Let \(\displaystyle{\left({x}_{{{1}}},{y}_{{{1}}}\right)}\) be the total consumption and Nation income is 2004.

Let \(\displaystyle{\left({x}_{{{2}}},{y}_{{{2}}}\right)}\) be the total consumption and Nation income in 2009.

Since, we have to write C as a linear function of I, \(\displaystyle{x}_{{{1}}}=\${8},{289},{y}_{{{1}}}=\${9},{938}\) and \(\displaystyle{x}_{{{2}}}=\${10},{086},{y}_{{{2}}}=\${12},{027}\).

From this state we arrive at the point (8,289,9,938)(10,086,12,027)

To find slope using two point formula:

\(\displaystyle{m}={\frac{{{y}_{{{2}}}-{y}_{{{1}}}}}{{{x}_{{{2}}}-{x}_{{{1}}}}}}\) ...(1)

Slope of the linear function is \(\displaystyle{y}={m}{x}+{b}\) ...(2)

Substitute \(\displaystyle{m}={0.9070}\) in (2)

\(\displaystyle{y}={0.9070}{x}+{b}\) ...(3)

To find b:

Substitute \(\displaystyle{m}={0.9070}\) and \(\displaystyle{x}=\${8},{289},{y}=\${9},{938}\) in (3)

\(\displaystyle{9},{938}={0.9070}{\left({8},{289}\right)}+{b}\)

\(\displaystyle{b}={2419}\)

Substitute \(\displaystyle{m}={0.9070}\) and \(\displaystyle{b}={2419}\) in (3)

\(\displaystyle{y}={0.9070}{x}+{2419}\)

Express C as an exact linear function of I.

\(\displaystyle{C}={0.9070}{I}+{2419}\)

Part b):

From part a) slope of the linear function is 0.9070.

Hence, the marginal propensity to consume for country A from 2004-2009 is $0.9070.

Let \(\displaystyle{\left({x}_{{{2}}},{y}_{{{2}}}\right)}\) be the total consumption and Nation income in 2009.

Since, we have to write C as a linear function of I, \(\displaystyle{x}_{{{1}}}=\${8},{289},{y}_{{{1}}}=\${9},{938}\) and \(\displaystyle{x}_{{{2}}}=\${10},{086},{y}_{{{2}}}=\${12},{027}\).

From this state we arrive at the point (8,289,9,938)(10,086,12,027)

To find slope using two point formula:

\(\displaystyle{m}={\frac{{{y}_{{{2}}}-{y}_{{{1}}}}}{{{x}_{{{2}}}-{x}_{{{1}}}}}}\) ...(1)

Slope of the linear function is \(\displaystyle{y}={m}{x}+{b}\) ...(2)

Substitute \(\displaystyle{m}={0.9070}\) in (2)

\(\displaystyle{y}={0.9070}{x}+{b}\) ...(3)

To find b:

Substitute \(\displaystyle{m}={0.9070}\) and \(\displaystyle{x}=\${8},{289},{y}=\${9},{938}\) in (3)

\(\displaystyle{9},{938}={0.9070}{\left({8},{289}\right)}+{b}\)

\(\displaystyle{b}={2419}\)

Substitute \(\displaystyle{m}={0.9070}\) and \(\displaystyle{b}={2419}\) in (3)

\(\displaystyle{y}={0.9070}{x}+{2419}\)

Express C as an exact linear function of I.

\(\displaystyle{C}={0.9070}{I}+{2419}\)

Part b):

From part a) slope of the linear function is 0.9070.

Hence, the marginal propensity to consume for country A from 2004-2009 is $0.9070.