Research

# Local Economy Model v1.0

Pingree's Potato Patch
Studying the Economics of Detroit

For the economy of a defined community, let H represent households, L represent locally-owned firms, N represent branches of national chains located in the community, and Y represent the Gross Municipal Product of the community. We'll ignore price levels and inflation, and we'll ignore the government sector for the moment. We'll use the subscripts i and e to represent Income received by and Expenditures made by H,L and N. We'll define the economy thusly:

First, we define the Gross Municipal Product as the sum of the revenues of the three sectors. ⚠ {$Y = H_i + L_i + N_i$}

Next, we define the revenue streams for each sector.

⚠ {$H_i = alpha_0L_e + alpha_1N_e$}

⚠ {$L_i = beta_0L_e + beta_1N_e + beta_2H_e$}

⚠ {$N_i = (1-alpha_0-beta_0)L_e + (1-alpha_1-beta_1-r)N_e + (1-beta_2)H_e$}

All of the coefficients on H, L and N (the αn and βn terms and the sums involving them) represent fractions of the whole, and thus are restricted to between 0 and 1. Also, the coefficients on the household and local expenditures terms add up to 1: household expenditures get split between local firms and national chains, and local firm expenditures are split between households, other local firms, and national chains. However, take note of the r in the expression for national expenditures. Unlike households and local firms, the national chains direct some of their expenditures out of the local economy in the form of licensing fees, profits, and so forth, which is represented by the fraction r; it basically acts as a "leak" out of an otherwise sealed system.

Now, the final piece of the puzzle is to recognize that at equilibrium expenditures equal revenues for each sector. This means we can drop the e and i subscripts and then regroup the terms. The end result is a revised system of equations:

⚠ {$Y - H - L - N = 0$}

⚠ {$H - alpha_0L - alpha_1N = 0$}

⚠ {$-(beta_2)/(1-beta_0)H + L -(beta_1)/(1-beta_0)N = 0$}

⚠ {$(1-beta_2)/(alpha_1 + beta_1 + r)H + (1 - alpha_0 - beta_0)/(alpha_1 + beta_1 + r)L - N = 0$}

The main questions we are interested in relate to how expenditure choices in the different sectors affect the size of the economy and of household income. In order to perform those calculations, we need to totally differentiate the four equations above (with respect to Y, H, L, N, β2 and r) and then construct the coefficient matrices from the resulting equations, thusly: {${:(A=[(1,-1,-1,-1),(0,1,-alpha_0,-alpha_1),(0,-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(0,(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]), (A[(dY),(dH),(dL),(dN)] = [(0,0),(0,0),(0,H/(1-beta_0)),((X)/((alpha_1 + beta_1 + r)^2), H/(alpha_1 + beta_1 + r))] [(dr),(dbeta_2)]):}$}

where X=(1-β2)H + (1-α0-β0)L (the total expenditure of households and local business on national chains).

## Household Choices

Now we can start exploring the consequences of this model. We'll start by looking at the effects of houshold spending choices on GMP, i.e. ∂Y/∂β2. We can solve for this and similar partial derivatives by using Cramer's Rule. Basically we need to find the determinant of the matrix formed by replacing the dY column of A with the dβ2 column of the right hand side coefficient matrix (I'll call the resulting matrix Yβ). {${:(Y_beta = [(0,-1,-1,-1),(0,1,-alpha_0,-alpha_1),(H/(1-beta_0),-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(H/(alpha_1 + beta_1 + r),(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]), (|Y_beta| = -((1+alpha_0)Hr)/((1-beta_0)(alpha_1 + beta_1 + r))):}$}

We then divide that by the determinant of A. After substantial amounts of algebra, this reduces down to {${:(|A| = -(r(1-beta_0-alpha_0beta_2))/((1-beta_0)(alpha_1 + beta_1 + r))), ((del_Y)/(del_beta_2)=(|Y_beta|)/(|A|)=((1+alpha_0)H)/(1-beta_0-alpha_0beta_2)):}$}

We can see that ∂Y/∂β2 is positive, which means that GMP rises as the percentage of household income spent on local businesses rises. To switch our attention to the effect of houshold spending choices on household income, i.e. ∂H/∂β2, we perform the same calculation, but replace the dH column of A with the dβ2 column {${:(H_beta = [(1,0,-1,-1),(0,0,-alpha_0,-alpha_1),(0, H/(1-beta_0),1,-(beta_1)/(1-beta_0)),(0,H/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]), (|H_beta| = -(alpha_0Hr)/((1-beta_0)(alpha_1 + beta_1 + r))), ((del_H)/(del_beta_2)=(|H_beta|)/(|A|)=(alpha_0H)/(1-beta_0-alpha_0beta_2)):}$}