An Overview of the Dynamic Calibration Examples

A balanced social accounting matrix (SAM) might be considered a snapshot of an economy on a long-run equilibrium path. Furthermore, we might assume that the economy, measured by the SAM, is on a steady-state growth path. The steady-state conditions provide information that allows us to specify the entire dynamic equilibrium. To begin, assume a classical Ramsey model of optimal growth (Barrow and Sala-I-Martin, 1995). Two features of the steady-state equilibrium facilitate calibration. First, all quantities grow at the exogenous labor growth rate and, second, all present-value prices decay at the interest rate (which equals the rate of time preference). This gives a template for projecting the SAM along the dynamic path.

Presented below are methods for calibrating an example set of social accounts. As a goal for the calibration exercise, we derive trajectories for the variables in the dynamic model as a function of the original data and assumed parameters. These trajectories can then be used to check the model’s formulation. The first three calibration techniques (sections A-1 through A-3) are for the basic single-sector Ramsey model, but they are differentiated by assumptions about the discount rate and where adjustments (if any) are to be made to the static social accounts. In Section A-4 we expand the analysis to a multi-sector Ramsey model, and in the final section we incorporate sector-specific growth projections to calibrate a non-balanced growth model. GAMS/MPSGE programs that correspond to each of the techniques outlined in sections A-1 through A-5 can be found on the world-wide-web at dyncal.html.

Assume that the following rectangular SAM represents the observed economic flows. Columns represent zero-profit conditions for activities, while rows indicate supply and demand balances for commodities traded at the associated price.

 Output(Y) Investment(I) Final Demand Output Price (PY) Y0 = 200 -20 -C0 = -180 Wage (PL) -150 LS0 = 150 Return to Capital (RK) -50 KS0 = 50 Price of Capital (PK) I0 = 20 -20

All of the conditions for a static equilibrium are met in the above data: (1) economic profits for each operating activity are zero; (2) supply equals demand for commodities with a non-zero price; and (3) total income is exhausted in final demand.

A-1 Calibration with No Assumption about the Value of the Discount Rate

In this calibration the equilibrium associated with the SAM is one in which output, investment, and final demand grow at the exogenous growth rate (g), and the market-clearing prices (measured in present value) decline at the interest rate (r), which equals the discount rate ( ). In addition, the level of investment must optimally support an unmeasured capital stock ( ). The underlying programming problem is as follows:

Max:       The infinitely lived representative agent seeks to maximize a constant-relative-risk-aversion utility function with an inter-temporal elasticity of substitution indicated by By using the information about the steady-state solution to this problem, we can find the dynamic equilibrium prices and quantities as a function of the original SAM. First, we need to find the value of the initial capital stock. The level of investment observed in the SAM ( ) must cover steady-state capital growth and depreciation ( ): (1)

We can solve for the initial level of capital: (2)

To find the reference price path, we must solve for the interest rate that is consistent with the return received by capital services in the SAM. The return to one unit of this capital stock ( ) is the total rental return flowing to capital (capital supply ) divided by the capital stock: (3)

The equilibrium interest rate is the rate of return implied by net of depreciation: (4)

The price of a unit of benchmark capital ( ) is given by: (5)

Equations 2, 4, and 5 can be used to define the dynamic equilibrium. For convenience, we define the calibrated equilibrium in terms of the reference growth and present value price paths: (6) (7)

Substituting equations 2, 3, and 4 into equation 7 we get: (8)

The dynamic equilibrium quantities and prices can now be found as a function of the original SAM and the assumed growth rate. The SAM measures value flows, so as a convention, we normalize all first-period prices to one (except , which is given by Equation 5):         A GAMS/MPSGE program that carries out this calibration to the above SAM can be found and downloaded from the Web at: dyn1.html.

