The representative firm's revenue is a decreasing function of the quantity of capital employed by the industry, and an increasing function of the marginal product of capital and the capital employed by the firm:
\(\pi (K_t)k_t - I_t - C(I_t)\)
from which investment and the costs of implementing the investment is subtracted. Depreciation is assumed zero and interest rate is constant over time. An increase in capital is equal to the firm's investment in that period, \(\dot k_t = I_t\).
The firm's profit maximization problem (in discrete time) is as follows:
\(\mathcal{L} = \sum \limits_{t = 0} ^\infty (\frac{1}{(1 + r)^t}) [\pi (K_t)k_t - I_t - C(I_t)] + \frac{q_t}{(1 + r)^t} (k_t - k_{t - 1} - I)\)
Using the definition for the marginal value of increasing the constraint by a tiny amount (lambda):
\(\lambda _t = \frac{q_t}{(1 + r)^t}\)
Taking first-order conditions with respect to the endogenous (choice) variables:
\(\frac{\partial L}{\partial k_t} = \frac{\pi(K_t)}{(1 + r)^r} + \frac{q_{t + 1}}{(1 + r)^{t + 1}} - \frac{q_t}{(1 + r)^t} = 0\)
\(\frac{\partial L}{\partial I_t} = - \frac{1}{(1 + r)^r} + \frac{C'(I_t)}{(1 + r)^{t + 1}} - \frac{q_t}{(1 + r)^t} = 0\)
Solving for change in q, \(q_{t + 1} - q_t\):
\(\Delta q = r q_t - \pi(K_t)(1 + r)\)
In equilibrium, the change in q should be equal to zero, thus:
\(q_t = \frac{1 + r}{r} \pi (K_t)\)
Since \(\pi '_K < 0\) from the law of diminishing returns, the slope of the equation will be negative. The next equation comes from solving for investment, where we find that \(q_t = 1\) in equilibrium. This means the market value of the assets are the firm are equal to the replacement value, thus the firm has no incentive to change its investments. 
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