Question

Adrienne wins a lottery prize of $5,000. The current interest rate is 5%. She could spend the money now or invest it for 10 years and spend it then. Draw her intertemporal budget constraint and illustrate some of her choices. Also create a table for some of her choices. How would a higher interest rate alter her budget constraint?

Answer 1

Consider the given problem here “Adrienne” wins a lottery of “$5000”. Let’s assume that “C1=consumption is the current period” and “C10=future consumption”.

So, given the rate of interest the intertemporal budget line is given below.

=> C1 + C10/(1+r)^10 = Y = 5,000, where r=rate of interest rate = 5%.

=> C1 + C10/(1.05)^10 = 5000 …………….(1).

So, if “Adrienne” consume it today will get “C1=$5,000” and “C10=0”, on the other hand she can invest it for 10 years at 5% rate of return, so the future consumption is “C1=0 and C10=$5,000*1.05^10=$8,144.47”.

Consider the following fig.

Now, as we can see that there are 2 possibility available for her the 1^st option is to consume it today or to invest it for 10 years. So, the optimum choice depends on the utility function.

Let’s assume that the utility function is “U=aC1+bC10” with “MRS=a/b”. Now, we also have the budget line with slope “(1+r)^10 = 1.05^10=1.63”. So, the “MRS=a/b < (1+r)^10 =1.63”, => “future consumption” is relatively more valuable, similarly if “MRS=a/b > (1+r)^10 =1.63”, => “present consumption” is more valuable. If “MRS=a/b = (1+r)^10 =1.63”, she will be indifferent between “current” and “future” consumption.

Consider the following table, here the optimum choice corresponding to the different possible cases is given.

Now, we have derived the equation of intertemporal budget line mentioned below.

=> C1 + C10/(1+r)^10 = Y = 5,000, where r=rate of interest rate, with absolute slope “(1+r)^10”. So, as the rate of interest will increase leads to increase in the slope, => the budget line will rotates upward as the “r” will increase, it will be more steeper with same horizontal intercept “C1=5000”.