  function purchasePrice(
    SD59x18 _amount,
    SD59x18 _emissionRate,
    SD59x18 _k,
    SD59x18 _decayConstant,
    SD59x18 _timeSinceLastAuctionStart
  ) internal pure returns (SD59x18) {
    if (_amount.unwrap() == 0) {
      return SD59x18.wrap(0);
    }
    SD59x18 topE = _decayConstant.mul(_amount).div(_emissionRate);
    topE = topE.exp().sub(ONE);
    SD59x18 bottomE = _decayConstant.mul(_timeSinceLastAuctionStart);
    bottomE = bottomE.exp();
    SD59x18 result;
    if (_emissionRate.unwrap() > 1e18) {
      result = _k.div(_emissionRate).mul(topE).div(bottomE);
    } else {
      result = _k.mul(topE.div(_emissionRate.mul(bottomE)));
    }
    return result;
  }
