A Firm's Investment Problem: Suppose that a firm produces output according to the following production function:...

 

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A Firm's Investment Problem: Suppose that a firm produces output according to the following production function: Y₁ = A₂K, 0<a<1 The production function looks the same in period t + 1: Yt+1 = At+1Kt+1 At and At+1 are exogenous and known by the firm. Capital accumulates according to a standard law of motion, except that there is full depreciation (so d = 1). This means that tomorrow's capital is today's investment: Kt+1 = It The period t capital stock, Kt, is taken as given by the firm. The firm does not use labor. The firm must borrow to finance investment from a financial intermediary at interest rate rt + ft, where ft is an exogenous credit spread variable. Period t and t + 1 dividends are therefore: Dt = Yt Dt+1=Yt+1-(1 + rt + ft) It The firm's objective is pick It to maximize its value (present discounted value of dividends) subject to the capital accumulation equation and production function: max V₁ = D₁ + s.t. Dt+1 1 + rt Kt+1 = It D₁ = Yt Dt+1=Yt+1-(1 + rt + ft) It Use calculus to derive an optimal demand function for investment. Show that investment is decreasing in both rt and ft, and increasing in At+1 (but not a function of At). A Firm's Investment Problem: Suppose that a firm produces output according to the following production function: Y₁ = A₂K, 0<a<1 The production function looks the same in period t + 1: Yt+1 = At+1Kt+1 At and At+1 are exogenous and known by the firm. Capital accumulates according to a standard law of motion, except that there is full depreciation (so d = 1). This means that tomorrow's capital is today's investment: Kt+1 = It The period t capital stock, Kt, is taken as given by the firm. The firm does not use labor. The firm must borrow to finance investment from a financial intermediary at interest rate rt + ft, where ft is an exogenous credit spread variable. Period t and t + 1 dividends are therefore: Dt = Yt Dt+1=Yt+1-(1 + rt + ft) It The firm's objective is pick It to maximize its value (present discounted value of dividends) subject to the capital accumulation equation and production function: max V₁ = D₁ + s.t. Dt+1 1 + rt Kt+1 = It D₁ = Yt Dt+1=Yt+1-(1 + rt + ft) It Use calculus to derive an optimal demand function for investment. Show that investment is decreasing in both rt and ft, and increasing in At+1 (but not a function of At).

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