# Local Economy Model v1.0

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 *α*_{n} and *β*_{n} 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, *β*_{2} 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-*β*_{2})H + (1-*α*_{0}-*β*_{0})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/∂*β*_{2}. 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 d*β*_{2} column of the right hand side coefficient matrix (I'll call the resulting matrix Y_{β}).
{${:(Y_beta = [(0,-1,-1,-1),(0,1,-alpha_0,-alpha_1),(H/(1-beta_0),-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(H/(alpha_1 + beta_1 + r),(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]),
(|Y_beta| = -((1+alpha_0)Hr)/((1-beta_0)(alpha_1 + beta_1 + r))):}$}

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/∂*β*_{2} 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/∂*β*_{2}, we perform the same calculation, but replace the dH column of A with the d*β*_{2} column
{${:(H_beta = [(1,0,-1,-1),(0,0,-alpha_0,-alpha_1),(0, H/(1-beta_0),1,-(beta_1)/(1-beta_0)),(0,H/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]),
(|H_beta| = -(alpha_0Hr)/((1-beta_0)(alpha_1 + beta_1 + r))),
((del_H)/(del_beta_2)=(|H_beta|)/(|A|)=(alpha_0H)/(1-beta_0-alpha_0beta_2)):}$}

So, we see that ∂H/∂*β*_{2} is positive, and thus household income also rises as the percentage of household income spent on local businesses rises. We can also look at the effect of a $1 shift in household spending form national firms to local firms, which requires that d*β*_{2} = 1/H.
{${:((del_Y)/(del_beta_2)dbeta_2 =(1+alpha_0)/(1-beta_0-alpha_0beta_2) > 1),
((del_H)/(del_beta_2)dbeta_2 =(alpha_0)/(1-beta_0-alpha_0beta_2) > 0)
:}$}
So, we can say that shifting $1 of household spending from national businesses to local businesses boosts gross municipal product by more than $1, but we can't say anything concrete about the effect on household incomes except that it is positive.

*r* Stands For Rakeoff

Let's move on to considering the effects of *r*. Basically the analysis is the same as for the effects of *β*_{2}, but applying Cramer's Rule with the coefficient column for *r*. We'll start with the effects on Gross Municipal Product.
{${:(Y_r = [(0,-1,-1,-1),(0,1,-alpha_0,-alpha_1),(0,-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(X/(alpha_1 + beta_1 + r),(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]),
(|Y_r| = (-X)/((alpha_1 + beta_1 + r)^2)xx|A|),
((del_Y)/(del_r)=(|Y_r|)/(|A|)=(-[(1-beta_2)H +(1-alpha_0-beta_0)L])/((alpha_1 + beta_1 + r)^2))
:}$}

This means a couple of things.

*r*negatively impacts gross municipal product, but since one can see that the second derivative is positive, the effect of increases in*r*tapers off.- The larger the share of national chain expenditures which go to national chains (i.e. the smaller the denominator), the larger the negative effect of a given increase in
*r*.

Next we'll look at the effects on household income. {${:(H_r = [(1,0,-1,-1),(0,0,-alpha_0,-alpha_1),(0,0,1,-(beta_1)/(1-beta_0)),(0,X/((alpha_1+beta_1+r)^2),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]), (|H_r| = (X[alpha_0beta_1 + alpha_1(1-beta_0)])/((1-beta_0)(alpha_1 + beta_1 + r)^2)), ((del_H)/(del_r)=(|H_r|)/(|A|)=(-[(1-beta_2)H +(1-alpha_0-beta_0)L][alpha_0beta_1 + alpha_1(1-beta_0)])/(r(1-beta_0-alpha_0beta_2)(alpha_1 + beta_1 + r))):}$}

Not too surprisingly, we see that *r* has a negative effect on household income. Now, it's pretty clear that *r* has a negative effect on N as well, (I'll refrain from posting the math), which means that we should be able to find an "optimal" level of *r* if one wanted to maximize outflow (as might be useful if we treated it as a fee to a franchiser). To perform this analysis, we'll need to augment our original set of equilibrium equations and resulting coefficient matrices to include what we'll term D for dividend, and we'll put a dot over the A to represent the modified matrix.
`{:(D - rN = 0),(dot A=[(1,0,0,0,-r),(0,1,-1,-1,-1),(0,0,1,-alpha_0,-alpha_1),(0,0,-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(0,0,(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]),(dot A[(dD),(dY),(dH),(dL),(dN)] = [(N,0),(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)]),(|dot A| = |A|):}`

Now we can calculate ∂D/∂*r*, again using Cramer's rule, and use it to find the dividend maximizing level of *r*.
{${:(D_r = [(N,0,0,0,-r),(0,1,-1,-1,-1),(0,0,1,-alpha_0,-alpha_1),(0,0,-(beta_2)/(1-beta_0),1,-(beta_1)/(1-beta_0)),(X/(alpha_1 + beta_1 + r),0,(1-beta_2)/(alpha_1+beta_1+r),(1-alpha_0-beta_0)/(alpha_1+beta_1+r),-1)]),
(|D_r| = N|dot A| - (X|dot A|)/(alpha_1 + beta_1 + r)),
((del_D)/(del_r)=(|D_r|)/(|dot A|)=N - X/(alpha_1 + beta_1 + r)),
((del_D)/(del_r)=0 => r = X/N - (alpha_1 + beta_1)):}$}

Now, since X was defined as the total expenditure of households and local business on national chains, D is maximized when *r* is set to equal the spread between the share of N coming from local sources and the share of N spent on local sources (i.e. *α*_{1} + *β*_{1}). So, essentially *r* should rise the more asymmetric the relationship between the chains and the local economy.

So, to recap, we've found that under the assumptions of this model:

- Spending at national chains tends to have a negative effect on the local economy when compared to the effects of spending at locally owned businesses.
- The household sector can increase its income by redirecting spending away from national chains and toward locally-owned businesses.
- Rational national chains will tend to export a larger fraction of their earnings the less they depend upon the local economy for resources and the larger the fraction of local consumption they represent.

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