=============================== Behavioral Equations GL06INSOUT =============================== -------------- Step Equations -------------- 1. Actual Inventory Sales Ratio Calculate the actual inventory-sales ratio (beginning-of-period). .. math:: :label: actual_inventory_sales_ratio :nowrap: \begin{align} \sigma(t) = inv(t-1) / s(t) \end{align} 2. Advances Demand Calculate bank demand for central bank advances. When the tentative liquidity ratio is below floor, bank borrows from CB. .. math:: :label: advances_demand :nowrap: \begin{align} A^d(t) = \max(bot \cdot (M1_s + M2_s) - B_b^T, 0) \cdot z_3 \end{align} 3. Bank Bill Holdings Calculate actual bank bill holdings after advances. .. math:: :label: bank_bill_holdings :nowrap: \begin{align} B_b(t) = B_b^T(t) + A^d(t) \end{align} 4. Bank Bills Tentative Calculate tentative bank bill holdings (residual of bank balance sheet). .. math:: :label: bank_bills_tentative :nowrap: \begin{align} B_b^T(t) = M1_s + M2_s - L_s - Hb_s \end{align} 5. Bank Liquidity Ratio Calculate the actual bank liquidity ratio. .. math:: :label: bank_liquidity_ratio :nowrap: \begin{align} BLR(t) = B_b(t) / (M1_s(t) + M2_s(t)) \end{align} 6. Bank Liquidity Ratio Tentative Calculate the tentative bank liquidity ratio. .. math:: :label: bank_liquidity_ratio_tentative :nowrap: \begin{align} BLR^T(t) = B_b^T(t) / (M1_s(t) + M2_s(t)) \end{align} 7. Bank Profit Margin Calculate the bank profit margin as a share of the prior deposit base. .. math:: :label: bank_profit_margin :nowrap: \begin{align} BPM(t) = \frac{FBP(t) + FBP(t-1)}{M1_s(t-1) + M1_s(t-2) + M2_s(t-1) + M2_s(t-2)} Uses LaggedM1Supply(t-1)=M1_s(t-1) and prior LaggedM1Supply(t-2)=M1_s(t-2). \end{align} 8. Bank Profits Calculate bank profits from prior-period stocks and rates. .. math:: :label: bank_profits :nowrap: \begin{align} FBP(t) = r_l(t-1) \cdot L_s(t-1) + r_b(t-1) \cdot B_b(t-1) - r_m(t-1) \cdot M2_s(t-1) - r_a(t-1) \cdot A_s(t-1) \end{align} 9. Bills Central Bank Calculate central bank bill holdings as residual. .. math:: :label: bills_central_bank :nowrap: \begin{align} B_{cb}(t) = B_s(t) - B_h(t) - B_b(t) \end{align} 10. Bills Demand Calculate bills demand via Tobin portfolio choice. .. math:: :label: bills_demand :nowrap: \begin{align} B^d(t) = V^e_{nc} \cdot (\lambda_{30} + \lambda_{31} \cdot r_m + \lambda_{32} \cdot r_b + \lambda_{33} \cdot ERr_{bl}) + \lambda_{34} \cdot YD^e \end{align} 11. Bills Supply Calculate bill supply as residual government financing. .. math:: :label: bills_supply :nowrap: \begin{align} B_s(t) = B_s(t-1) + PSBR(t) - (BL_s(t) - BL_s(t-1)) \cdot p_{bl}(t) \end{align} 12. Bond Price Calculate the bond price as the inverse of the bond yield. .. math:: :label: bond_price :nowrap: \begin{align} p_{bl}(t) = \frac{1}{r_{bl}(t)} \end{align} 13. Bonds Demand Calculate bond demand (in number of bonds) via Tobin portfolio choice. .. math:: :label: bonds_demand :nowrap: \begin{align} BL^d(t) = \frac{V^e_{nc} \cdot (\lambda_{40} + \lambda_{41} \cdot r_m + \lambda_{42} \cdot r_b + \lambda_{43} \cdot ERr_{bl}) + \lambda_{44} \cdot YD^e}{p_{bl}(t)} \end{align} 14. Bonds Supply Set bond supply equal to household bond demand. .. math:: :label: bonds_supply :nowrap: \begin{align} BL_s(t) = BL_h(t) \end{align} 15. Capital Gains Calculate capital gains on bond holdings. .. math:: :label: capital_gains :nowrap: \begin{align} CG(t) = (p_{bl}(t) - p_{bl}(t-1)) \cdot BL_h(t-1) \end{align} 16. Cash Demand