4. Consider the Q model of investment with adjustment costs. Equilibrium suggests that capital K(t) evolves...

 

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4. Consider the "Q" model of investment with adjustment costs. Equilibrium suggests that capital K(t) evolves as K(t) C-¹(q(t) 1) (normalizing the number of firms N = 1 and assuming no depreciation), while the marginal value of capital, q(t) evolves as à(t) rg(t) (K(t)), where r is the real interest rate. Note that the capital adjustment cost function, C(I(t)) satisfies C(0) = 0, C'(0) = 0, and C"(-) > 0 and the real profit function, (K(t)), satisfies '(-) < 0. Assume the transversality condition limeeg(t)x(t) = 0, where x(t) is the representative firm's capital stock. = (a) Draw the phase diagram for this model, explaining the location of the saddle path. (b) Use the phase diagram to show what happens given a sudden permanent drop in demand for output in a given industry. Explain what happens to q and K in that industry. What happens to the relative price of the industry's output, as well as profits and the market value of capital, both on impact and over time? (c) Compare your results in part (b) to what would happen if the drop in demand were only temporary. Does q jump by more or less than in the case of a permanent drop? Why can't there be any anticipated jump in q after the initial fall? (d) Comparing the results for parts (b) and (c), what do they imply about the effects of future changes in output on current investment for this industry? (Hint: in which case is the accelerator effect larger?) 4. Consider the "Q" model of investment with adjustment costs. Equilibrium suggests that capital K(t) evolves as K(t) C-¹(q(t) 1) (normalizing the number of firms N = 1 and assuming no depreciation), while the marginal value of capital, q(t) evolves as à(t) rg(t) (K(t)), where r is the real interest rate. Note that the capital adjustment cost function, C(I(t)) satisfies C(0) = 0, C'(0) = 0, and C"(-) > 0 and the real profit function, (K(t)), satisfies '(-) < 0. Assume the transversality condition limeeg(t)x(t) = 0, where x(t) is the representative firm's capital stock. = (a) Draw the phase diagram for this model, explaining the location of the saddle path. (b) Use the phase diagram to show what happens given a sudden permanent drop in demand for output in a given industry. Explain what happens to q and K in that industry. What happens to the relative price of the industry's output, as well as profits and the market value of capital, both on impact and over time? (c) Compare your results in part (b) to what would happen if the drop in demand were only temporary. Does q jump by more or less than in the case of a permanent drop? Why can't there be any anticipated jump in q after the initial fall? (d) Comparing the results for parts (b) and (c), what do they imply about the effects of future changes in output on current investment for this industry? (Hint: in which case is the accelerator effect larger?)