Consider the problem of a value maximizing firm whose profit function at time t is given...

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Consider the problem of a value maximizing firm whose profit function at time t is given by II(K₁) =eK! where e is the natural exponent, A denotes fixed the productivity level, K, denotes capital and is a parameter representing the elasticity of output with respect to capital. Assume that there is no depreciation of capital, so the law of motion of capital is given as K₁+1 = K₁+I₂ where I, is investments in capital at time t. The price of a unit of capital good is p and investment is subject to a smooth convex installation cost given by C(I. K) = R₁ where is a constant denoting the adjustment cost parameter. Time is discrete and runs to infinity, t = 0, 1... oo. Firm manager discounts future values with the factor 3. Based on the above information, answer the following two questions: 1.1 Write down the Bellman equation and derive the optimal investment decision condition. Define marginal Q and provide an economic interpretation. How is average Q related to marginal Q for this firm? [20%] 1.2 Solve the Bellman equation using dynamic programming. You can calibrate model param- eters as follows • 8 = 0.95, 0 = 0.7, y = 0.2 and p = 1.2. Discretize capital grid with 301 uniformly spaced points in the interval [30,80]. Constant Productivity Level is Ā= 1.5. Plot value function and investment policy functions. Interpret these graphs. Explain how optimal investment responds to changes in the adjustment cost parameter. • changes in how the firm manager values time. Answer in not more than 500 words. [40%] Consider the problem of a value maximizing firm whose profit function at time t is given by II(K₁) =eK! where e is the natural exponent, A denotes fixed the productivity level, K, denotes capital and is a parameter representing the elasticity of output with respect to capital. Assume that there is no depreciation of capital, so the law of motion of capital is given as K₁+1 = K₁+I₂ where I, is investments in capital at time t. The price of a unit of capital good is p and investment is subject to a smooth convex installation cost given by C(I. K) = R₁ where is a constant denoting the adjustment cost parameter. Time is discrete and runs to infinity, t = 0, 1... oo. Firm manager discounts future values with the factor 3. Based on the above information, answer the following two questions: 1.1 Write down the Bellman equation and derive the optimal investment decision condition. Define marginal Q and provide an economic interpretation. How is average Q related to marginal Q for this firm? [20%] 1.2 Solve the Bellman equation using dynamic programming. You can calibrate model param- eters as follows • 8 = 0.95, 0 = 0.7, y = 0.2 and p = 1.2. Discretize capital grid with 301 uniformly spaced points in the interval [30,80]. Constant Productivity Level is Ā= 1.5. Plot value function and investment policy functions. Interpret these graphs. Explain how optimal investment responds to changes in the adjustment cost parameter. • changes in how the firm manager values time. Answer in not more than 500 words. [40%]

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11 The Bellman equation for the firms problem can be written as follows VK max K CL K VK where VK represents the value function K is the profit function CL K is the convex installation cost is the dis... View the full answer