Calculate household cash demand proportional to consumption. .. math:: :label: cash_demand :nowrap: \begin{align} Hh^d(t) = \lambda_c \cdot C(t) \end{align} 17. Central Bank Profits Calculate central bank profits from prior-period stocks and rates. .. math:: :label: central_bank_profits :nowrap: \begin{align} FCB(t) = r_b(t-1) \cdot B_{cb}(t-1) + r_a(t-1) \cdot A_s(t-1) \end{align} 18. Deposit Rate Calculate the endogenous deposit rate based on bank liquidity. .. math:: :label: deposit_rate :nowrap: \begin{align} r_m(t) = r_m(t-1) + \zeta_m \cdot (z_4 - z_5) + \zeta_b \cdot (r_b(t) - r_b(t-1)) \end{align} where :math:`z_4 = 1` if :math:`BLR^T(t-1) < bot`, :math:`z_5 = 1` if :math:`BLR^T(t-1) > top`. 19. Employment Calculate employment from real output and labor productivity. .. math:: :label: employment :nowrap: \begin{align} N(t) = \frac{y(t)}{pr} \end{align} 20. Expected Inventories Calculate expected end-of-period inventories via partial adjustment. .. math:: :label: expected_inventories :nowrap: \begin{align} inv^e(t) = inv(t-1) - \gamma \cdot (inv(t-1) - inv^T(t)) \end{align} 21. Expected Nominal Wealth Calculate expected nominal wealth for portfolio allocation. .. math:: :label: expected_nominal_wealth :nowrap: \begin{align} V^e(t) = V(t-1) + YD^e(t) - C(t) \end{align} 22. Expected Non Cash Wealth Calculate expected non-cash wealth available for portfolio allocation. .. math:: :label: expected_non_cash_wealth :nowrap: \begin{align} V^e_{nc}(t) = V^e(t) - Hh^d(t) \end{align} 23. Expected Real Disposable Income Calculate expected real disposable income via adaptive expectations. .. math:: :label: expected_real_disposable_income :nowrap: \begin{align} yd_r^e(t) = \varepsilon \cdot yd_r(t-1) + (1 - \varepsilon) \cdot yd_r^e(t-1) \end{align} 24. Expected Return On Bonds Calculate the expected return on bonds (static expectations). .. math:: :label: expected_return_on_bonds :nowrap: \begin{align} ERr_{bl}(t) = r_{bl}(t) \end{align} 25. Expected Sales Calculate expected real sales using adaptive expectations. .. math:: :label: expected_sales :nowrap: \begin{align} s^e(t) = \beta \cdot s(t-1) + (1 - \beta) \cdot s^e(t-1) \end{align} 26. Firm Profits Calculate firm profits. .. math:: :label: firm_profits :nowrap: \begin{align} FP(t) = S(t) - TX(t) - WB(t) + (INV(t) - INV(t-1)) - r_l(t-1) \cdot L_s(t-1) \end{align} 27. Government Debt Calculate total government debt. .. math:: :label: government_debt :nowrap: \begin{align} GD(t) = B_s(t) + p_{bl}(t) \cdot BL_s(t) \end{align} 28. Government Spending Calculate nominal government spending. .. math:: :label: government_spending :nowrap: \begin{align} G(t) = g(t) \cdot p(t) \end{align} 29. Haig Simons Disposable Income Calculate Haig-Simons disposable income (including capital gains). .. math:: :label: haig_simons_disposable_income :nowrap: \begin{align} YD_{hs}(t) = YD_r(t) + CG(t) \end{align} 30. High Powered Money Calculate high-powered money supply. .. math:: :label: high_powered_money :nowrap: \begin{align} H_s(t) = Hh(t) + Hb_s(t) \end{align} 31. Inflation Rate Calculate the inflation rate. .. math:: :label: inflation_rate :nowrap: \begin{align} \pi(t) = \frac{p(t) - p(t-1)}{p(t-1)} \end{align} 32. Loan Demand Set firm loan demand equal to nominal inventories. .. math:: :label: loan_demand :nowrap: \begin{align} L^d(t) = INV(t) \end{align} 33. Loan Rate Calculate the endogenous loan rate based on bank profit margin. .. math:: :label: loan_rate :nowrap: \begin{align} r_l(t) = r_l(t-1) + \zeta_l \cdot (z_6 - z_7) + (r_b(t) - r_b(t-1)) \end{align} where :math:`z_6 = 1` if :math:`BPM(t) < bot_{pm}`, :math:`z_7 = 1` if :math:`BPM(t) > top_{pm}`. 