In an economic model with a pay-as-you-go social security scheme, where the government can levy lump-sum taxes and give lump-sum transfers, the key feature is the redistribution of resources from the young to the old. This redistribution is achieved through the taxation of the young workers and the provision of transfers to the retired elderly.
When considering the steady state level of capital (K) in the presence of a pay-as-you-go social security scheme (with T > 0, where T is the lump-sum tax on the young), it's important to consider how this taxation and transfer system affects incentives for saving and capital accumulation.
In the case of T = 0 (no social security), the individuals save for their retirement based on their own savings decisions, and the economy reaches a steady state capital level where the marginal product of capital equals the rate of time preference (interest rate).
However, when T > 0 (social security is in place), the taxation of the young reduces their disposable income and thereby their incentive to save. Since part of their income is being transferred to the old through lump-sum transfers, the young individuals might reduce their savings, resulting in a lower steady state level of capital compared to the case with no social security (T = 0).
In summary, in the presence of a pay-as-you-go social security scheme (T > 0), the steady state level of capital is likely to be lower than in the no-social security case (T = 0) due to the reduced savings incentives caused by the taxation and transfer system. The redistribution of resources from the young to the old can lead to decreased capital accumulation and potentially lower economic growth in the long run.