A firm lives for two periods, t=0,1. In each period, it operates the technology F (K-1)...

 

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A firm lives for two periods, t=0,1. In each period, it operates the technology F (K₁-1) = A₂K-1 where A, is the firm's total factor productivity, K, is its capital and labor inputs respectively, and 0 < a <1. The firm owns its own capital and discounts the future at the rate r. Capital depreciates at the rate & and costs på each period. In period. 0, the firm chooses how much to invest Io. Investment is subject to quadratic adjustment costs. At the end of period 1, the firm sells its undepreciated capital stock in the capital market for its current price. Given an initial capital stock K_₁.the firm maximizes the present value of profits, subject to max F (K-1) - la-(₁) ³ K-1 + ² adjustment costs Ko=(1-6) K-1 + 10 F(Ko)+(1-6) p Ko 1+r and let An be the Lagrange Multiplier on this constraint. 1. Write down a Lagrangian for the firm's problem, and compute the first order conditions for the firm's choices of investment and capital. Show that ² (1+61) A₁akg-¹ + (1 - 6) p² 1+r = and interpret the two sides of the equation. 2. Define Tobin's Q as the ratio Q = Ao/p. Find an expression for Tobin's Q in terms of pf.r. A₁.a, Ko and show that if Q> 1 the firm's investment is positive, and negative otherwise. 3. Suppose A₁ rises, all else fixed. What is the impact on Tobin's Q? What about on investment In? 4. Suppose you were asked to predict whether a given company is likely to begin investing in new capital based on the model above. (a) Describe some challenges in finding an empirical value for Tobin's Q and how one typically calculates it using data on market values of debt and equity and on the replacement cost of capital. List some sources for this data. (b) How well do empirically estimated values of Tobin's Q actually predict firm investment? A firm lives for two periods, t=0,1. In each period, it operates the technology F (K₁-1) = A₂K-1 where A, is the firm's total factor productivity, K, is its capital and labor inputs respectively, and 0 < a <1. The firm owns its own capital and discounts the future at the rate r. Capital depreciates at the rate & and costs på each period. In period. 0, the firm chooses how much to invest Io. Investment is subject to quadratic adjustment costs. At the end of period 1, the firm sells its undepreciated capital stock in the capital market for its current price. Given an initial capital stock K_₁.the firm maximizes the present value of profits, subject to max F (K-1) - la-(₁) ³ K-1 + ² adjustment costs Ko=(1-6) K-1 + 10 F(Ko)+(1-6) p Ko 1+r and let An be the Lagrange Multiplier on this constraint. 1. Write down a Lagrangian for the firm's problem, and compute the first order conditions for the firm's choices of investment and capital. Show that ² (1+61) A₁akg-¹ + (1 - 6) p² 1+r = and interpret the two sides of the equation. 2. Define Tobin's Q as the ratio Q = Ao/p. Find an expression for Tobin's Q in terms of pf.r. A₁.a, Ko and show that if Q> 1 the firm's investment is positive, and negative otherwise. 3. Suppose A₁ rises, all else fixed. What is the impact on Tobin's Q? What about on investment In? 4. Suppose you were asked to predict whether a given company is likely to begin investing in new capital based on the model above. (a) Describe some challenges in finding an empirical value for Tobin's Q and how one typically calculates it using data on market values of debt and equity and on the replacement cost of capital. List some sources for this data. (b) How well do empirically estimated values of Tobin's Q actually predict firm investment?

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1 The Lagrangian for the firms problem can be written as L FK A KFR 1K I where FK A represents the firms production function K is the firms capital st... View the full answer