Assume that the utility function of a consumer is given by

U = ln c1 + ln c2
The interest rate is given by 5% and the endowments are given by (y1, y2) = (100, 120).
(a) Draw the budget constraint.
(b) Calculate the consumption and saving in period 1.
(c) Draw the indifference curve that included the optimal bundle in part (a).
(d) Suppose that interest rate decreased to 4%, draw the Hicksian substitution effect and income
effect on your graph.
(e) Suppose that interest rate decreased to 4%, draw the sky substitution effect and income effect
on your grap

(a) To draw the budget constraint, we need to find the combinations of consumption in period 1 (c1) and period 2 (c2) that a consumer can afford given their endowments and the interest rate.

The budget constraint equation is given by:
c1 + (1+r)c2 = y1 + (1+r)y2

Substituting the given values, we get:
c1 + 1.05c2 = 100 + 1.05(120)
c1 + 1.05c2 = 100 + 126
c1 + 1.05c2 = 226

To draw the budget constraint, we can rearrange the equation for c2:
c2 = (226 - c1) / 1.05

Using this equation, we can plot the budget constraint on a graph with c1 on the x-axis and c2 on the y-axis.

(b) To calculate the consumption and saving in period 1, we need to find the intersection point of the budget constraint and the highest attainable indifference curve (optimal bundle).

From the graph, we can find the coordinates of the intersection point. Let's assume the values of c1 and c2 to be (x1, x2).

(c) To draw the indifference curve that includes the optimal bundle, we need to find the combination of c1 and c2 that maximizes the utility function U = ln(c1) + ln(c2), subject to the budget constraint.

(d) With a decreased interest rate of 4%, the consumer's budget constraint will change. The new budget constraint equation is:
c1 + 1.04c2 = 226

We can plot this new budget constraint on the same graph.

To draw the Hicksian substitution effect and income effect, we need to find the new optimal bundle with the new interest rate. This can be done by finding the new intersection point between the indifference curve and the new budget constraint.

(e) To draw the sky substitution effect and income effect, we need to compare the new optimal bundle with the old optimal bundle.

The sky substitution effect measures how the consumer's consumption of one good changes when the price of that good changes while maintaining the same level of utility.

The income effect measures how the consumer's consumption of one good changes when their income changes while the prices of goods remain constant.

(a) To draw the budget constraint, we need to understand the consumer's budget set. The budget set represents all the combinations of consumption goods that a consumer can afford to purchase given their income and prices.

In this case, the consumer's endowments are given as (y1, y2) = (100, 120), which means they have 100 units of good 1 (c1) and 120 units of good 2 (c2). We also know that the interest rate is 5%, which means the consumer can lend or borrow at this rate.

The budget constraint can be expressed as:
c1 + (1 + r) * c2 = y1 + (1 + r) * y2

Substituting the given values:
c1 + (1 + 0.05) * c2 = 100 + (1 + 0.05) * 120
c1 + 1.05 * c2 = 100 + 126

Simplifying the equation:
c1 + 1.05 * c2 = 226

Now we can plot this constraint on a graph. Assume c1 is represented on the X-axis, and c2 is represented on the Y-axis. The intercepts can be found by setting c1 or c2 to zero:

When c1 = 0: 0 + 1.05 * c2 = 226
c2 = 226/1.05 ≈ 215.24

When c2 = 0: c1 + 1.05 * 0 = 226
c1 = 226

Connecting these two points will give us the budget constraint line.

(b) To calculate the consumption and saving in period 1, we need to find the optimal bundle that maximizes the consumer's utility.

Given the utility function U = ln c1 + ln c2, we can set up the consumer's optimization problem:

Maximize U = ln c1 + ln c2, subject to the budget constraint c1 + 1.05 * c2 = 226.

To solve this problem, we can use the method of Lagrange multipliers. The Lagrangian function is:

L = ln c1 + ln c2 + λ(226 - c1 - 1.05 * c2)

Taking the derivatives with respect to c1, c2, and λ and setting them equal to zero will give us the optimal values.

∂L/∂c1 = 1/c1 - λ = 0
∂L/∂c2 = 1/c2 - 1.05λ = 0
∂L/∂λ = 226 - c1 - 1.05 * c2 = 0

Solving these equations simultaneously will give us the optimal values of c1, c2, and λ.

(c) To draw the indifference curve that includes the optimal bundle found in part (a), we need to understand the consumer's preferences as represented by the utility function U = ln c1 + ln c2.

Since U represents the consumer's level of satisfaction, all combinations of c1 and c2 that yield the same level of utility will be on the same indifference curve.

To visualize this, we can plot several indifference curves on a graph where c1 is represented on the X-axis and c2 on the Y-axis. Each indifference curve represents a different level of utility.

Using the optimal bundle found in part (a), we can plot the indifference curve that includes this bundle. This curve will represent all combinations of c1 and c2 that yield the same level of utility as the optimal bundle.

(d) To draw the Hicksian substitution effect and income effect on the graph with a decreased interest rate of 4%, we need to analyze the changes in the consumer's behavior.

The Hicksian substitution effect refers to the change in consumption that results from a change in relative prices, assuming that the consumer's utility remains constant. The income effect refers to the change in consumption that results from a change in purchasing power, assuming that relative prices remain constant.

To depict the Hicksian substitution effect, we keep the consumer at the same level of utility (indifference curve) as before the interest rate decrease. We adjust the budget constraint line to reflect the new interest rate of 4%. The new budget constraint will have a different slope, representing the change in relative prices due to the lower interest rate.

To illustrate the income effect, we need to determine the change in the consumer's purchasing power resulting from the interest rate decrease. If the interest rate decreases, the consumer's income in period 2 (y2) will decrease.

We adjust the budget constraint to reflect the change in purchasing power caused by the decrease in income in period 2. The new budget constraint will have a parallel shift inward, representing the reduction in income.

(e) To draw the sky substitution effect and income effect on the graph with a decreased interest rate of 4%, we need to analyze the changes in both relative prices and purchasing power and their combined effect on the consumer's behavior.

The sky substitution effect considers both the change in consumption due to a change in relative prices (similar to the Hicksian substitution effect) and the change in consumption due to a change in purchasing power (similar to the income effect).

To illustrate this effect, we need to adjust both the slope of the budget constraint line to reflect the new relative prices (lower interest rate of 4%) and the position of the budget constraint parallel to reflect the reduction in income in period 2.

By doing so, we can show the combined changes in both relative prices and purchasing power, indicating the overall effect these changes have on the consumer's consumption decisions.