Question # In macroeconomic theory, total consumption expenditure on goods and services, C,

Decimals
ANSWERED In macroeconomic theory, total consumption expenditure on goods and services, C, is assume to be a linear function of national income, I. The table gives the values of C and I for 2004 and 2009 in country A(in billions of dollars).
a. Find the formula for C as a function of 1.
b. The slope of the linear function is called the marginal propensity to consume. What is the marginal propensity to consume for country A from 2004-2009?
a. FInd the formula for C as a function of I.
$$\displaystyle{C}=?$$
$$\begin{array}{|c|c|} \hline \text{Year}&2004&2013 \\ \hline \text{Total consumption (C)}&8,289&10,086\\ \hline \text{National income(f)}&9,938&12,027\\ \hline \end{array}$$ 2021-09-22
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.