Behavioral Equations GL06INSOUT

Step Equations

1. Actual Inventory Sales Ratio

Calculate the actual inventory-sales ratio (beginning-of-period).

\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.

\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.

\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).

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\begin{align} BL_s(t) = BL_h(t) \end{align}

15. Capital Gains

Calculate capital gains on bond holdings.

\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.

\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.

\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.

\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 \(z_4 = 1\) if \(BLR^T(t-1) < bot\), \(z_5 = 1\) if \(BLR^T(t-1) > top\).

19. Employment

Calculate employment from real output and labor productivity.

\begin{align} N(t) = \frac{y(t)}{pr} \end{align}

20. Expected Inventories

Calculate expected end-of-period inventories via partial adjustment.

\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.

\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.

\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.

\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).

\begin{align} ERr_{bl}(t) = r_{bl}(t) \end{align}

25. Expected Sales

Calculate expected real sales using adaptive expectations.

\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.

\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.

\begin{align} GD(t) = B_s(t) + p_{bl}(t) \cdot BL_s(t) \end{align}

28. Government Spending

Calculate nominal government spending.

\begin{align} G(t) = g(t) \cdot p(t) \end{align}

29. Haig Simons Disposable Income

Calculate Haig-Simons disposable income (including capital gains).

\begin{align} YD_{hs}(t) = YD_r(t) + CG(t) \end{align}

30. High Powered Money

Calculate high-powered money supply.

\begin{align} H_s(t) = Hh(t) + Hb_s(t) \end{align}

31. Inflation Rate

Calculate the inflation rate.

\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.

\begin{align} L^d(t) = INV(t) \end{align}

33. Loan Rate

Calculate the endogenous loan rate based on bank profit margin.

\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 \(z_6 = 1\) if \(BPM(t) < bot_{pm}\), \(z_7 = 1\) if \(BPM(t) > top_{pm}\).

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.

\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.

\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.

\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.

\begin{align} C(t) = c(t) \cdot p(t) \end{align}

38. Nominal Expected Disposable Income

Calculate the nominal expected disposable income.

\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.

\begin{align} INV(t) = inv(t) \cdot UC(t) \end{align}

40. Nominal Output

Calculate nominal output (sales at price + inventory change at cost).

\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.

\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.

\begin{align} W(t) = W(t-1) \cdot (1 + \Omega_3 \cdot (\omega^T(t-1) - \omega(t-1))) \end{align}

where \(\omega(t-1) = W(t-1) / p(t-1)\).

43. Nominal Wealth

Calculate nominal household wealth.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\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.

\begin{align} s(t) = c(t) + g(t) \end{align}

53. Real Wage

Calculate the current real wage.

\begin{align} \omega(t) = W(t) / p(t) \end{align}

54. Real Wealth

Calculate real household wealth.

\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.

\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.

\begin{align} RR(t) = ro_1 \cdot M1_s(t) + ro_2 \cdot M2_s(t) \end{align}

57. Target Inventories

Calculate target inventories.

\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.

\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.

\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.

\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.

\begin{align} FD(t) = FP(t) + FBP(t) \end{align}

62. Unit Cost

Calculate the unit cost of production.

\begin{align} UC(t) = \frac{WB(t)}{y(t)} \end{align}

63. Wage Bill

Calculate the nominal wage bill.

\begin{align} WB(t) = N(t) \cdot W(t) \end{align}