A-2 Calibration Assuming a Given Discount Rate (Adjusting Investment)

In the previous calibration we made no assumption about the interest (discount) rate. Using actual social accounts can lead to unrealistically high, calibrated interest rates. One solution is to specify a discount rate and then reconcile the SAM by adjusting the ratio of output devoted to investment versus consumption. Given , the reference path for prices is determined. Equation 5 immediately gives us the price of a unit of benchmark capital, and we can resolve Equation 4 for the return per-unit of capital: (4a)

Rearranging Equation 3 gives us the initial capital stock consistent with capital's return: (3a)

Again, the initial level of investment should just cover the growth rate and depreciation, but this will be inconsistent with the observed investment in the SAM: , (2a)

To adjust the social accounts, we shift the excess investment into consumption: (9)

The dynamic equilibrium quantities and prices can now be specified exactly as they were in the previous example, except the consumption and investment paths are based on the newly calibrated levels, and the reference prices are based on the assumed interest rate. A GAMS/MPSGE program that carries out this calibration to the above SAM can be found and downloaded from the Web at: dyn2.html.

A-3 Calibration Assuming a Given Discount Rate (Adjusting Capital’s Value Share in Production)

As an alternative to the previous calibration, it may be more appropriate to adjust the value share returned to capital in production rather than adjusting investment. We use equations 5 and 4a to calculate the price of capital and the return per unit of capital. We also use Equation 2 directly to calculate the initial capital stock from the given growth and depreciation rates. Equation 3 will no longer hold unless the value returned to capital (capital supply) is adjusted: (3b)

To maintain balance in the social accounts we must shift the adjustment in capital's payment to labor: (10)

A GAMS/MPSGE program that carries out this calibration to the above SAM can be found and downloaded from the Web at: dyn3.html.

A-4 Calibrating the Multi-Sector Ramsey Model

The multi-sector model is calibrated in the same way except that we need to track the individual sectors, and if we choose to adjust capital’s value share (the previous example), then this adjustment must be allocated across sectors. The utility maximization problem is expanded to include the following constraints:   is the function that aggregates the individual sector outputs into the macroeconomic composite commodity. This function can be interpreted as the transitory utility function. identifies the individual sector production functions. Assume the following expanded SAM with two production sectors:

 Output(X1) Output(X2) Output(Y) Investment(I) Final Demand Output Price (PY) Y0 = 200 -20 -C0 = -180 Output Price (PX1) X10 = 120 -120 Output Price (PX2) X20 = 80 -80 Wage (PL) -LD1 =-100 -LD2 =-50 LS0 = 150 Return to Capital (RK) -KD1 = -20 -KD2 =-30 KS0 = 50 Price of Capital (PK) I0 = 20 -20

Assuming that we have arrived at a new supply of capital consistent with the equilibrium from equation 3b, we must now derive some method of distributing the capital payment adjustment across the sectors. One method is to use a weighted least-squares routine that minimizes a function of the sum of squares. This is flexible enough to account for an arbitrary number of sectors. In this example, we minimize the squared deviation in sector capital demands weighted by the initial capital demand in each sector. Any number of other objectives might be considered as valid rules for distributing the adjustment:

Min:    A GAMS/MPSGE program that carries out this calibration to the above SAM can be downloaded from the Web at: dyn4.html.

A-5 Calibrating to Sector-Specific Growth Projections

When used in applied work, the above calibration techniques might be criticized on the grounds that the growth paths are uniform. It might be useful to incorporate the output shifts as predicted by a reputable forecaster. One technique for generalizing the calibration to a non-balanced growth model is to assume that growth across sectors differs because of sector-specific factor productivity changes. This technique has three distinct steps: 1) calibrate the multi-sector balanced growth model; 2) find the sector-specific ad valorem subsidy to factor demand that satisfies a constraint on output to be equal to the forecasted projection, and correct for the induced income effects; and 3) use the value of the subsidy to calculate an output coefficient that scales productivity.

Consider a balanced-growth dynamic equilibrium calibrated using the technique outlined in the previous section. The second step is to use the model to find an appropriate set of endogenous subsidies that satisfy the output projections. We want to meet an arbitrary growth path for each of the output sectors, such that . Introduce an ad valorem subsidy (or tax) on the capital input to each sector ( ) and a variable that tracks the lump-sum income effects generated by paying the subsidy ( ). With the balanced growth model calibrated, a new equilibrium can be computed in which the subsidy adjusts to meet the constraint: Income effects are eliminated by adding to the agent’s endowment where: The third step is to remove the subsidy and adjustment from the model while introducing a productivity shift parameter, which is determined by the computed subsidy: Multiplying by the value of capital devoted to sector in period yields the desired non-balanced equilibrium. A GAMS/MPSGE program that carries out this calibration to the above SAM can be downloaded from the Web at: dyn5.html.