34. M1 Demand Tentative Calculate tentative M1 demand as the portfolio residual. M1 is the buffer-stock asset that absorbs whatever *actual* wealth remains after the three Tobin-allocated assets (M2, Bills, Bonds). The R sfcr reference uses actual non-cash wealth (``Vnc = V - Hhh``), not expected, for this residual. .. math:: :label: m1_demand_tentative :nowrap: \begin{align} M1^{dt}(t) = V_{nc}(t) - M2^d(t) - B^d(t) - p_{bl}(t) \cdot BL^d(t) \end{align} 35. M2 Demand Calculate M2 demand via Tobin portfolio choice. .. math:: :label: m2_demand :nowrap: \begin{align} M2^d(t) = V^e_{nc} \cdot (\lambda_{20} + \lambda_{21} \cdot r_m + \lambda_{22} \cdot r_b + \lambda_{23} \cdot ERr_{bl}) + \lambda_{24} \cdot YD^e \end{align} 36. M2 Household Calculate household M2 holdings. When z1=1 (normal), M2 = M2Demand. When z2=1, M2 absorbs the residual non-cash wealth after bills and bonds. .. math:: :label: m2_household :nowrap: \begin{align} M2_h(t) = M2^d \cdot z_1 + (V_{nc} - B_h - p_{bl} \cdot BL^d) \cdot z_2 \end{align} 37. Nominal Consumption Calculate nominal consumption. .. math:: :label: nominal_consumption :nowrap: \begin{align} C(t) = c(t) \cdot p(t) \end{align} 38. Nominal Expected Disposable Income Calculate the nominal expected disposable income. .. math:: :label: nominal_expected_disposable_income :nowrap: \begin{align} YD^e(t) = yd_r^e(t) \cdot p(t) + \pi(t) \cdot \frac{V(t-1)}{p(t)} \end{align} 39. Nominal Inventories Calculate nominal inventories valued at current unit cost. .. math:: :label: nominal_inventories :nowrap: \begin{align} INV(t) = inv(t) \cdot UC(t) \end{align} 40. Nominal Output Calculate nominal output (sales at price + inventory change at cost). .. math:: :label: nominal_output :nowrap: \begin{align} Y(t) = p(t) \cdot s(t) + UC(t) \cdot (inv(t) - inv(t-1)) \end{align} 41. Nominal Sales Calculate nominal sales. .. math:: :label: nominal_sales :nowrap: \begin{align} S(t) = s(t) \cdot p(t) \end{align} 42. Nominal Wage Calculate the nominal wage via partial adjustment to target real wage. .. math:: :label: nominal_wage :nowrap: \begin{align} W(t) = W(t-1) \cdot (1 + \Omega_3 \cdot (\omega^T(t-1) - \omega(t-1))) \end{align} where :math:`\omega(t-1) = W(t-1) / p(t-1)`. 43. Nominal Wealth Calculate nominal household wealth. .. math:: :label: nominal_wealth :nowrap: \begin{align} V(t) = V(t-1) + YD_{hs}(t) - C(t) \end{align} 44. Non Cash Wealth Calculate actual non-cash wealth available for portfolio allocation. In the R sfcr reference, M1 (the buffer-stock residual) is computed against actual non-cash wealth (``Vnc = V - Hhh``), not expected. Portfolio *demands* (M2d, Bhd, BLd) use expected wealth (VncE), but the realized residual uses actual wealth so that the portfolio identity holds. .. math:: :label: non_cash_wealth :nowrap: \begin{align} V_{nc}(t) = V(t) - Hh^d(t) \end{align} 45. Normal Historic Unit Cost Calculate the normal historic unit cost. A weighted average of current and prior unit cost, with inventory financing cost applied to the prior component. .. math:: :label: normal_historic_unit_cost :nowrap: \begin{align} NHUC(t) = (1 - \sigma^T) \cdot UC(t) + \sigma^T \cdot (1 + r_l(t-1)) \cdot UC(t-1) \end{align} 46. Price Level Calculate the price level as a markup over normal