor-Proportions Model Factor Prices and Costs • The unit isoquant in Figure 5P-1 shows combinations of capital and labor that can be used to produce one unit of the good. • It’s increasingly difficult to substitute capital for labor as the capital-labor ratio increases, and vice-versa. • Producers choose the mix of capital and labor that minimizes their cost. • Such as point E, the point at which the unit isoquant is tangent to a line whose slope equals –w/r.

  3. Fig. 5P-1: Efficient Production

  4. The Factor-Proportions Model (cont.) • The cost of production equals the sum of the cost of capital and labor inputs, where the input coefficients (aK and aL) have been chosen to minimize costs. c = aK r + aL w • Because the mix of factors was chosen to minimize cost, an infinitesimal change in the capital-labor ratio must have no effect on cost. 0 = r daK + w daL

  5. The Factor-Proportions Model (cont.) • A change in the factor prices has two effects: • It will change the choice of factor mix (aK and aL) , and it will change the cost of production c. • The cost-minimizing labor-capital ratio depends on the ratio of the price of labor to that of capital. • For small changes in factor prices, the change in production cost is the following (the last two terms sum to zero): dc = aKdr + aLdw + rdaK + wdaL = aKdr + aLdw

  6. is share of capital in total production costs The Factor-Proportions Model (cont.)

  7. The price of each good must equal production cost: Equations for the rate of change for factor prices: The Factor-Proportions Model (cont.)

  8. The economy’s factors must be fully employed: For given prices, equations for rate of change for outputs: The Factor-Proportions Model (cont.)

  9. The Factor-Proportions Model (cont.) Goods Prices and Factor Prices • If the price of food rises relative to the price of cloth, then the real price of capital rises in terms of both goods, while the real price of labor falls in terms of both goods. • In particular, if the price of F were to rise with no change in the price of C, the wage rate would fall.

  10. The Factor-Proportions Model (cont.) Factor Supplies and Output • If the prices of the goods stay constant, while the supply of capital rises relative to the supply of labor, then the output of food grows relative to the supply of capital and the output of cloth shrinks relative to the supply of labor. • In particular, if the supply of K were to rise with no change in the supply of L, the output of cloth would fall.

  11. Fig. 6P-1: Consumption Effects of a Price Change

  12. The Monopolistic Competition Model • Consider the effects of changes in the size of the market on equilibrium in a monopolistically competitive industry. • Each firm has the total cost C = F + cX, where c is marginal cost, F a fixed cost, and X the firm’s output. • This implies an average cost AC = C/X = F/X + c. • Each firm faces a demand curve X = S [1/n – b (P – P)], where S is total industry sales (taken as given), n is the number of firms, and P is the average price charged by other firms (taken as given).

  13. The Monopolistic Competition Model (cont.) • Profits p = PX – C = (PS – c) [1/n – b (P – P)] – F • Each firm chooses its price to maximize profts by setting dp /dP = X – SbP + Sbc = 0 • Since all firms are symmetric, P = P and X = S/n so P = 1/bn + c and AC = Fn/S + c

  14. The Monopolistic Competition Model (cont.) • In zero-profit equilibrium, the price charged by a typical firm must equal its average cost P = AC. 1/bn + c = Fn/S + c • Solving implies n = sqrt (S/bF). • An increase in the size of the market, S, will lead to an increase in the number of firms, n, but not in proportion.

  15. The Monopolistic Competition Model (cont.) • The price charged by the representative firm is P = 1/bn + c = c + sqrt (S/bF). • An increase in the size of the market leads to lower prices. • The sales per firm equal X = S/n = sqrt (S/bF). • The scale of each individual firm also increases with the size of the market.

  16. Fig. 21P-1: Indifference Curves for Uncertain Consumption Costs

  17. Fig. 21P-2: Maximizing Expected Utility

  18. Fig. 21P-3: Nondiversified Portfolios

  19. Fig. 21P-4: Effects of a Rise in H1 on Consumption

  20. Fig. 21P-5: Effects of a Rise in H1 on Portfolio Shares

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