historic unit cost. .. math:: :label: price_level :nowrap: \begin{align} p(t) = (1 + \tau) \cdot (1 + \phi) \cdot NHUC(t) \end{align} 47. Public Sector Borrowing Requirement Calculate the public sector borrowing requirement. .. math:: :label: public_sector_borrowing_requirement :nowrap: \begin{align} PSBR(t) = G(t) + r_b(t-1) \cdot B_s(t-1) + BL_s(t-1) - T(t) - FCB(t) \end{align} 48. Real Consumption Calculate real consumption. .. math:: :label: real_consumption :nowrap: \begin{align} c(t) = \alpha_0 + \alpha_1 \cdot yd_r^e(t) + \alpha_2 \cdot v(t-1) \end{align} 49. Real Inventories Calculate real end-of-period inventories. .. math:: :label: real_inventories :nowrap: \begin{align} inv(t) = inv(t-1) + y(t) - s(t) \end{align} 50. Real Output Calculate real output as expected sales plus inventory adjustment. .. math:: :label: real_output :nowrap: \begin{align} y(t) = s^e(t) + inv^e(t) - inv(t-1) \end{align} 51. Real Regular Disposable Income Calculate real regular disposable income with inflation erosion. .. math:: :label: real_regular_disposable_income :nowrap: \begin{align} yd_r(t) = \frac{YD_r(t)}{p(t)} - \pi(t) \cdot \frac{V(t-1)}{p(t)} \end{align} 52. Real Sales Calculate real sales as consumption plus real government spending. .. math:: :label: real_sales :nowrap: \begin{align} s(t) = c(t) + g(t) \end{align} 53. Real Wage Calculate the current real wage. .. math:: :label: real_wage :nowrap: \begin{align} \omega(t) = W(t) / p(t) \end{align} 54. Real Wealth Calculate real household wealth. .. math:: :label: real_wealth :nowrap: \begin{align} v(t) = V(t) / p(t) \end{align} 55. Regular Disposable Income Calculate regular (non-capital-gains) disposable income. With indirect taxation, households receive firm/bank dividends (which already net out the production tax) plus interest income. No explicit tax subtraction at the household level. .. math:: :label: regular_disposable_income :nowrap: \begin{align} YD_r(t) = WB(t) + FD(t) + r_m(t-1) \cdot M2_h(t-1) + r_b(t-1) \cdot B_h(t-1) + BL_h(t-1) \end{align} 56. Required Reserves Calculate required reserves. .. math:: :label: required_reserves :nowrap: \begin{align} RR(t) = ro_1 \cdot M1_s(t) + ro_2 \cdot M2_s(t) \end{align} 57. Target Inventories Calculate target inventories. .. math:: :label: target_inventories :nowrap: \begin{align} inv^T(t) = \sigma^T(t) \cdot s^e(t) \end{align} 58. Target Inventory Sales Ratio Calculate the target inventory-sales ratio. .. math:: :label: target_inventory_sales_ratio :nowrap: \begin{align} \sigma^T(t) = \sigma_0 - \sigma_1 \cdot r_l(t) \end{align} 59. Target Real Wage Calculate the target real wage from productivity and employment. .. math:: :label: target_real_wage :nowrap: \begin{align} \omega^T(t) = \exp(\Omega_0 + \Omega_1 \log(pr) + \Omega_2 \log(N(t) / N_{fe})) \end{align} 60. Taxes Calculate indirect production tax on sales. .. math:: :label: taxes :nowrap: \begin{align} TX(t) = \frac{\tau}{1 + \tau} \cdot S(t) \end{align} 61. Total Dividends Calculate total dividends distributed to households. All firm and bank profits are distributed as dividends. .. math:: :label: total_dividends :nowrap: \begin{align} FD(t) = FP(t) + FBP(t) \end{align} 62. Unit Cost Calculate the unit cost of production. .. math:: :label: unit_cost :nowrap: \begin{align} UC(t) = \frac{WB(t)}{y(t)} \end{align} 63. Wage Bill Calculate the nominal wage bill. .. math:: :label: wage_bill :nowrap: \begin{align} WB(t) = N(t) \cdot W(t) \end{align}